The Approximability of Multiple Facility Location on Directed Networks with Random Arc Failures

Abstract

We introduce and study the maximum reliability coverage problem, where multiple facilities are to be located on a network whose arcs are subject to random failures. Our model assumes that arcs fail independently with non-uniform probabilities, and the objective is to locate a given number of facilities, aiming to maximize the expected demand serviced. In this context, each demand point is said to be serviced (or covered) when it is reachable from at least one facility by an operational path. The main contribution of this paper is to establish tight bounds on the approximability of maximum reliability coverage on bidirected trees as well as on general networks. Quite surprisingly, we show that this problem is NP-hard on bidirected trees via a carefully-constructed reduction from the partition problem. On the positive side, we make use of approximate dynamic programming ideas to devise an FPTAS on bidirected trees. For general networks, while maximum reliability coverage is $(1 - 1/e + \epsilon )$-inapproximable as an extension of the max k-cover problem, even estimating its objective value is #P-complete, due to generalizing certain network reliability problems. Nevertheless, we prove that by plugging-in a sampling-based additive estimator into the standard greedy algorithm, a matching approximation ratio of $1 - 1/e - \epsilon$ can be attained.

Appendices

Appendix 1: Additional Proofs

1.1 Proof of Claim 2.6

To obtain the desired inequality, note that

$$\begin{aligned} \prod _{i \in I} \left( e^{-a_i/A} + \frac{ 1 }{ K } \right)= \, & {} e^{- \sum _{i \in I} a_i/A} + \sum _{S \subsetneq I} \frac{ e^{-\sum _{i \in S} a_i/A} }{ K^{|I| - |S|} } \\\le\, & {} e^{- \sum _{i \in I} a_i/A} + \sum _{s = 0}^{|I|-1} {|I| \atopwithdelims ()s } \cdot \frac{ 1 }{ K^{|I| - s} } \\= \, & {} e^{- \sum _{i \in I} a_i/A} + \left( 1 + \frac{ 1 }{ K } \right) ^{ |I| } - 1 \\\le\, & {} e^{- \sum _{i \in I} a_i/A} + e^{ |I| / K } - 1 \\\le\, & {} e^{- \sum _{i \in I} a_i/A} + \frac{ 2 \cdot |I| }{ K } \\\le\, & {} e^{- \sum _{i \in I} a_i/A} + \frac{ 1 }{ 100A^2 } . \end{aligned}$$

Here, the first inequality is obtained by observing that $e^{- \sum _{i \in S} a_i / A} \le 1$. In the third inequality we are using $e^x \le 1 + 2x$ for $x \in [0,1]$. Finally, the last inequality holds since $|I| \le n$ and $K = 200 nA^2$.

1.2 Proof of Claim 2.8

The proof is similar to that of Claim 2.6, and we provide it for completeness. To this end, note that

$$\begin{aligned} \prod _{i \in I} \left( e^{-a_i/A} - \frac{ 1 }{ K } \right)= \, & {} e^{- \sum _{i \in I} a_i/A} + \sum _{S \subsetneq I} \cdot \frac{ e^{-\sum _{i \in S} a_i/A} }{ (-K)^{|I| - |S|} } \\\ge \, & {} e^{- \sum _{i \in I} a_i/A} - \sum _{S \subsetneq I} \frac{ 1 }{ K^{|I| - |S|} } \\= \, & {} e^{- \sum _{i \in I} a_i/A} - \sum _{s = 0}^{|I|-1} {|I| \atopwithdelims ()s } \cdot \frac{ 1 }{ K^{|I| - s} } \\= \, & {} e^{- \sum _{i \in I} a_i/A} - \left( 1 + \frac{ 1 }{ K } \right) ^{ |I| } + 1 \\\ge \, & {} e^{- \sum _{i \in I} a_i/A} - e^{ |I| / K } + 1 \\\ge \, & {} e^{- \sum _{i \in I} a_i/A} - \frac{ 2 \cdot |I| }{ K } \\\ge \, & {} e^{- \sum _{i \in I} a_i/A} - \frac{ 1 }{ 100A^2 } . \end{aligned}$$

1.3 Properties of $\varvec{{\mathcal {E}}}$

Lemma 5.1

The expected demand coverage function ${{\mathcal {E}}} : 2^V \rightarrow {\mathbb {R}}_+$is monotone and submodular.

