Abstract
The problem of pollution control has been mainly studied in the environmental economics literature where the methodology of game theory is applied for the pollution control. To the best of our knowledge this is the first time this problem is studied from the computational point of view. We introduce a new network model for pollution control and present two applications of this model. On a high level, our model comprises a graph whose nodes represent the agents, which can be thought of as the sources of pollution in the network. The edges between agents represent the effect of spread of pollution. The government who is the regulator, is responsible for the maximization of the social welfare and sets bounds on the levels of emitted pollution in both local areas as well as globally in the whole network. We first prove that the above optimization problem is NPhard even on some special cases of graphs such as trees. We then turn our attention on the classes of trees and planar graphs which model realistic scenarios of the emitted pollution in water and air, respectively. We derive approximation algorithms for these two kinds of networks and provide deterministic truthful and truthful in expectation mechanisms. In some settings of the problem that we study, we achieve the best possible approximation results under standard complexity theoretic assumptions. Our approximation algorithm on planar graphs is obtained by a novel decomposition technique to deal with constraints on vertices. We note that no known planar decomposition techniques can be used here and our technique can be of independent interest. For trees we design a two level dynamic programming approach to obtain an FPTAS. This approach is crucial to deal with the global pollution quota constraint. It uses a special multiple choice, multidimensional knapsack problem where coefficients of all constraints except one are bounded by a polynomial of the input size. We furthermore derive truthful in expectation mechanisms on general networks with bounded degree.
Keywords
Algorithmic mechanism design Approximation algorithms Planar graphs Pollution control1 Introduction
The advance of technology and commercial freedom have fused and accelerated the development process in an unprecedented scale. Environmental degradation however has accompanied this progress, resulting in global water and air pollution. In many developing countries, this has caused wide public concerns. As an example, in 2012, China discharged 68.5 billion tons of industrial wastewater and the SO\(_2\) emissions reached 21.2 million tons (National Bureau of Statistics of China, 2013). China has become one of the most polluted countries in the world with industrial emissions as the main source of its pollution. The recent annual State of the Air report of the American Lung Association finds that \(47\%\) of the Americans live in counties with frequently unhealthy levels of either ozone or particulate pollution [4]. The latest assessment of air quality, by the European Environment Agency, finds that around \(90\%\) of city inhabitants in the European Union are exposed to one of the most damaging air pollutants at harmful levels [1]. Environmental research suggests that water pollution is one of the very significant factors affecting water security worldwide [57].
It is the role of regulatory authorities to make efficient environmental policies in balancing economic growth and environment protection. Pollution control regulations are inspired by the managerial approaches in environment policies, where models based on game theory are proposed and analysed. Kwerel [38] proposed a mechanism where firms, potential polluters, report their cleanup cost information to the regulator. The regulator sells a fixed number of pollution licences at a fixed price per licence and offers a subsidy for those licences which firms hold in excess of emission, based on the cost information provided by firms. In Kwerel’s mechanism truthtelling by all firms implies a Nash equilibrium. Kwerel’s scheme maintains a mild level of pollution by optimizing the social welfare (sum of the global cleanup cost and damage cost of emitted pollution).
From a different point of view, Dasgupta et al. [15] focus on minimizing the sum of pollution damages, abatement costs and individual rationality for consumers. Spulber [50] develops a market model of environmental regulation with interdependent production, pollution abatement costs and heterogeneous firms who have private information about costs and pursue BayesNash strategies in communication with the regulator. Their paper illustrates that the full information optimum cannot be attained unless gains from trade in the product market net of external damages exceed the information rents earned by firms and aggregate output and externality levels are lower at the regulated equilibrium than at the full information social optimum.
In a given geographic area, there are owners of pollution sources (e.g., factories, cars). The owners of these sources are interested in buying licences, i.e., permits, for the emission of pollution. The government, as a regulator, is responsible for allocating the licences to the owners in such a way that the amount of emissions does not exceed certain levels both globally and locally, i.e., the regions around the pollution sources. This allocation aims at the maximization of owners’ satisfaction, that is, the social welfare is maximized. See Sect. 3 for details.
Pollution has a diffusion nature: emitted from one source, it will have an effect on its neighbours at some decreased level. We consider two applications using a network model. In the first application, the vertices represent pollution sources and edges are routes of pollution transition from one source to another, similar to Belitskaya [11]. Our model measures the pollution diminishing transition by arbitrary weights on the edges, which are also present in the model of Montgomery [41]. The polluters’ privately known cleanup cost and damage of the emitted pollution in our model are inspired by Kwerel [38]. In the second application, the vertices represent mayors of cities and the edges represent the roads between cities. The percentage of cars moving from one city to another is represented by the weight of the corresponding edge. Note, that in this application, although cars are physical sources of pollution, vertices, i.e., cities to which cars drive, are factual pollution sources. Thus, also in this application, the vertices (cities) can be regarded as pollution sources.
Our model covers both aforementioned applications with details given in Sect. 3. The government, as the regulator, can decide to either shut down or keep open a pollution source taking into account the diffusion nature of pollution. It sets bounds on the global and local levels of pollution, while trying to optimize the social welfare. The emissions that exceed the amount of pollution allowed by the licences, if any, must be cleanedup, incurring an additional cost to the agent, called agent’s cleanup cost. Furthermore, in the second application of our model, the regulator is allowed to auction pollution licences for cars to mayors. In this case, the pollution level of an agent (mayor), i.e., the number of allocated licences, is set by the regulator together with the prices that the agent pays to get them.
Furthermore we study water pollution in rivers modelled by tree networks. In water pollution the government decides which pollution sources should be shut down so that the effluent level in water is as low as possible. Water pollution cost sharing was introduced in [42] where the network is a path (single river). This model was extended to tree networks (a system of rivers) in [18]. We model a system of rivers as a tree, but study a different pollution control model.
As a variant of the first application described above, we also consider the case in which the government is allowed to sell licences to the pollution sources instead of deciding to shut them down or keep them open. This is a widely used approach to control pollution levels by auctioning a fixed number of licences or pollution allowances. For instance, the European Emissions Trading System sells EU Emission Allowances (EUAs), each one representing the right to emit one ton of CO\(_2\). In such an auction, firm’s bid is a number of EUAs and per EUA a price. The auction ranks all the bids in descending order of per EUA price and determines the per EUA clearing price. The clearing price is the first bid price such that the total volume of EUAs in the bids (demand) in this descending order meets the total volume of EUAs offered by the regulator (supply). All the bids above this clearing price are awarded and they all pay the clearing price, see, e.g., [2]. This very simple auction does not take into account the diffusion relations between polluters, etc.
Finding an optimal social welfare solution to our problem, which we call Pollution Game (PG), is NPhard, that is why we study polynomial time approximation algorithms which can lead to incentive compatible (truthful) mechanisms. We study linear cost and damage functions and derive approximation algorithms and truthful mechanisms focusing on planar network topologies. In contrast, Belitskaya [11] assumes quadratic cost functions and linear damage functions deriving optimal social welfare and Nash equilibria solutions by explicit analytic formulas. We focus our study on planar network topologies which model realistic scenarios.
Most of the cited economics papers derive equilibria by closed analytic formulas. Some of these papers provide computational mechanisms without guaranteeing polynomial running time. Our approach is algorithmic and focuses on efficiently computing these solutions. We also analyze the computational complexity/hardness, of computing the social optimum in our model. To the best of our knowledge, this work is the first attempt to algorithmically analyze pollution control from the perspective of regulators by a network game model with information asymmetry between regulators and polluters.
We mainly study linear objective functions on trees and planar graphs. When the network is a directed tree, a somehow nonstandard two level dynamic programming approach is designed to obtain an FPTAS for our pollution game (PG). This approach is crucial to deal with the global pollution quota constraint. It uses a special multiple choice, multidimensional knapsack problem where coefficients of all constraints except one are bounded by a polynomial of the input size.
Baker’s shifting and treewidth decomposition techniques, see, e.g., [8, 30], are used for designing PTASs for various problems on planar graphs. It seems unlike to design a PTAS for PG with binary variables on planar graphs by adapting these techniques. That is because they deal with constraints on edges (e.g., for the independent set problem), but PG’s constraints are imposed on vertices from its neighbouring vertices. More precisely, in the independent set problem, the number of nodes is bounded (at most one) per each edge, while in the PG problem the number of selected neighbouring nodes is bounded, per each node. Furthermore, given two optimal solutions on two subgraphs, with common boundary vertices, of the planar graph, combining them together may not result in a feasible solution for PG on the whole graph. This is due to the possible infeasibility of local constraints of the boundary vertices of these two subgraphs. We overcome this major difficulty by introducing a new decomposition technique of planar graphs, which we call an \((\alpha ,\beta )\)decomposition, see Sect. 6.1 for details. This new technique is of independent interest and it may have further applications for the problems with constraints on vertices rather than on edges.
To obtain our PTAS for planar graphs on PG that may violate local quota constraints, we first use known rounding techniques (e.g., [13, 35]) to make all the coefficients polynomially bounded. Then, we design a dynamic programming approach to solve PG on any graph with a bounded treewidth tree decomposition. Finally, we combine a special tree decomposition of kouterplanar graphs, called a nice tree decomposition, see [34], Baker’s shifting technique and our twolevel dynamic programming approach for dealing with the global constraint, obtaining our PTAS.
Our results. TiE/DT: truthful in expectation/deterministic truthful mechanism. PG(poly) is PG with polysize integer variables, PG(general) without this assumption
General objective function  Linear objective function  

