Fast core pricing algorithms for path auction

Abstract

Path auction is held in a graph, where each edge stands for a commodity and the weight of this edge represents the prime cost. Bidders own some edges and make bids for their edges. The auctioneer needs to purchase a sequence of edges to form a path between two specific vertices. Path auction can be considered as a kind of combinatorial reverse auctions. Core-selecting mechanism is a prevalent mechanism for combinatorial auction. However, pricing in core-selecting combinatorial auction is computationally expensive, one important reason is the exponential core constraints. The same is true of path auction. To solve this computation problem, we simplify the constraint set and get the optimal set with only polynomial constraints in this paper. Based on our constraint set, we put forward two fast core pricing algorithms for the computation of bidder-Pareto-optimal core outcome. Among all the algorithms, our new algorithms have remarkable runtime performance. Finally, we validate our algorithms on real-world datasets and obtain excellent results.

Appendices

Appendix A: Summary of main notation

$G=(V,E)$ :

A directed graph that consists of a edge set E and a vertex set V

$v_0$ :

The starting vertex

$v_n$ :

The ending vertex

$e_i= (v_{i-1},v_i)$ :

The edge that starts in the vertex $v_{i-1}$ and ends in $v_{i}$, also represents a bidder owning edge $e_i$

$c_{e_i}$ :

The cost of the edge $e_i$

$\pi _{e_i}$ :

The utility of the bidder $e_i$

$\Pi$ :

Social welfare of the auction

0:

The auctioneer

N :

The total set of players, including the bidders and auctioneer

W(L):

The social welfare of the subset L of N

P :

The payment set of the auction

$p_{e_i}$ :

The payment to the bidder $e_i$

$\text {core}$ :

The total set of core outcomes

$W_G(v_i,v_j)$ :

A walk from $v_i$ to $v_j$ in graph G

$P_G(v_i,v_j)$ :

The shortest path from $v_i$ to $v_j$ in graph G

$V_G(v_i,v_j)$ :

The vertex set of $P_G(v_i,v_j)$

$E_G(v_i,v_j)$ :

The edge set of $P_G(v_i,v_j)$

$P_w(v_0,v_n)$ :

The path that is selected as the winner path in the auction

$E_w(v_0,v_n)\; or\; E_w$ :

The edge set of $P_w(v_0,v_n)$

$V_w(v_0,v_n)\; or\; V_w$ :

The vertex set of $P_w(v_0,v_n)$

$P_w(v_i,v_j)$ :

A subpath of $P_w(v_0,v_n)$ that is from $v_i$ to $v_j$

$E_w(v_i,v_j)$ :

The edge set of $P_w(v_i,v_j)$

$V_w(v_i,v_j)$ :

The vertex set of $P_w(v_i,v_j)$

$d_G(v_i,v_j)$ :

The cost of the shortest path from $v_i$ to $v_j$ in graph G

Appendix B: Proof of the correctness of CCG-VCG algorithm for path auction

Theorem 15

The outcome of CCG-VCG algorithm is a bidder-Pareto-optimal core outcome.

Proof

Consider the constraint added into CCG-SET

$$\begin{aligned} \sum _{e_i \in E_w \backslash z}p_{e_i} \le \sum _{e_i \in E_{w'} \backslash z}c_{e_i} \end{aligned}$$

(65)

$E_{w'}$ is the new winner set in a new graph where we change the cost of each edge in $E_w$ from $p^{t-1}_{e_i}$ to $p^{t}_{e_i}$ in G. Denote by $G_1$ this graph and $P_{w'}$ the path corresponding to the edge set $E_{w'}$. $P_{w'}$ is the shortest path in $G_1$. We first prove that the constraint (65) is a standard constraint of (C1).

In $G_1$, we remove the edges in $E_w \backslash z$ and change the costs of the edges in z from $p^{t}_{e_i}$ to $c_{e_i}$. Denote by $G_2$ this graph. $P_{w'}$ also exists in $G_2$ because it doesn’t include any edge in $E_w \backslash z$. Compared with $G_1$, the cost of $P_{w'}$ reduces by $\sum _{e_i\in z}p^t_{e_i} - c_{e_i}$⁶ in $G_2$. As to other paths in $G_2$, their costs reduce by $\sum _{e_i\in z}p^t_{e_i} - c_{e_i}$ at most, so $P_{w'}$ is also the shortest path in $G_2$. Note that $G_2$ is just the graph $G\backslash (E_w \backslash z)$, from the constraint (65), we have

$$\begin{aligned} \sum _{e_i \in E_w \backslash z}p_{e_i}&\le \sum _{e_i \in E_{w'} \backslash z}c_{e_i}\\&= d_{G_2}(v_0,v_n) - \sum _{e_i\in z}c_{e_i}\\&= d_{G\backslash (E_w\backslash z)}(v_0,v_n)- \sum _{e_i\in z}c_{e_i} \end{aligned}$$

(66)

Recall the constraint in (C1) as

$$\begin{aligned} (C1):\sum _{e_i \in x}p_{e_i} \le d_{G\backslash x}(v_0,v_n) - (d_G(v_0,v_n) -\sum _{e_i\in x}c_{e_i}),\forall x\in E_w \end{aligned}$$

(67)

Let $x= E_w\backslash z$, we have

$$\begin{aligned} \sum _{e_i \in E_w\backslash z}p_{e_i}&\le d_{G\backslash (E_w\backslash z)}(v_0,v_n) - \left(d_G(v_0,v_n) -\sum _{e_i\in (E_w\backslash z)}c_{e_i}\right)\\&= d_{G\backslash (E_w\backslash z)}(v_0,v_n) - \sum _{e_i\in z}c_{e_i} \end{aligned}$$

(68)

The constraint (68) is (C1) is just the same as the constraint (66), so the constraint we add into CCG-SET during each iteration is a standard constraint of (C1). Then the constraint set CCG-SET is a subset of the constraint set (C1). In each iteration, CCG-VCG algorithm adds a constraint of (C1). The number of constraints in (C1) is limited so that this algorithm must stop in a limited number of steps.