Proof

We begin by observing that it suffices to prove that each of the functions $\{ \pi _v \}_{v \in V}$ is monotone and submodular, since ${{\mathcal {E}}}( F ) = \sum _{v \in V} d_v \cdot \pi _v( F )$ is a non-negative weighted sum of these functions. To this end, recall that $\pi _v( F )$ stands for the probability that node v is covered by at least one facility in F. Put differently, letting ${{\mathcal {D}}}_{F \rightsquigarrow v}$ be the event where a least one of the directed paths connecting a facility in F to the node v survives, we have $\pi _v( F ) = {\mathrm{Pr}} [ {{\mathcal {D}}}_{F \rightsquigarrow v} ]$. With this representation, we derive the desired properties as follows:

• Monotonicity of $\pi _v$: For two subsets of facilities $F_1 \subseteq F_2$, since ${{\mathcal {D}}}_{F_1 \rightsquigarrow v} \subseteq {{\mathcal {D}}}_{F_2 \rightsquigarrow v}$,

  $$\begin{aligned} \pi _v( F_1 ) = {\mathrm{Pr}} \left[ {{\mathcal {D}}}_{F_1 \rightsquigarrow v} \right] \le {\mathrm{Pr}} \left[ {{\mathcal {D}}}_{F_2 \rightsquigarrow v} \right] = \pi _v( F_2 ) . \end{aligned}$$

• Submodularity of $\pi _v$: For two subsets of facilities $F_1$ and $F_2$,

  $$\begin{aligned} \pi _v( F_1 \cup F_2 )= \, & {} {\mathrm{Pr}} \left[ {{\mathcal {D}}}_{F_1 \cup F_2 \rightsquigarrow v} \right] \\= \, & {} {\mathrm{Pr}} \left[ {{\mathcal {D}}}_{F_1 \rightsquigarrow v} \cup {{\mathcal {D}}}_{F_1 \rightsquigarrow v} \right] \\= \, & {} {\mathrm{Pr}} \left[ {{\mathcal {D}}}_{F_1 \rightsquigarrow v} \right] + {\mathrm{Pr}} \left[ {{\mathcal {D}}}_{F_2 \rightsquigarrow v} \right] - {\mathrm{Pr}} \left[ {{\mathcal {D}}}_{F_1 \rightsquigarrow v} \cap {{\mathcal {D}}}_{F_2 \rightsquigarrow v} \right] \\\le\, & {} {\mathrm{Pr}} \left[ {{\mathcal {D}}}_{F_1 \rightsquigarrow v} \right] + {\mathrm{Pr}} \left[ \mathcal{D}_{F_2 \rightsquigarrow v} \right] - {\mathrm{Pr}} \left[ {{\mathcal {D}}}_{F_1 \cap F_2 \rightsquigarrow v} \right] \\= \, & {} \pi _v( F_1 ) + \pi _v( F_2 ) - \pi _v( F_1 \cap F_2 ) \ , \end{aligned}$$

  where the inequality above holds since ${{\mathcal {D}}}_{F_1 \cap F_2 \rightsquigarrow v} \subseteq ({{\mathcal {D}}}_{F_1 \rightsquigarrow v} \cap {{\mathcal {D}}}_{F_2 \rightsquigarrow v})$.

$\square$

Appendix 2: APX-Hardness for Undirected Graphs

Theorem 6.1

The maximum reliability coverage problem on undirected graphs is APX-hard.

Proof

We describe a gap-preserving reduction from the minimum-cardinality vertex cover problem on cubic graphs (henceforth, VCC), which is known to be APX-hard [2]. In other words, for some constant $\alpha > 0$, it is NP-hard to distinguish between graphs with $\tau (G) \le k$ and those with $\tau (G) \ge (1+\alpha )k$, where $\tau (G)$ stands for the minimum size of a vertex cover in G. Given an instance of VCC, consisting of a cubic graph $G=(V,E)$ on n vertices and a parameter $k \ge |E|/3$, we construct an instance of maximum reliability coverage on the same underlying graph as follows:

• Each vertex has a demand of 1.

• Each edge has a survival probability of 1/2.

• At most k facilities can be located.

Under this reduction, letting $F^*$ be an optimal set of facilities, we proceed by proving the following claims:

• 1. If $\tau (G) \le k$ then ${{\mathcal {E}}}( F^* ) \ge \frac{7n}{8} + \frac{k}{8}$.