Bounded Degree \(\Delta \)  Trees  Planar  
Lower bound  \(\varOmega \left( \frac{\Delta }{\log \Delta ^2}\right) \)  NPhard  Strongly NPhard (\(\delta \) violation)  
PG(poly)  \(O(\Delta )^\mathrm{a}\)  FPTAS TiE  O(1) DT  PTAS (\(\delta \) violation) 
PG(general)  \(O(\Delta )\) TiE \(^\mathrm{b}\)  FPTAS TiE [6]\(^\mathrm{c}\)  O(1) TiE [5] 
2 Literature Overview
An invaluable source of pollution control regulations comes from the managerial approaches in environment policies. The majority of literature in this field deals with symmetric information. This problem however shows a fundamental asymmetry between the regulatory bodies and pollutants. That is because the regulator (i.e. the government) and the rest of the players (i.e. the owners of pollution sources) have different incentives and therefore expressed through different utility functions, as explained in detail in Sect. 3. The research contributions considering environmental policy with asymmetric information and the diffusion nature of pollution have been limited until recently.
In order to control pollution, an incentive mechanism that is environmentally friendly and resource efficient needs to be designed and deployed by regulatory authorities. However, it is not obvious how to design such a mechanism in the presence of asymmetric information; just as Hurwicz [26] put it: the firms know that information will be used by the regulator to design a policy which will affect their profits. Hence, they have an incentive to manipulate reported information in order to influence the content of the policy. In this context, Farell [21] discusses the relevance of the Coase Theorem. This theorem basically asserts that bargaining will lead to an efficient outcome regardless of the initial allocation of property if negotiation and trade in presence of externality are possible and the transaction costs are sufficiently low. Considering the problems of incomplete information, that paper shows that voluntary negotiation does not lead to the firstbest outcome that maximizes joint surplus in the presence of twosided private information. That is to say, centralised economic institutions such as government control and intervention, and decentralised institutions such as bargaining and ownership rights, should be viewed as complementary to each other. Therefore, a necessary condition for the government when designing an optimal pollution control plan is the truthful information about firms.
Kwerel [38], Dasgupta et al. [15] and Spulber [50] have proposed mechanisms that implement truth telling by firms to maintain a mild level of pollution. Under this assumption the firms can communicate with the regulator but not with each other. In Kwerel’s scheme [38] firms are informed in advance that their messages will be translated into pollution taxes. The regulator issues a fixed number of transferable pollution licences and offers a subsidy for those licences which firms hold in excess of emission. Both the number of licences to be issued and the subsidy rate offered are calculated on the basis of the cost information provided by firms.
Kim and Chang [33] constructed an optimal incentive tax/subsidy scheme in an oligopoly market with pollution and suggested a differential damages mechanism, which leads to an optimal emission level. McKitrick [40] proposes that a Cournot Mechanism for pollution control under asymmetric information, in which a Nash Equilibrium exists, is stable and can be reached by iterative computations. Because firms may attempt to manipulate the pollution level allocation to their own advantage, the adjustment rule is exogenous and depends on the actions of the firms. The approach by Karp and Livernois [28] is related to that in Conrad and Wang [14]. The authors examined the steadystate properties of a tax adjustment mechanism in situations where the government has no information about firms’ abatement costs.
These prior studies provide an overall framework in the administrative approach to control pollution. However, those models are only a first level of approximation in characterizing the reality. Although, there is some literature studying an economics environment consisting of firms or countries with geographical distinction, few of them take the diffusion nature of air and water pollution into consideration. For instance, Petrosjan and Zaccour [44] study the problem of allocation over time of total cost incurred by countries in a cooperative game of pollution reduction. Segerson [48] develops a general incentive scheme for controlling nonpoint source pollution^{1} that considers the diffusion nature, in which rewards for environmental quality above a given standard are combined with penalties for substandard quality. Based on the work of Petrosjan and Zaccour [44], Belitskaya [11] develops an nperson network game model of emission reduction. Dorner et al. [19] create a multiobjective modeling system using Bayesian probability networks to study nonpoint source pollution. Both the work of Belitskaya [11] and Dorner et al. [19] are different from the setting of ours, in either model assumption or function settings. In addition to these works built on the network framework, Dong et al. [18] models the water pollution problem as a cost sharing problem on a tree network. However, none of the literature mentioned above takes into account the role of governments in pollution abatement, more specifically how to make policies assuming information asymmetry. A model that adequately takes both factors into account is what we need to tackle such problems in reality.
Few other papers have studied air pollution in relation to network models. Singh and Datta [49] use artificial neural network method to identify unknown pollution sources in the groundwater. Gianessi et al. [24] analyze the national water pollution control policies. And, finally, Trujillo and Hugh [54] study multiobjective air pollution monitoring network design. These papers use networks in a very different context from ours.
Turning into current practice, emission trading is a marketbased approach used to control pollution by providing economic incentives for achieving reductions in the emissions of pollutants. Various countries have adopted emission trading systems as one of the strategies for mitigating climatechange by addressing international greenhousegas emission [52]. Usually a governmental body sets a limit or cap on the amount of a pollutant that may be emitted. The limit or cap is allocated and/or sold by the central authority to firms in the form of emission permits which represent the right to emit or discharge a specific volume of the specified pollutant [51]. Permits (and possibly also derivatives of permits) can then be traded on secondary markets. For example, the European Union Emissions Trading Scheme (EU ETS) trades primarily in European Union Allowances (EUAs), the Californian scheme in California Carbon Allowances, the New Zealand scheme in New Zealand Units and the Australian scheme in Australian Units [53]. Firms are required to hold a number of permits (or allowances or carbon credits) equivalent to their emissions. The total number of permits cannot exceed the cap, limiting total emissions to that level. Firms that need to increase their volume of emissions must buy permits from those who require fewer permits [51, 52]. Currently a simple auction mechanism for selling EUAs is adopted in Europe, see, e.g., [3]. Furthermore in order to limit the automobile pollution, governments use policies of car taxation [22, 27]. A radical transport policy introduced in the UK and first applied in Central London resulting in 19% reduction of CO\(_2\) emissions (see Table 2 in [9]).
3 Model and Applications
We first describe the general model of a Pollution Game (PG) and then explain how our two suggested applications fit into it. We are given an area of pollution sources (e.g., factories, cars) each owned by an agent. The government acts as a regulator restricting the levels of emitted pollution, while aiming to maximize the social welfare.
More formally, we are given a weighted digraph \(G=(V,E)\), where V is the set of n pollution sources, also called players or agents, and E represents the set of neighbouring nodes, i.e., \((u,v) \in E\) if and only if an amount of pollution can be transferred by u to v. In this model no geometric assumptions are made. For each \((u,v) \in E\), the weight \(w_{(u,v)} = w_{uv}\) denotes the pollution transfer factor from node u to v. Without loss of generality we may suppose that \(w_{uv} \in (0,1]\), \(\forall (u,v)\in E\). Intuitively, \(w_{uv}\) is percentage of the pollution emitted at u that reaches v.
We note here, that the game theoretic ingredients of our PG model will be defined in Sect. 3.3.
3.1 Application 1: Regulation of Pollution Sources
In our first application the pollution sources are factories and the agents in this case are their owners. In the weighted digraph \(G=(V,E)\), V is the set of n factories and E is the set of neighbouring ones, i.e., \((u,v) \in E\) if and only if the pollution emitted by u affects v. The weight \(w_{(u,v)} = w_{uv}\) denotes a discount factor of the pollution discharged by agent u affecting its neighbour v. This can be intuitively understood as a percentage of pollution emitted at u that reaches, via air, v. The government has to decide which factories must remain open and which must be shut down in order for the local and global constraints to be fulfilled. This fact is denoted by the value of \(x_v\). If the owner of factory \(v\in V\) is given a licence then we set \(x_v=1\), otherwise the factory must be shut down and we set \(x_v=0\). As a result, in this first application we assume that \(x_v \in \{0,1\}\) and \(q_v=1,\)\(\forall v \in V\), and \(b_v : \{0,1\} \longrightarrow \mathbb {R}_{\ge 0}\). In this case the global constraint corresponds to the maximum number of awarded licences, or, equivalently, the maximum number of factories that can remain open in the whole area.
Although the pollution sources might not emit the same amount of effluents, they are treated as equal. That is because the imposed constraints by the government in every subarea take into account the total amount of effluents emitted by the pollution sources and not by each one independently, i.e., the decision of shutting down a factory depends also on the structure of the neighborhood graph.
3.2 Application 2: Allocation of Pollution Licences
In the second application formulated by the above convex program we consider an area of n cities each one administered by its mayor, who is the agent in this case. In every city statistical observations are used to measure the car traffic to the neighboring cities. More precisely we consider a network represented by a weighted digraph \(G=(V,E,w)\), where V is the set of n agents (mayors of the cities), E is the set of roads connecting cities such that \((u,v) \in E\) if and only if u and v are neighboring cities. Then, \(w: E \rightarrow {\mathbb {R}}\) represents the percentage of cars entering a city from a neighboring one, i.e., \(w_{uv}\) denotes the percentage of cars driving from u to v in some time interval measured by observations. The duty of the regulator is to allocate a number of licences to the agents (mayors) such that the total welfare is maximized while fulfilling a number of constraints. The agent with \(x_u\) licences gains a benefit of \(b_u(x_u)\) which is a monetary income coming from selling these \(x_u\) licences to car drivers, one licence per car. Our model does not model this explicitly but just assumes for simplicity that all \(x_u\) licences are sold.
Naturally a percentage of cars with licences from city u remains in u and the rest is split and drives into the neighboring cities. We denote by \(w_u\) the percentage of cars remaining in u and \(w'_{vu}\) the percentage of cars entering u from neighbouring city v. The maximum number of cars (maximum number of licences) allowed at any moment in city u is bounded by \(p'_u\) also given in the input. This is represented by the local constraint: \(w_u x_u+\sum _{v \in \delta (u)}w'_{vu}x_v \le p'_u\). If \(w_u \ne 0\) the last inequality can equivalently be written as \(x_u+\sum _{v \in \delta (u)}w_{vu} x_v \le p_u\), where \(w_{vu}=w'_{vu}/w_u\) and \(p_u=p'_u/w_u\).
