Earliest Arrival Transshipments in Networks with Multiple Sinks

Abstract

We study a classical flow over time problem that captures the essence of evacuation planning: given a network with capacities and transit times on the arcs and sources/sinks with supplies/demands, an earliest arrival transshipment (EAT) sends the supplies from the sources to the sinks such that the amount of flow which has reached the sinks is maximized for every point in time simultaneously. In networks with only a single sink earliest transshipments do exist for every choice of supplies and demands. This is why so far a lot of effort has been put into the development of efficient algorithms for computing EATs in this class of networks, whereas not much is known about EATs in networks with multiple sinks, aside from the fact that they don’t exist in general.

We make huge progress regarding EATs in networks with multiple sinks by formulating the first exact algorithm that decides whether a given tight EAT problem has solution and that computes the EAT in case of existence. Our algorithm only works on the originally given network without requiring any form of expansion and thus just requires polynomial space. Complementing this algorithm we show that in multiple sink networks it is, already for tight instances, $\mathcal {NP}$-hard to decide whether an EAT does exist for a specific choice of supplies and demands.

This work was supported by the DFG SPP 1736 “Algorithms for Big Data”.

A Appendix

Preliminaries for the Proof of Lemma 1. To prove Lemma 1 we need the notion of static lex-max flows. Let $\mathcal {N}$ be a dynamic network and $\prec$ a total order on $S^+\cup S^-$. A static lexicographically maximum (lex-max) flow x in $\mathcal {N}$ with respect to $\prec$ is a static flow which maximizes the net amount of flow sent out of $S^+ \cup S^-$ in increasing order $\prec$. For the sinks this means that the flow into them is maximized in decreasing order $\prec$. If we denote by ${\text {max}}_{\mathcal {N}}(S,X)$ the value of a static maximum flow from $S \subseteq S^+$ to $X \subseteq S^-$ in network $\mathcal {N}$, then it is due to Minieka [29] that

$$\begin{aligned} \begin{aligned} {\text {net}}_{x}(&\{s'\in S^+\cup S^-\mid s' \preceq s\})\\&= {\text {max}}_{\mathcal {N}}(\{s'\in S^+ \mid s' \preceq s\},\{s'\in S^- \mid s \preceq s'\}) \text { }\forall \text { }s \in S^+\cup S^-. \end{aligned} \end{aligned}$$

(2)

The proof of Lemma 1 strongly relies on Lemma 3, a connection between flows over time in $\mathcal {N}$ satisfying $\gamma ^{\theta }(X)$ for some rational $\theta = p,q$ with $p,q \in \mathbb {Z}_{>0}$ and some $X \subseteq S^-$, and certain static lex-max flows in the time-expanded network. We consider the time-expanded network $\mathcal {N}^\theta$ corresponding to our given dynamic network $\mathcal {N}$ with only a single source s and denote by $s^*$ the single super source of $\mathcal {N}^\theta$. Additionally, in each layer $\theta '$ we attach a super sink $t_X^{\theta '}$ to nodes $t^{\theta '}$ with $t\in X$ by an arc $(t^{\theta '},t^{\theta '}_X)$ with infinite capacity. Similarly, we attach a super sink $t^{\theta '}_{S^- \setminus X}$ and moreover we add an overall super sink $t_*^{\theta '}$ to each layer by infinite capacity arcs $(t^{\theta '}_{S^- \setminus X},t_*^{\theta '})$ and $(t^{\theta '}_{X},t_*^{\theta '})$. We consider $\{t_X^{1/q},t_{S^-\setminus X}^{1/q}, \ldots ,t_X^{p/q},t_{S^-\setminus X}^{p/q} \}$ or $\{t_*^{1/q}, \ldots ,t_*^{p/q}\}$ as sets of sinks of $\mathcal {N}^T$.

Lemma 3

A flow over time f in a dynamic network $\mathcal {N}$ with only a single source s with rational time horizon $T=p/q$, $p,q\in \mathbb {Z}_{>0}$, that satisfies $\gamma ^T(X)$ for some $X \subseteq S^-$ fulfills the following properties: (1) We have ${\text {net}}_{f}(X,\theta ) = -\gamma ^{\theta }(X)$ for all $\theta \le T$. (2) The flow over time f induces a static lex-max flow in $\mathcal {N}^T$ with respect to the total order $s^* \prec t^{p/q}_{S^- \setminus X} \prec t^{p/q}_{X} \prec t^{(p-1)/q}_{S^- \setminus X} \prec t^{(p-1)/q}_{X} \prec \ldots \prec t^{1/q}_{S^- \setminus X} \prec t^{1/q}_{X}$.

The proof of Lemma 3 mostly relies on (1) and (2) and we skip it here.