• 2. If $\tau (G) \ge (1+\alpha )k$ then ${{\mathcal {E}}}( F^* ) \le \frac{7n}{8} + \frac{k}{8} - \frac{\alpha n}{64}$.

These claims imply that, unless ${\mathrm {P}} = {\mathrm {NP}}$, maximum reliability coverage on undirected graphs cannot be approximated within factor greater than

$$\begin{aligned} \frac{ \frac{7n}{8} + \frac{k}{8} - \frac{\alpha n}{64} }{ \frac{7n}{8} + \frac{k}{8} } = 1 - \frac{\alpha n}{ 8\cdot (7n+k)} \le 1 - \frac{\alpha }{ 64} . \end{aligned}$$

Proof of Claim 1 Since $\tau (G) \le k$, there exists a vertex cover $U \subseteq V$ with $|U| = k$. Now, when we locate facilities at U, each $v \in U$ is covered with probability $\pi _v( U ) = 1$, and each vertex $v \notin U$ is covered with probability $\pi _v( U ) = 7/8$. To understand the latter claim, note that since U is a vertex cover, when $v \notin U$ its set of neighbors N(v) is necessarily a subset of U, in which case $\pi _v( U ) = 1 - (1/2)^{ |N(v)| } = 1 - (1/2)^3$. As a result,

$$\begin{aligned} {{\mathcal {E}}}( F^* ) \ge {{\mathcal {E}}}( U ) = |U| \cdot 1 + |V {\setminus } U| \cdot \frac{ 7 }{ 8 } = k + (n-k) \cdot \frac{ 7 }{ 8 } = \frac{7n}{8} + \frac{k}{8} . \end{aligned}$$

Proof of Claim 2 With respect to the optimal set of facilities $F^*$, as before, each $v \in F^*$ is covered with probability $\pi _v( F^* ) = 1$. On the other hand, each $v \notin F^*$ has $| N(v) \cap F^* |$ facilities within its set of neighbors as well as $| N(v) {\setminus } F^*|$ facility-free neighbors. Therefore, to derive a simple bound on $\pi _v( F^* )$, note that this probability can only increase when we replace the two neighbors (different from v) of each $u \in N(v) {\setminus } F^*$ by two auxiliary vertices that are connected only to u, while locating facilities in both. In this setting, it is easy to verify that we obtain an upper bound of

$$\begin{aligned} \pi _v( F^* ) \le 1 - \left( \frac{ 1 }{ 2 } \right) ^{ | N(v) \cap F^*| } \cdot \left( 1 - \frac{ 1 }{ 2 } \cdot \frac{ 3 }{ 4 } \right) ^{ | N(v) {\setminus } F^*| } = 1 - \frac{ 1 }{ 8 } \cdot \left( \frac{ 5 }{ 4 } \right) ^{ | N(v) {\setminus } F^*| } . \end{aligned}$$

Consequently,

$$\begin{aligned} {{\mathcal {E}}}( F^* )\le\, & {} |F^*| \cdot 1 + \sum _{v \in V {\setminus } F^*} \left( 1 - \frac{ 1 }{ 8 } \cdot \left( \frac{ 5 }{ 4 } \right) ^{ | N(v) {\setminus } F^*| } \right) \\\le\, & {} |F^*| + (|V| - |F^*| - \alpha k) \cdot \frac{ 7 }{ 8 } + \alpha k \cdot \frac{ 27 }{ 32 } \\= \, & {} \frac{7n}{8} + \frac{k}{8} - \frac{\alpha k}{32} \\\le\, & {} \frac{7n}{8} + \frac{k}{8} - \frac{\alpha n}{64} . \end{aligned}$$

The second inequality holds since the number of vertices $v \in V {\setminus } F^*$ for which $N(v) {\setminus } F^* \ne \emptyset$ is at least $\alpha k$. Otherwise, by adding these vertices to $F^*$ we obtain a vertex cover in G with cardinality strictly smaller than $|F^*| + \alpha k \le (1+\alpha )k$, which contradicts the case hypothesis, $\tau (G) \ge (1+\alpha )k$. The last inequality follows by recalling that $k \ge |E|/3 = n/2$, as G is a cubic graph. $\square$