Planar graphs are close to real applications, and it is natural to study our second application on planar networks [56]. Imagine a collection of cities (each being a contiguous geographic area) and roads connecting them. This defines a planar map where we only consider edges (roads) between neighbouring cities, which implies a planar graph. We disregard other roads and we consider only frequent driving patterns in a time interval measured by observations. They correspond to frequent commuters, e.g., between house and work, which typically are neighbouring cities.
3.3 Basic Definitions
Let \(I=(G,\mathbf {b,d,p,q})\) be an instance of PG, where \({\mathbf {b}}=(b_v)_{v \in V}\), \({\mathbf {d}}=(d_v)_{v \in V}\), \({\mathbf {p}}=(p_v)_{v \in V}\) and \({\mathbf {q}}=(q_v)_{v \in V}\) (\(b_v\) is assumed private information of v and the other parameters public). Let \({\mathcal {I}}\) be the set of all instances, and \({\mathcal {X}}\) the set of feasible allocations. Given a digraph \(G=(V,E)\) the undirected graph \(G^{un}=(V,E^{un})\) is such that \(E^{un} = \{(u,v) : (u,v) \in E \text{ or } (v,u) \in E\}\).
A mechanism \(\phi =(X,P)\) consists of an allocation \(X:{\mathcal {I}}\rightarrow {\mathcal {X}}\) and payment function \(P:{\mathcal {I}}\rightarrow \mathbb {R}^{V}_{\ge 0}\) (X(I) satisfies (6)–(8)). For any vector x, \(x_{u}\) denotes vector x without its uth component. We also denote by \((y,x_{u})\) vector x that has y at position u, in particular, \((x_u, x_{u})=x\). Note, \(r_v(X(I))=b_v(X_v(I))d_v(X_v(I)+\sum _{u\in \delta ^{}_G(v)}w_{uv}X_u(I))\) is the welfare of player v under X(I). A mechanism \(\phi =(X,P)\) is truthful, if for any \(b_{v}\), \(b_v\) and \(b'_v\), \(r_v(X(b_v,b_{v}))P_v(b_v ,b_{v})\ge r_v(X(b'_v,b_{v}))P_v(b'_v,b_{v})\). A randomized mechanism is truthful in expectation if for any \(b_{v}\), \(b_v\) and \(b'_v\), \(\mathbb {E}(r_v(X(b_v,b_{v}))P_v(b_v ,b_{v}))\)\(\ge \mathbb {E}(r_v(X(b'_v,b_{v}))P_v(b'_v,b_{v}))\), where \(\mathbb {E}(\cdot )\) is over the algorithm’s random bits. Note, that the utility of player v is defined as \(u_v=r_v(X(b_v,b_{v}))P_v(b_v ,b_{v})\) and the expected utility as \(\mathbb {E}(r_v(X(b_v,b_{v}))P_v(b_v ,b_{v}))\).
A mechanism is individually rational if each agent v has nonnegative utility when he declares \(b_v\), regardless of the other agents’ declarations.
In the following we will denote by \(OPT^{fr}_G(PG)\) the value of the optimal fractional solution of PG on G. Similarly \(OPT^{in}_G(PG)\) denotes the optimal integral solution. The integrality gap of PG on G is defined as \(\frac{OPT^{fr}_G(PG)}{OPT^{in}_G(PG)}\). The approximation ratio of an algorithm \({\mathcal {A}}\) with respect to \(OPT^{in}_G(PG)\) (\(OPT^{fr}_G(PG)\) respectively) is \(\rho ^{in}({\mathcal {A}})=\frac{OPT^{in}_G(PG)}{R({\mathcal {A}})}\) (\(\rho ^{fr}({\mathcal {A}})=\frac{OPT^{fr}_G(PG)}{R({\mathcal {A}})}\)), where \(R({\mathcal {A}})\) is the objective value of the \({\mathcal {A}}\)’s solution. Unless stated otherwise, the approximation \(\rho \) will be with respect to \(OPT^{fr}_G(PG)\). An FPTAS (PTAS, EPTAS, respectively)^{3} for a problem \({\mathcal {P}}\) is an algorithm \({\mathcal {A}}\) that for any \(\epsilon >0\) and any instance I of \({\mathcal {P}}\), outputs a solution with the objective value at least \((1\epsilon ) OPT^{in}_I({\mathcal {P}})\) and terminates in time \(poly(\frac{1}{\epsilon },I)\) (\((\frac{1}{\epsilon }I)^{g(\frac{1}{\epsilon })}\) and \(g(\frac{1}{\epsilon })poly(I)\), respectively), where g is a function independent from I. We also let \([n]=\{1,\ldots ,n\}\).
Proposition 1
[10, 45] There is a polynomial time deterministic algorithm for kcolumnsparse linear packing programming problem with binary variables, achieving the approximation ratio \(\rho ^{fr}=\gamma _{k}\).
Proposition 2
[10] There is a polynomial deterministic algorithm for kcolumnsparse convex submodular packing programming problem with binary variables, achieving the approximation ratio \(\rho ^{fr}=\frac{e\gamma _{k}}{e1}\).
Proposition 3
[39] For any linear packing programming problem, if there is a polynomial deterministic algorithm with the approximation ratio \(\rho ^{fr}\) for this problem, then there is a polynomial, randomized, individually rational, \(\rho ^{fr}\)approximation mechanism for the same problem that is truthful in expectation.
4 Hardness
Theorem 1
Finding a feasible solution which satisfies constraints (10) and (7) to PG when \(p_v=1 \forall v \in V\) and \(w_{uv}>0\) for any \((u, v)\in E\) is NPcomplete.
Proof
It is straightforward that the problem is in NP. Consider now a formula of monotone 1in3 SAT where an instance of this problem consists of n Boolean variables and m clauses. A YES instance is one in which an assignment to its Boolean variables is such that exactly one literal from each clause is true. The problem is known to be NPComplete even when there are no negations [47]. The following proof is inspired by the reduction of 3SAT to Independent Set (p. 248 [16]).
Consider now an instance of 1in3 SAT which is satisfiable and let \(G=(V,E)\) be the corresponding graph constructed as above, letting \(p_u=1\) and \(p=m\) for every vertex \(u \in V\). Suppose that we have a truth assignment which satisfies all the clauses. Then this means we choose p vertices in G without violating any of the constraints. Indeed if two vertices have the same label then they are not connected. If they have different labels, say x from clause \(C_1\) and y from clause \(C_2\) and they are connected, this is because their corresponding clauses have a common literal, either x or y. Thus if one of them has value true, the other will have value false for the formula to be satisfiable. Finally if two vertices belong to the same clause, only one of them will have the value true.
Suppose now that there is a solution for graph G with m vertices when \(p_v=1, \)\(\forall v \in V\) (recall that \(p = m\)). Then setting the literal in the set \(\{v:x_v=1\}\) is a solution of 1in3 SAT. The argument is as follows, in each triangle, there is exactly one vertex such that its value is one since at most one vertex in each triangle can be selected and there are m triangles and \(p=m\). By the construction of G, these vertices cannot be connected to each other (and thus, form a solution of 1in3 SAT), due to (7) and the fact that \(p_v=1, \forall v \in V\). \(\square \)
Theorem 2
It is strongly NPhard to find an optimal solution to Pollution Game (PG) when \(p_v\) is any constant number \(\ge 1\) , \(b_v(x_v)\) is linear and \(d_v(y)\) is piecewise linear (with two pieces) \(\forall v \in V\) and \(w_{vu}\) is positive constant for any \((v, u)\in E\).
Proof
First we only consider undirected graphs, however, our reduction also applies to directed graphs. Let \(G=(V,E)\) be a graph with degree \(d(G) \le d\). Next construct a bipartite graph \(G'=(V',U',E')\) with \(V'=V\) and \(U'=E\), where each vertex of \(V'\) corresponds to a vertex of V and each vertex of \(U'\) corresponds to an edge of E. Connect a vertex \(v \in V'\) with a vertex \(u\in U'\) if the corresponding vertex of v is incident to the corresponding edge of u in G. It can easily be seen that every \(v \in V'\) has degree at most d and every \(u \in U'\) has degree 2. Let \(b_v(x_v)=x_v\) and \(d_v(x_v)=0, \forall v \in V'\). Furthermore, for any \(u\in U'\), let \(b_u(x_u)=0\) and \(d_u(y)=\frac{V(y\max \{w_{vu},w_{v'u}\})}{\min \{1,w_{vu}+w_{v'u}\}\max \{w_{vu},w_{v'u}\}}\) if \(y > \max \{w_{vu}, w_{v' u}\}\) and \(d_u(y)=0\) otherwise, where \((v,u)\in E'\) and \((v',u)\in E'\). The intuition behind the definition of this damage function is that it basically allows the second claim below to hold.
Let W be an independent set of G. Let \(W=k \le p\). Then the welfare of W for PG on \(G'\) is k. Suppose now there is a better solution \(W'\) with larger welfare for PG. We declare the following two claims:
Claim
\(W' \cap U'= \emptyset \)
If \( W' \cap U'\ne \emptyset \), suppose a vertex \(u\in U'\) is included in \(W'\), then \(r_u\le 0\). Hence, removing u from \(W' \cap U'\) will not decrease the total welfare.
Claim
Any two vertices \(u,v \in W'\) are not connected to the same vertex in \(U'\).
If there exist two vertices \(u, v \in W'\) that are connected to the same vertex in \(U'\), suppose they are connected to \(u'\in U'\). Then we know \(r_u=r_v=1\). However, since for the local level of pollution in \(u'\) is \(y\ge w_{vu}+ w_{v' u}>\max \{w_{vu}, w_{v' u}\}\), we have \(r_{u'}= d_{u'}(y)\le V\). Hence, the total welfare achieved by \(W'\) is at most \(W'\cap V'V\le 0< k\) . Removing either u or v from \(W'\) will increase welfare by \(V1\).
Therefore, \(W'\) corresponds to an independent set in G with size larger than W. Thus, any independent set W gives a welfare of W in \(G'\). As a consequence, if we can find a solution of PG in \(G'\) with welfare at least k, then we can easily find an independent set in G of size at least k. And, in the other direction, an independent set in G of size k corresponds directly to a PG solution in \(G'\) with welfare k. \(\square \)
From [23] it is strongly NPhard to find the maximum independent set on a planar graph with degree at most 3.
Corollary 1
For a planar graph \(G=(V,E)\) with degree at most 3, the problem of finding an optimal solution in PG setting as in Theorem 2 is strongly NPhard.
Proof
For any planar graph \(G=(V,E)\), the constructed graph \(G'=(V',U',E')\) in the proof of Theorem 2 is planar. To see this, just add one vertex to the center of each edge in G representing the edge vertex in \(U'\). The resulting graph is planar and the same as \(G'\). The corollary follows from the reduction in the proof of Theorem 2. \(\square \)
In the next theorem we use a by now commonly used complexity theoretic Unique Games conjecture, see [31].
Theorem 3
PG is Unique Gameshard to approximate within \(n^{1\epsilon }\) and within \(\frac{\Delta }{\log ^2\Delta }\) for graph G with degree \(\Delta \) when \(p_v\) is any constant number \(\ge 1\) , \(b_v(x_v)\) is linear and \(d_v(y)\) is piecewise linear (with two pieces) \(\forall v \in V\) and \(w_{vu}\) is positive constant for any \((v, u)\in E\).
Proof
According to [32], maximum independent set is Unique Gameshard to approximate within \(n^{1\epsilon }\) in general graphs and within \(\frac{\Delta }{\log ^2\Delta }\) for graph G with degree \(\Delta \). The theorem follows from the reduction in the proof of Theorem 2. \(\square \)
Theorem 4
There is no EPTAS for PG with binary variables on the directed planar graph \(G=(V,E)\) when \(b_v\) and \(d_v\) are both linear functions, for any \(v\in V\).
Proof
Consider PG on the following simple planar graph. There are \(n+2\) vertices labeled as \(\{o_1,o_2,1,2,...,n\}\) and the edge set \(E=\{(i, o_j),i\in [n],j\in \{1,2\}\}\) with weights \(w_{io_j}\), \(i\in [n],j\in \{1,2\}\). For any two dimensional knapsack problem, there exists an instance of PG with binary variables without the global constraint on such a simple graph exactly corresponding to this two dimensional knapsack problem. According to [37], there is no EPTAS for two dimensional knapsack. Hence, there is no EPTAS for PG on this simple planar graph (Fig. 3). \(\square \)
5 Directed Trees
We present a truthful in expectation FPTAS for PG on directed trees by a two level dynamic programming approach and a 3approximation deterministic truthful mechanism.
5.1 Truthful in Expectation Mechanisms
A digraph G is called a directed tree if the undirected graph \(G^{un}\) is a tree. We consider rooted trees where arcs are directed towards the leaves. We obtain our truthful in expectation FPTAS for PG with binary variables on any directed trees by a twolevel dynamic programming (DP) approach. The first bottom–up level is based on a careful application of the standard singledimensional knapsack FPTAS. The second level is by an interesting generalization of an FPTAS of [12] for a special multiple choice multidimensional knapsack problem with a constant number of constraints most of which have poly(I) size of coefficients. This FPTAS generalizes the results in [12], where the authors consider the one dimensional knapsack problem with cardinality constraint.