To prove the theorem, we just need to prove that the outcome of CCG-VCG algorithm is in the core. Assuming that the outcome of CCG-VCG algorithm isn’t in the core. Thus, there is at least one constraint in (C1) which is not satisfied by this result. Without loss of generality, let $x=E_w\backslash z'$ is the corresponding set, then the constraint becomes

$$\begin{aligned} \sum _{e_i \in E_w\backslash z'}p_{e_i}&> d_{G\backslash (E_w\backslash z')}(v_0,v_n) - \left(d_G(v_0,v_n) -\sum _{e_i\in (E_w\backslash z')}c_{e_i}\right)\\&= d_{G\backslash (E_w\backslash z)}(v_0,v_n) - \sum _{e_i\in z}c_{e_i} \end{aligned}$$

(69)

Then we have

$$\begin{aligned} \sum _{e_i \in E_w}p_{e_i}&= \sum _{e_i \in E_w\backslash z'}p_{e_i} +\sum _{e_i \in z'}p_{e_i}\\&> d_{G\backslash (E_w\backslash z)}(v_0,v_n)+\sum _{e_i \in z'}p_{e_i}-\sum _{e_i \in z'}c_{e_i} \end{aligned}$$

(70)

where $d_{G\backslash (E_w\backslash z)}(v_0,v_n)$ is the cost of the shortest path from $v_0$ to $v_n$ in graph $G\backslash (E_w\backslash z)$. This path still exists in graph which changes the cost of edges in $E_w$ from $c_{e_i}$ to $p_{e_i}$. This change makes the cost of this path increase by $\sum _{e_i \in z'}p_{e_i}-\sum _{e_i \in z'}c_{e_i}$ at most. So the cost of this path is no more than $d_{G\backslash (E_w\backslash z)}(v_0,v_n)+\sum _{e_i \in z'}p_{e_i}-\sum _{e_i \in z'}c_{e_i}$, which means it is shorter than the sum of outcome in CCG-VCG algorithm. This produces a contradiction with terminal condition in CCG-VCG algorithm, so this theorem is established. $\square$

Appendix C: Proof of Theorem 9

Proof

In $G'$, the edge in $E_w$ is converted into a reverse edge with negative original cost. As we know, each edge is not cut edge for the connectivity from $v_0$ to $v_n$, that is, there exist a path from $v_0$ to $v_n$ after removing this edge in graph G. We use the mathematical induction by proving the following two propositions.

• 1. There exist a path from $v_0$ to $v_1$ in $G'$.

• 2. If there exists a path from $v_0$ to $v_i$ in $G'$, then there exists a path from $v_0$ to $v_{i+1}$ in $G'$ ($0< i < n$).

It is obvious that Theorem 9 is established if these two propositions is correct. Note that $V_w(v_1,v_n)$ is the vertex set including $v_1, v_2,\dots , v_n$. In proposition 1, since $(v_0,v_1)$ is not a cut edge, there must exist a path from $v_0$ to any vertex of $V_w(v_1,v_n)$ in the graph $G\backslash E_w$. Otherwise, there will not exist a path from $v_0$ to any vertex of $V_w(v_1,v_n)$ in graph $G\backslash (v_0,v_1)$, this is because compared with $G\backslash E_w$, the extra edges in $G\backslash (v_0,v_1)$ is useless for the connectivity between $\{v_0 \}$ and $V_w(v_1,v_n)$. This means $(v_0,v_1)$ is a cut edge, which is contradictory. Therefore, there exists a path from $v_0$ to any vertex of $V_w(v_1,v_n)$ in $G\backslash E_w$. This path also exists in $G'$ and once this path could arrive at any vertex of $V_w(v_1,v_n)$ from $v_0$, it could arrive at $v_1$ along the negative edges in $G'$. Thus, proposition 1 is true.

In proposition 2, since there exists a path from $v_0$ to $v_i$ in $G'$, we can arrive at any vertex of $V_w(v_0,v_i)$ by just lengthening this path along the negative edges. Based on that $(v_i,v_{i+1})$ is not a cut edge, similarly, we can draw a conclusion that there exists a path from any vertex of $V_w(v_0,v_i)$ to any vertex of $V_w(v_{i+1},v_n)$. Then there also exists a path from any vertex of $V_w(v_0,v_i)$ to any vertex of $V_w(v_{i+1},v_n)$ in $G'$. Denote these two vertices by $v_a,v_b$ and we have a path from $v_0$ to $v_{i+1}$ as $v_0 \rightarrow v_i \rightarrow v_a \rightarrow v_b \rightarrow v_{i+1}$, like Fig. 12. Therefore, proposition 2 is proved.

Fig. 12

Path $v_0 \rightarrow v_i \rightarrow v_a \rightarrow v_b \rightarrow v_{i+1}$

Above all, Theorem 9 is established. $\square$