Proof of Lemma 1. Lemma 3, (1) and (2), imply for an integral $T \ge 0$,

Note that ${\text {max}}_{\mathcal {N}^T}(s^*,\{t^1_*,\ldots ,t^i_*\})$ is independent of X and thus in order to show submodularity, it suffices to show that $g^\theta :2^{S^-} \!\! \rightarrow \mathbb {R}$ with

$$\begin{aligned} g^\theta (X) := {\text {max}}_{\mathcal {N}^T}(s^*,\{t^1_*,\ldots ,t^\theta _*, t_X^\theta \}) \text { for } X \subseteq S^-, \end{aligned}$$

is submodular on $S^-$ for all $\theta \in \{1,\ldots ,T\}$. To show this fact, fix some integral $\theta$, $A \subseteq B \subseteq S^-$, and $v \in S^- \setminus B$. We redefine the time-expanded network $\mathcal {N}^\theta$. In time layer $\theta$ we now attach a super sink $t_v^\theta$ to $v^\theta$, a super sink $t_A^\theta$ to the copies of sinks in A in layer $\theta$, a super sink $t_{B \setminus A}^\theta$ to the copies of the sinks in $B \setminus A$, and a super sink $t^\theta _{S^-\setminus (B \cup \{v\})}$ to the copies of the remaining sinks in layer $\theta$.

Then, $g^{\theta }(A \cup \{v\}) - g^{\theta }(A)$ is exactly the amount of flow that reaches $t_v^{\theta }$ in a lex-max flow in $\mathcal {N}^{\theta }$ with respect to the order $\prec$ given by

$$\begin{aligned} s^* \prec t^{\theta }_{S^-\setminus (B \cup \{v\})} \prec t^{\theta }_{B \setminus A} \prec t_v^{\theta } \prec t^{\theta }_A \prec t^{\theta -1}_* \prec \ldots \prec t^1_*. \end{aligned}$$

Similarly, $g^{\theta }(B \cup \{v\}) - g^{\theta }(B)$ is exactly the amount of flow that reaches $t_v^{\theta }$ in a static lex-max flow in $\mathcal {N}^{\theta }$ with respect to

$$\begin{aligned} s^* \prec t^{\theta }_{S^-\setminus (B \cup \{v\})} \prec t^{\theta }_v \prec t^{\theta }_{B \setminus A} \prec t^{\theta }_A \prec t^{\theta -1}_* \prec \ldots \prec t^1_*. \end{aligned}$$

Since the sink $t_v^\theta$ has a higher priority with respect to $\prec$ in the first order, we obtain $g^{\theta }(B \cup \{v\}) - g^{\theta }(B) \le g^{\theta }(A \cup \{v\}) - g^{\theta }(A)$, and thus submodularity. The submodularity for arbitrary rational time horizons follows by using a finer discretization of time and for irrational time horizons by continuity.    $\square$

Preliminaries for the Proof of Lemma 2. It turns out that Lemma 2 is a direct consequence of the following lemma which is the reverse direction of Lemma 3.

Lemma 4

Let $\mathcal {N}$ be a dynamic network and $T = p/q$ with $p,q \in \mathbb {Z}_{>0}$ a rational time horizon. A static lex-max flow x in ${\mathcal {N}}^T$ with respect to the total order

$$\begin{aligned} s^* \prec t^{p/q}_{S^- \setminus X} \prec t^{p/q}_{X} \prec t^{(p-1)/q}_{S^- \setminus X} \prec t^{(p-1)/q}_{X} \prec \ldots \prec t^{1/q}_{S^- \setminus X} \prec t^{1/q}_{X}, \end{aligned}$$

induces a flow over time with time horizon T satisfying $\gamma ^\theta (X)$ for each $\theta \in [0,T)$.

Proof

For the simplification of notation we only consider integral time horizons in our proof. Let f be the flow over time induced by the static lex-max flow x. By Lemma 3, (1) and (2) we obtain for our flow over time f,

By definition of f and (1) it also holds that $|f|_\theta = o^{\theta }(\{s\})$ for all $\theta \in \{0,1,\ldots ,T\}$. The function $\theta \mapsto o^{\theta }(\{s\})$ is piecewise linear [29, 40] and since all transit times are integral, breakpoints of this function only occur at integral points in time. However, by the construction of f from x (see [25]) the function $\theta \mapsto |f|_\theta$ is also a piecewise linear function with breakpoints only occurring at integral points in time. Thus, we have $|f|_\theta = o^{\theta }(\{s\})$ for all $\theta \le T$ and hence by our arguing above the flow over time f satisfies $\gamma ^\theta (X)$ for each integral $\theta \in \{1,2,\ldots ,T\}$. By the first statement of Lemma 3, $\gamma ^\theta (X)$ is satisfied by f for all $\theta \in [0,T)$.    $\square$

We can now prove Lemma 2.

Proof of Lemma 2. Let $\prec$ be given by $t_1\prec t_2\prec \ldots \prec t_k$ for $S^- = \{t_1,t_2,\ldots ,t_k\}$. Lemma 4 implies that a static lex-max flow in $\mathcal {N}^T$ with respect to a total order $\prec '$ given by $s^* \prec ' t^{p/q}_1 \prec ' \ldots \prec ' t^{p/q}_k \prec ' t^{(p-1)/q}_1 \prec ' \ldots \prec ' t^{(p-1)/q}_{k} \prec ' \ldots \prec ' t^{1/q}_1 \prec ' \ldots \prec ' t^{1/q}_k$ induces a flow over time f with the required properties.    $\square$