J is the number of items available for selection,

K denotes the number of different classes of items where at most one item can be chosen from each class,

\(N=O(1)\) is the number of dimensions of the constraints or items,

\(C_{jk}\) denotes the profit of item j from class k,

\(B_i=poly(I)\), \(\forall \)\(i\in [N]\) is the capacity (size) of the ith dimension,

\(A_{ijk}\) is the size of jth item in dimension i from class k and

\(A'_{jk}\) is the size of the jth item from class k of dimension \(N+1\).
Lemma 1
There is a pseudo polynomial optimal algorithm for SKP, terminating in \(O(CJKB^{N})\) time.
Proof
Theorem 5
\({\mathcal {A}}^{scaled}\) is an FPTAS for SKP, terminating in \(O(\frac{J^3KB^N}{\epsilon })\) time.
Proof
We will also need the following tool from mechanism design for packing problems. An integer linear packing problem with binary variables is a problem of maximising a linear objective function over a set of linear packing constraints, i.e., constraints of form \(a \cdot x\le b\) where \(x \in \{0,1\}^{n}\) is a vector of binary variables, and \(a, b \in \mathbb {R}_{\ge 0}^{n}\).
Proposition 4
[20] Given an FPTAS for an integer linear packing problem with binary variables, there is a truthful in expectation mechanism that is an FPTAS.
We first present an FPTAS on directed trees without global constraint which captures our main technical ingredients.
5.1.1 FPTAS Without Global Constraint
The algorithm uses a dynamic programming approach and the FPTAS for knapsack problem as a subroutine. Note that on a star, any instance of knapsack can be reduced to a PG instance without global constraints. Thus, an FPTAS is the best we can hope for such PG unless \(P=NP\).
5.1.2 FPTAS with Global Constraint
Theorem 6
There is a truthful in expectation mechanism for PG with binary variables on directed trees, which is an FPTAS.
For general \(x_v\in \mathbb {Z}\), we can replace each \(x_v\) by \(q_v\) duplicated variables \(x_{vj}\), \(j=1,\ldots ,q_v\), i.e., \(\{x_v \in \{0,1,\ldots , q_v\}\}=\{\sum _{j\in [q_v]}jx_{vj}\,\, \sum _{j\in [q_v]}x_{vj}\le 1, x_{vj}\in \{0,1\}\}\). Note that this transforms a polynomial size integer constraint into a multiple choice, one dimensional knapsack constraint. Hence, for directed trees, by a DP approach, we can construct a pseudo polynomial time algorithm to compute the exact optimal value of PG with integer variables, in time \(poly(V,q,OPT^{in}(PG))\). In addition, we can remove \(OPT^{in}(PG)\) from the running time by a loss of \(\epsilon \) of the optimal value using scaling techniques. Thus, there is a \((1\epsilon )\)approximation algorithm for PG with integer variables with running time in \(poly(V,q,1/\epsilon )\). Finally, by Proposition 4, we obtain the following:
Theorem 7
There is a truthful in expectation mechanism for PG with polynomial size integer variables on directed trees, which is an FPTAS.
5.2 Deterministic Truthful Mechanisms on Directed Trees
We will use a maximal in range (MIR) mechanism, see, e.g., [17], to obtain a \((3+\epsilon )\) approximate deterministic truthful mechanism for PG with polynomial size integer variables on directed trees. As we know, by transformation from integer constraint into multiple choice and one dimensional knapsack constraint, we only need to show such an approximation algorithm for binary variables. Our mechanism is based on a recent deterministic truthful PTAS for 2 dimensional knapsack problem^{4} [13, 17, 35]. We will first need the following:
Definition 1
A vertex in a rooted directed tree is called at level i if the distance between the vertex and the root is i in the undirected version of the tree.
Lemma 2
Lemma 3
 (1)
\(\max _{s_i\in {\mathcal {S}}_i}\{\omega (s_i)\}\ge (1\epsilon )OPT(PG_i)\).
 (2)
There exists an \(O(L_i\Delta ^{6+\frac{1}{\epsilon }})\) algorithm \({\mathcal {A}}_i\) that finds the optimal solution of the range \({\mathcal {S}}_i\) of \(PG_i\), for any \(\epsilon >0\).
Proof
Theorem 8
There is a deterministic \(\rho ^{in}\)approximate truthful mechanism for PG with polynomial size integer variables on directed trees, where \(\rho ^{in}=3+\epsilon \). For binary variables the mechanism terminates in \(O(V^2\Delta ^{6+\frac{1}{\epsilon }})\) time.
Proof
6 Planar Graphs
We present two algorithms for PG on planar graphs. The first has a constant approximation ratio, obtained by decomposing the plane and not violating any constraint. The second algorithm is a PTAS, obtained by a rounding of variables and a dynamic programming approach on a tree decomposition and violating the local constraints by a small value \(\delta > 0\).
6.1 Constant Approximation Without Violations
According to the following Lemma 4, we can obtain a constantfactor approximation for PG with integer variables for any graph with \((\alpha ,\beta )\)decomposition. Such a decomposition of planar graphs will be presented later.
Lemma 4
If a directed graph G has an \((\alpha ,\beta )\)decomposition, then there is a deterministic \((\rho ^{fr}=\alpha \gamma _{\beta +2}+1)\)approximation algorithm for PG with integer variables, and, a truthful in expectation mechanism for the same problem with the same approximation.
Proof
Planar graphs In the following, we will show that the integrality gap of PG on planar graphs is at least 4 as shown by a complete graph with four vertices. For a small \(\epsilon >0\), let \(w_{uv}=\epsilon \), for any \((u,v)\in E\), and \(p_v=\omega _{v}=1\), for any \(v\in V\). There is no global constraint. The optimal integer solution of PG on this graph is \(x_v=1\) for some \(v\in V\) and \(x_u=0\) for all \(u\ne v\), implying the optimal objective value 1. However, setting \(x_v=14\epsilon \), for any \(v\in V\) provides a feasible fractional solution, which gives the objective value \(416\epsilon \). Therefore, the integrality gap is at least 4, meaning that our LP relaxation cannot lead to better than 4 (e.g., PTAS) approximations.
We provide an \((\alpha ,\beta )\)decomposition of any planar graph, with \(\alpha =18\), \(\beta =6\). We did not attempt to optimize these two parameters.
Theorem 9
There is an \((\alpha ,\beta )\)decomposition of a directed planar graph \(G=(V,E)\), where \((\alpha ,\beta )=(18,6)\).
Proof
Lemma 5
For each \(v\in N_j\), the number of significant neighbours of v in \(N_{j1}\) with respect to \(N_j\) is at most two.
Proof
Suppose towards a contradiction that v has three significant neighbours with respect to \(N_j\). That is, suppose there exists \(v_1\ne v_2 \ne v_3 \in N_{j1}\cap \delta _{G'}(v)\) and \(v\ne u_1,u_2,u_3 \in N_j\) such that \((u_i,v_i)\in G'\), \(i\in \{1,2,3\}\) (see Fig. 5). By the definition of \(N_j\), there is a path from \(v_0\) to \(v_i\), \(i\in \{1,2,3\}\), and \((v,v_i)\in G'\), \(i\in \{1,2,3\}\). Suppose without loss of generality that \(v_2\) is inside the circle constructed from the path of \(v_0\) to \(v_1\), \(v_3\) and edges \((v,v_1)\) and \((v,v_3)\) in the planar embedding. Then \((u_2,v_2)\) will intersect this circle, which contradicts that \(G'\) is planar. \(\square \)
Next, we partition \(N_j\) into two sets \(N^1_j\) and \(N^2_j\) such that each vertex in \(N^i_j\) has at most two significant neighbours in \(N_{j+1}\) with respect to \(N^i_j\), \(i\in \{1,2\}\). We say two vertices \(v, u \in N_j\) are connected by a zigzag path if there exists a path \((v,v_1,v_2,v_3,\ldots ,v_s,u)\) in \(G'\) such that \(v_{i}\) and \(v_{i+1}\) alternatively belong to \(N_{j+1}\) and \(N_{j}\), i.e., \(v_1\in N_{j+1}\) and \(v_2\in N_j\). Note that s must be odd. We define the zigzag length of this zigzag path as \(\frac{s+1}{2}\). The zigzag distance between v and u, denoted \(d^{z}_{uv}\), is defined as the zigzag length of the shortest zigzag path between v and u if there exists one and \(\infty \) otherwise. Note that the zigzag distance of v to itself is zero. The partition algorithm PA\(_1\) (see Algorithm 1) partitions the set of vertices in layer \(N_j\) into sets \(N_j^1\) and \(N_j^2\). The algorithm proceeds in iterations. In each iteration a vertex \(v \in N_j\) is chosen arbitrarily and all vertices with odd zigzag distance from v are assigned to \(N_j^1\). Similarly all vertices with even zigzag distance are assigned to \(N_j^2\). Let \(N^1_j=A_1\) and \(N^2_j=A_2\), where \(A_1, A_2\) is the output of PA\(_1\). (Note that PA\(_1\) runs for each \(j \in [K]\).)
Lemma 6
For each \(v\in N^i_j\), v has at most two significant neighbours in \(N_{j+1}\) with respect to \(N^i_j\), \(i\in \{1,2\}\).
Proof
First, note that if v and u are selected in different iterations of the while loop in Algorithm 1, there is no zigzag path between them. Therefore, for a single iteration of the while loop, suppose \(v\in B\) is selected. We only need to show that for any \(u\in B_i\), u has at most two significant neighbours in \(N_{j+1}\) with respect to \(B_i\), \(i\in \{1,2\}\). First, note that \(v\in B_2\) (its zigzag distance to itself is 0). Since all the other vertices in \(B_2\) have zigzag distance to v at least two, v has no significant neighbours with respect to \(B_2\) in \(N_{j+1}\). Now fix \(i\in \{1,2\}\). Consider any two vertices \(u, u' \in B_i\), where u and \(u'\) connect to the same vertex in \(N_{j+1}\) only if they have the same zigzag distance to v. Suppose there exist three different vertices \(v_1\), \(v_2\), \(v_3 \in N_{j+1}\), such that they are significant neighbours of u with respect to \(B_i\) (see Fig. 6). By similar arguments as above, there exists zigzag paths from v to \(v_i\), \(i\in \{1,2,3\}\). Also note that edges \((u,v_i)\in G', i\in \{1,2,3\}\). Without loss of generality, suppose \(v_2\) is in the circle constructed from the zigzag paths v to \(v_1\), \(v_3\) and edges \((u,v_1)\) and \((u,v_3)\). Since \(G'\) is planar, there exists no edge between \(v_2\) and another vertex in \(B_i\) with the same zigzag distance to u. Therefore, u has at most two significant neighbours with respect to \(B_i\) in \(N_{j+1}\). \(\square \)
Lemma 7
For any \(k\in \{1,2,3\}\), and each \(v\in N^{ik}_j\), v has at most 2 neighbours in \(N_j\), or has no neighbours in \(N^{ik}_j\) nor significant neighbours with respect to \(N^{ik}_j\) in \(N_j\backslash N^{ik}_j\).
Proof
First, note that if v and u are selected in different iterations of while loop in Algorithm 2, there is no \(N_j\)path between them. Therefore, for a single iteration of the while loop, suppose \(v\in B\) is selected. Since \(v\in B_3\) (\(N_j\) distance to itself is 0), v has no neighbours in \(B_3\) nor significant neighbours with respect to \(B_3\) in \(N_j\backslash B_3\) by PA\(_2\). Now fix \(k\in \{1,2,3\}\). Consider any two vertices \(u_1, u_2 \in B_k\), \(u_1\) and \(u_2\) connect to the same vertex in \(N_j\) only if they have the same \(N_j\)distance to v. Next we will show for any \(u_1\ne v\) and \(u_1\in B_k\), for any k, \(u_1\) has at most two neighbours in \(N_j\). Suppose there exist three different vertices \(u_1\), \(u_2\), \(u_3 \in N_j\), such that \((u_1,u_2)\in G'\) and \((u_1,u_3)\in G'\). By similar arguments as above, there exists \(N_j\)paths from v to \(u_i\), \(k\in \{1,2,3\}\). Since \(u_i \in N_j\), \(i\in \{1,2,3\}\), there exist paths in \(G'\) from \(v_0\) to \(u_i\), \(i\in \{1,2,3\}\). We observe that it is only possible that \(u_1\) is in the circle constructed from the \(N_j\) paths v to \(u_2\), \(u_3\) and paths from \(v_0\) to \(u_2\) and \(u_3\) (the case where \(u_2\) or \(u_3\) is in the circle constructed by the other two vertices with v and \(v_0\) will violate the planarity of \(G'\)) (see Fig. 7). Since graph \(G'\) is planar, there exists no edge between \(u_1\) and another vertex in \(N_j\) (due to that such a vertex will have a path to v and \(v_0\) respectively). Therefore, \(u_1\) has at most two neighbours in \(N_j\). \(\square \)
Combining Lemmas 5, 6 and 7, \(\{N^{ik}_j\}_{ijk}\) is an \((\alpha ,\beta )\)decomposition of G with \((\alpha ,\beta )=(18,6)\), which finishes the proof of Theorem 9. \(\square \)
By Theorem 9 and Lemma 4, we have
Theorem 10
There is a randomized, individually rational and truthful in expectation \((18\gamma _8+1)\)approximation mechanism for PG on planar graphs with integer variables.
6.2 Better Approximation Under Some Mild Condition
Theorem 11
Suppose condition (12) holds and \(R(x^1)\ge 1\). There is a randomized, individually rational, \((\rho ^{fr}=6+\epsilon )\)approximation mechanism that is truthful in expectation for PG on planar graphs with integer variables, terminating in time \(poly(I,\log (\frac{1}{\epsilon }))\).
Proof
Note that if condition (12) holds, then every independent set is a feasible solution for PG with binary variables without global constraint. By 4color theorem [7, 46] for planar graphs, there is an independent set \(S\subset V\) such that \(4R(z_S)\ge R(x^1)\) where \(z_S\) is defined by \(z_v=1\) if \(v\in S\) and \(z_v=0\) otherwise. Further there is an \(O(V^2)\) algorithm finding \(z_S\) [46]. By Theorem 3 of [36] and \(R(x^1)\ge 1\), there is a deterministic \((\rho ^{fr}=5+\epsilon )\)approximation algorithm for PG with binary variables, running in \(poly(I,\log (\frac{1}{\epsilon }))\) time. Then there is a deterministic \((\rho ^{fr}=6+\epsilon )\)approximation algorithm for PG with integer variables, running in time \(poly(I,\log (\frac{1}{\epsilon }))\) [10]. By Proposition 3, this \((\rho ^{fr}=6+\epsilon )\)approximation mechanism is truthful in expectation for PG with integer variables. \(\square \)
6.3 A PTAS with \(\delta \) Violation of Constraints
 1.
Round PG to an equivalent problem \({\bar{PG}}_2\) with polynomial size integer variables.
 2.
Using the nice tree decomposition [34], we present a dynamic programming approach to solve \({\bar{PG}}_2\) optimally on a kouterplanar graph.
 3.
By a shifting technique similar to [8], we obtain a PTAS with \(1+\delta \) violation of local constraints for PG.
Lemma 8
Any feasible solution of \(PG'\) is feasible in \({\bar{PG}}_1\), and any feasible solution of \({\bar{PG}}_1\) is feasible for PG except violating each local constraint by a factor of \(1+\delta \).
Proof
Definition 2
 1.
\(\bigcup _{i\in I}X_i=V\),
 2.
for all edges \((v,w)\in E\), there exists an \(i\in I\) with \(v\in X_i\) and \(w\in X_i\),
 3.
for all i, j, k\(\in I\): if j is on the path from i to k in T, then \(X_i\cap X_k\subseteq X_j\).
Definition 3
 1.
if a node \(i\in I\) has two children j and k, then \(X_i=X_{j}=X_{k}\) (joint node),
 2.
if a node \(i \in I\) has one child j, then either \(X_i\subset X_j\), and \(X_i = X_j  1\) (forget node), or \(X_j \subset X_i\) and \(X_j =\)\(X_i  1\) (introduce node),
 3.
if node \(i \in I\) is a leaf of T, then \(X_i = 1\) (leaf node).
Lemma 9
[29] For any kouterplanar graph \(G=(V,E)\), there is an algorithm to compute a tree decomposition \((\{X_ii\in I\},\,T=(I,F))\) of G with treewidth at most \(3k1=O(k)\), and \(I=O(V)\) in O(kV) time.
Given a tree decomposition \((\{X_ii\in I\},\,T=(I,F))\) for \(G=(V,E)\) with treewidth k and \(I=O(V)\), we can obtain a nice tree decomposition with the same treewidth k and the same number of nodes O(kV) in \(O(k^2V)\) time [34]. Thus, for any kouterplanar graph \(G=(V,E)\), we can compute a nice tree decomposition \((\{X_ii\in I\},\,T=(I,F))\) of G with treewidth at most \(3k1=O(k)\), and \(I=O(kV)\) in \(O(k^2V)\) time. In the following, we will assume there is a nice tree decomposition for any kouterplanar graph.
Dynamic Programming (DP) We present a DP approach to solve \({\bar{PG}}_2\) on a directed kouterplanar graph using a nice tree decomposition of its undirected version. Note that a nice tree decomposition of an undirected version of a directed graph is also a nice tree decomposition of itself. Suppose we have a nice tree decomposition \((\{X_ii\in I\},\,T=(I,F))\) of a directed kouterplanar graph \(G=(V,E)\). We will use a bottom–up DP approach for \({\bar{PG}}_2\). In the following we will present our DP approach to the more general application of the allocation of pollution licences (application 2).

\(X_i\) is a leaf node or a start node, where \(t=1\). \(\varPsi _i(a^i_1,\ell ^i_1,Q^i)=\omega _{v^i_1}a^i_1\) if the triple \((a^i,\ell ^i,Q^i)\) is feasible, which can be verified easily e.g. \(Q^i=a^i_1\) and \(\ell ^i_1={\bar{w}}_{v^i_1v^i_1}(a^i_1)\). Let \(\varPsi _i(a^i_1,\ell ^i_1,Q^i)=\infty \) if the triple \((a^i,\ell ^i,Q^i)\) is not feasible.

\(X_i\) is a forget node and suppose its child is \(X_j=X_i\cup \{v^j_{t+1}\}\).
\(\varPsi _i(\mathbf {a^i},{\varvec{\ell }}^{\mathbf {i}},Q^i)=\max _{a^j_{t+1},\ell ^j_{t+1}}\varPsi _j(\mathbf {a^i},a^j_{t+1},{\varvec{\ell }}^{\mathbf {i}},\ell ^j_{t+1},Q^i)\)

\(X_i\) is an introduce node and suppose its child is \(X_j=X_i\backslash \{v^i_{t}\}\). Let \(a^j_s=a^i_s\) and \(\ell ^j_s=\ell ^i_s{\bar{w}}_{v^i_tv^i_s}(a^i_t)\), \(\forall s\in [t1]\). \(\varPsi _i(\mathbf {a^i},{\varvec{\ell }}^{\mathbf {i}},Q^i)=\varPsi _j(\mathbf {a^j},{\varvec{\ell }}^{\mathbf {j}}, Q^ia^i_{t})+\omega _{v^{i}_t}a^i_{t}\) if \(\sum _{s\in [t]}{\bar{w}}_{v^i_sv^i_{t}}(a^i_s)=\ell ^i_{t}\), and \(\varPsi _i(\mathbf {a^i},{\varvec{\ell }}^{\mathbf {i}},Q^i)=\infty \) otherwise.

\(X_i\) is a joint node and suppose its two children j and k are such that \(X_j=X_k=X_i\). \(\varPsi _i(\mathbf {a^i},{\varvec{\ell }}^{\mathbf {i}},Q^i)=\max _{A}\{\varPsi _j(\mathbf {a^j},{\varvec{\ell }}^{\mathbf {j}},Q^j)+\varPsi _k(\mathbf {a^k},{\varvec{\ell }}^{\mathbf {k}},Q^k)\}\), where the condition
\(A=\{(\mathbf {a^j},{\varvec{\ell }}^{\mathbf {j}},Q^j),(\mathbf {a^k},{\varvec{\ell }}^{\mathbf {k}},Q^k)\,\,\mathbf {a^j}+\mathbf {a^k}=\mathbf {a^i},{\varvec{\ell }}^{\mathbf {j}}+{\varvec{\ell }}^{\mathbf {k}}={\varvec{\ell }}^{\mathbf {i}}, Q^j+Q^k=Q^i\}\).

\(X_i\) is the root of T, \(OPT(Q^i)=\max _{\mathbf {a^i},{\varvec{\ell }}^{\mathbf {i}}}\{\varPsi _i(\mathbf {a^i},{\varvec{\ell }}^{\mathbf {i}},Q^i)\}\) is the optimal value (social welfare) of \({\bar{PG}}_2\) when the total scaled number of licences is exactly \(Q^i\), i.e., the global constraint satisfies \(\sum _{v\in V}x_v= Q^i\).
Claim
The DP approach on the nice tree decomposition of a kouterplanar graph gives the optimal solution to \({\bar{PG}}_2\).
Proof
We prove the claim by induction on the height h of the tree T.
Induction base In the case where \(h=0\) the root node is the leaf hence we can easily verify whether the triple \(\varPsi \) is feasible and set its value according to step 1 of the DP approach.

v is an introduce node and its child is \(u_1\). We assume that we have computed the optimal solution for the subtree rooted at \(u_1\) (with height \(h1\)). In this case the optimal value of v is given by the formula of the third step of the DP approach.

v is a forget node and its child is \(u_1\). The case is similar to the one given in the introduce node step with the difference that the optimal value of v is given by the formula of the second step of the DP approach.

v is a joint node and its children are \(u_1\) and \(u_2\). We assume again that we have computed the optimal solutions for the subtrees rooted at \(u_1\) and \(u_2\) (where at least one has height \(h1\)). Then the optimal value of v is given by the formula of the fourth step of the DP approach.
We note that the tree is rooted at an empty forget node. Furthermore, each vertex can be introduced multiple times but can only be forgotten once. Hence, the optimal solution of the root is also the optimal solution of \({\bar{PG}}_2\) when the global constrained, \(\sum _{v \in V} x_v = Q^i\), is satisfied.
For each node \(X_i\), we need to keep \(O(pq^{3k} \lceil \frac{2\rho }{\delta }\rceil ^{3k})=O(Vq^{3k+1} \lceil \frac{2\rho }{\delta }\rceil ^{3k})\) number of \(\varPsi _i\) values. Each \(\varPsi _i\) can be computed in \(O(Vq^{3k+1} \lceil \frac{2\rho }{\delta }\rceil ^{3k})\) time (this is the worst case running time when \(X_i\) is a joint node). There are O(kV) nodes in T. Therefore, the total running time of the DP approach (by multiplying the above three numbers) is \(O(kV^3q^{6k+2} \lceil \frac{2\rho }{\delta }\rceil ^{6k})\).
Based on the above DP approach, we can solve \({\bar{PG}}_2\) on any kouterplanar graph optimally for any fixed k (which includes any directed tree whose treewidth is 2). Therefore, for any \(\delta >0\) and fixed k, we can use VCG to get an optimal deterministic truthful mechanism for PG on any directed kouterplanar graph that violates each local constraint by a factor of \(\delta \) and runs in \(O(kV^3q^{6k+2} \lceil \frac{2\rho }{\delta }\rceil ^{6k})\) time (note that Theorem 12 also works for bounded treewidth graphs).
Theorem 12
For any \(\delta >0\) and fixed k, there is an optimal deterministic truthful mechanism for PG on any directed kouterplanar graph \(G=(V,E)\) that violates each local constraint by a factor of \(1+\delta \) and runs in \(O(kV^3q^{6k+2} \lceil \frac{2\rho }{\delta }\rceil ^{6k})\) time, where \(\rho =V(\lfloor \log _2(q)\rfloor +2)\).
Step 3: PTAS for planar graphs
Observe that when there are some boundary conditions on a kouterplanar graph, the above DP approach still works. For example, if the number of licences of any vertex in any first and last face (level 1 and level k face) of the kouterplanar graph is zero, we just modify the dynamic programming approach in a bottom–up manner to set \(\varPsi _i=\infty \) if any vertex v in any first and last face is a parameter of \(\varPsi _i\) and its number of licences \(a^i_v>0\). Then the modified DP approach is the desired algorithm for \({\bar{PG}}_2\) on the kouterplanar graph under this boundary condition.
Proposition 5
PG is strongly NPhard on planar graphs when we allow a \(\delta \) violation of local constraints.
Proof
Theorem 14 provides a PTAS for PG with \(q= poly(V)\) (in particular \(q_v =1\) and 13) and \((1+\delta ')\)violation, giving a tight approximation in this sense.
Theorem 13
For any fixed k and \(\delta >0\), there is an \(O(k^2V^3q^{6k+2} \lceil \frac{2\rho }{\delta }\rceil ^{6k})\) algorithm for PG with integer variables on directed planar graph \(G=(V,E)\) that achieves \(\rho ^{in}\)approximation and violates each local constraint by a factor of \(1+\delta \), where \(\rho =V(\lfloor \log _2(q)\rfloor +2)\) and \(\rho ^{in}=\frac{k}{k2}\).
Proof
Let \(\frac{2}{k}=\epsilon \) in Theorem 13. Also note that \(\rho =V(\lfloor \log _2(q)\rfloor +2)\). We have:
Theorem 14
7 General Objective Function for Bounded Degree Graphs
7.1 Approximation Algorithms
If R(x) is monotone, we present an algorithm with an approximation ratio of \(O(\Delta )\) for PG on a graph with maximum degree \(\Delta \).
Theorem 15
If R(x) with binary variables is monotone increasing, then there is an \((\rho ^{fr}=\frac{e\gamma _{\Delta +2}}{e1}+1)\)approximation algorithm for PG with integer variables.
Proof
If \(x_v\in \{0,1\}, \, \forall v\in V\), for any \(A\subseteq V\), we define \(g(A)=R(x)\) where \(x_v=1\), \(\forall v\in A\) and \(x_v=0\), for any \(v\notin A\). It is not difficult to see that R with binary variables is submodular if and only if g satisfies \(g(A\cup B)+g(A\cap B)\le g(A)+g(B)\), for any \(A, B \subset V\). For any \(A\subseteq V\), and \(v\in V\), denote by \(A+v\) the set \(A\cup \{v\}\). Let \(g_v(A)=g(A+v)g(A)\). Then it is not difficult to see that \(g(A\cup B)+g(A\cap B)\le g(A)+g(B)\), for any \(A, B \subset V\) if and only if for any \(A\subseteq B\subseteq V\) and \(v\in V\backslash B\), \(g_v(A)\ge g_v(B)\). Next we will prove that \(g_v(A)\ge g_v(B)\), which implies that R(x) is submodular.
For the graph with degree \(\Delta \), note that PG is \(\Delta +2\) column sparse. By Proposition 2, there is a randomized \(\rho ^{fr}=\frac{e\gamma _{\Delta +2}}{e1}\)approximate algorithm for PG with binary variables if R(x) is monotone increasing, because such R(x) is also submodular. This algorithm can be derandomized to be deterministic with the same approximation ratio. Now observe that for a concave function G(x) from \(\mathbb {R}_{\ge 0}\) to \(\mathbb {R}_{\ge 0}\), we have \(G(x+y)\le G(x)+G(y)\), for any x, \(y\in \mathbb {R}_{\ge 0}\) (without loss of generality let \(x\ge y>0\), by concavity of G, it holds that \(\frac{G(x+y)G(x)}{x+yx}\le G'(x)\le G'(y)\le \frac{G(y)G(0)}{y0}\)). By this property, since \(b_v(x)\) and \(d_v(x)\) are concave from \(\mathbb {R}_{\ge 0}\) to \(\mathbb {R}_{\ge 0}\), for any \(v\in V\), for any feasible solution \(x=\{x_v\}_v\) and \(y=\{y_v\}_v\), we have \(R(x+y)\le R(x)+R(y)\). By ellipsoid algorithm for convex programming problem in [25], we can get an optimal fractional solution of PG denoted as \(x^*=\{x^*_v\}_v\). Let \(z^*=\{z^*_v\}_v\) where \(z^*_v=\lfloor x^*_v\rfloor \), for any \(v\in V\) and \(x^*=z^*+y^*\). Let \(y'\) be an \(\frac{e\gamma _{\Delta +2}}{e1}\)approximate solution for PG with binary variables when R(x) is monotone increasing. Note that \(y^*\) is a feasible fractional solution for PG with binary variables. We know \(R(y^*)\le \frac{e\gamma _{\Delta +2}R(y')}{e1}\). Let \(x'\) be an solution of PG with integer variables such that \(x'=y'\) if \(R(y')\ge R(z^*)\) and \(x'=z^*\) otherwise. Therefore, we have \(R(x^*)\le R(z^*)+R(y^*)\le R(z^*)+\frac{e\gamma _{\Delta +2}R(y')}{e1}\le (\frac{e\gamma _{\Delta +2}}{e1}+1)R(x')\). \(\square \)
7.2 Truthful in Expectation Mechanisms
Lemma 10
Let \(x^*\) be an optimal fractional solution of PG under the condition that functions \(b_v\) and \(d_v\) satisfy constraints (16) and (17), then \(x^*\) can have the following property: for each \(v\in V\), the local level of pollution in v satisfies that \(x^*_v+\sum _{u\in \delta ^{}_G(v)} w_{u v}x^*_u\le y_v\).
Proof
We prove this lemma by contradiction. Suppose there exists \(v \in V\), such that \(x^*_v+\sum _{u\in \delta ^{}_G(v)} w_{u v}x^*_u> y_v\). If \(x^*_v>0\), by constraints (16), we can decrease the value of \(x^*_v\) by an amount of \(\alpha \) such that \(x^*_v\alpha +\sum _{u\in \delta ^{}_G(v)} w_{u v}x^*_u> y_v\). By simple calculation, the total social welfare increases by an amount of \((s^0_v s^2_v\sum _{u\in \delta ^{+}_G(v)}s^1_uw_{vu})\alpha \ge 0\). Thus, we can do this until either \(x^*_v=0\) or \(x^*_v\alpha +\sum _{u\in \delta ^{}_G(v)} w_{u v}x^*_u\le y_v\). If the first case holds and the second case does not hold, then there exists \(u\in \delta ^{}_G(v)\) with \(x^*_u>0\). Note that \(v\in \delta ^{+}_G(u)\). Since \(\sum _{u\in \delta ^{}_G(v)} w_{u v}x^*_u> y_v\), if we decrease the value \(x^*_u\) by \(\alpha \), by simple calculation, the total social welfare increases by at least \((s^0_u s^{i}_u\sum _{u'\in \delta ^{+}_G(u)\backslash \{v\}}s^{i}_{u'}w_{uu'} s^2_vw_{uv})\alpha \ge (s^0_u s^{1}_u\sum _{u'\in \delta ^{+}_G(u)\backslash \{v\}}s^{1}_{u'}w_{uu'} s^2_vw_{uv})\alpha \), which is nonnegative by constraints (17). Here, the value i is defined as follows, if total pollution in v is below \(y_u\) then \(s^{i}_u=s^{1}_u\), otherwise \(s^{i}_u=s^{2}_u\), with the same argument for \(s^{i}_{u'}\). By this operation, we can decrease the value of v without loss of total social welfare until the total level of pollution in v does not reach \(y_v\). \(\square \)
Lemma 11
If PG functions \(b_v\) and \(d_v\) satisfy constraints (16) and (17) then there is a deterministic polynomial time algorithm with approximation ratio \(\rho ^{fr}=\gamma _{\Delta +2}\).
Proof
This modified PG has the same optimal fractional solution as PG. In the modified PG, \(R(x)=\sum _{v\in V}\omega _vx_v\), where \(\omega _v=s^0_vs^1_v\sum _{u\in \delta ^{}_G(v)}s^1_uw_{vu}\). By Proposition 1, there is a deterministic polynomial time algorithm for the modified PG with approximation ratio \(\rho ^{fr}=\gamma _{\Delta +2}\). This algorithm is also an algorithm for PG with the same approximation ratio. \(\square \)
With Lemma 11, we now present a truthful in expectation mechanism for PG with approximation ratio \(\gamma _{\Delta +2}=(e+o(1))(\Delta +2)\).
Theorem 16
Suppose the bidding strategy \(s^0_v\) of each player \(v \in V\) satisfies constraints (16) and (17). There is a randomized, individually rational, \((\rho ^{fr}=\gamma _{\Delta +2})\)approximation mechanism that is truthful in expectation for PG on G with degree at most \(\Delta \).
Proof
Since the bidding strategy \(s^0_v\) of each player v satisfies constraints (16) and (17), by Proposition 3 and Lemma 11, there is a randomized, individually rational, \(\gamma _{\Delta +2}\)approximation mechanism that is truthful in expectation for the modified PG, which is also a truthful in expectation mechanism for PG with the same approximation ratio. \(\square \)
Corollary 2
If \(b_v\) and \(d_v\) are linear functions for any v and the bidding strategy \(s^0_v\) of player v is arbitrary, then there is a randomized, individually rational, \((\rho ^{fr}=\gamma _{\Delta +2})\)approximation mechanism that is truthful in expectation for PG.
Proof
Since \(d_v\) is linear, it is equivalent to the above piecewise linear function with \(s^2_v=+\infty \) and \(y_v=+\infty \), for any \(v\in V\). By Theorem 16, there is a randomized, individually rational, \(\gamma _{\Delta +2}\)approximation mechanism that is truthful in expectation for PG. \(\square \)
Remark
We cannot anticipate an algorithm with constant approximation ratio for PG on the graph with average degree \(\Delta \) (the average degree of a graph G is \(\frac{\sum _{v\in V}\delta _{G^{un}}(v)}{V}\)) even if \(\Delta =1\). Consider a graph \(G'\) consisting of a complete graph with n vertices and \(n^2n\) isolated vertices with valuation 0. Note that \(G'\)’s average degree is \(\frac{n^2}{n^2}= 1\). PG on G cannot be approximated within \(n^{1\epsilon }\) unless Unique Game conjecture fails. Thus, PG cannot be approximated within \((n^2)^{\frac{1\epsilon }{2}}\) on \(G'\), where \(n^2\) is the number of vertices of \(G'\).
8 Open Problems
We presented a new network model for the pollution control problem and studied planar and tree networks which model realistic scenarios. These networks can be applied to model air and water pollution from diffuse sources. Our main technical results include a constant approximation algorithm and a PTAS with a small violation in the constraints for the case of planar graphs and an FPTAS which is truthful in expectation and a 3approximate deterministic truthful mechanism for the case of trees. We obtained these results by introducing novel algorithmic techniques for planar and tree graphs which could be of independent interest.
Many interesting open problems arise from this new model. Our main open question is to determine whether PG with binary variables on planar graphs admits a PTAS or whether it is APXhard, when no local constraint is volated. Another direction would be to study lower bounds on truthful (deterministic, universal, truthful in expectation) mechanisms for PG. Can externality be used to obtain such lower bounds? Furthermore it would be interesting to generalize our results to other graphs, e.g., Euclidean graphs.
Footnotes
 1.
Nonpoint source (NPS) pollution refers to both water and air pollution from diffuse sources, that is sources without a specified fixed location. For instance, nonpoint source water pollution affects a water body from sources such as polluted runoff from agricultural areas draining into a river, or windborne debris blowing out to sea. This work deals mainly with point source pollution.
 2.
The author of [38] uses cost function rather than benefit function, which can be viewed as \(M_vb_v(x_v)\), with \(M_v\) a large constant for any \(v\in V\). The author assumes that cost function is convex decreasing and it is equivalent to \(b_v(x_v)\) being a concave increasing function. We use benefit function rather than cost function for ease of analysis.
 3.
Fully Polynomial Time Approximation Scheme, Polynomial Time Approximation Scheme and Efficient Polynomial Time Approximation Scheme respectively.
 4.
This PTAS also works for multiple choice and constant dimensional knapsack problem, which will be used for PG with polynomial size integer variables.
Notes
Acknowledgements
X. Deng was supported by the National Science Foundation of China (Grant No. 61173011) and a Project 985 grant of Shanghai Jiaotong University. P. Krysta and J. Zhang were supported by the Engineering and Physical Sciences Research Council under Grant EP/K01000X/1. M. Li was partly supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 117913). H. Qiao was supported by the National Science Foundation of China (Grant No. 71373262).
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