A 2-Approximation Algorithm for Flexible Graph Connectivity

We present a 2-approximation algorithm for the Flexible Graph Connectivity problem [AHM20] via a reduction to the minimum cost r-out 2-arborescence problem.

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02/27/2022

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1 Introduction

In this paper, we consider the Flexible Graph Connectivity (FGC) problem which was introduced by Adjiashvili, Hommelsheim and Mühlenthaler [AHM20]. In an instance of FGC, we have an undirected connected graph , a partition of into unsafe edges and safe edges , and nonnegative costs on the edges. The graph may have multiedges, but no self-loops. A subset of edges is feasible for FGC if for any unsafe edge , the subgraph is connected. We seek a (feasible) solution minimizing . The motivation for studying FGC is two-fold. First, FGC generalizes many well-studied survivable network design problems. Most notably, the minimum-cost -edge connected spanning subgraph (2ECSS) problem corresponds to an instance of FGC where all edges are unsafe. Second, FGC captures a non-uniform model of survivable network design problems where a subset of edges never fail, i.e., they are always safe. Adjiashvili et al. [AHM20] gave a -approximation algorithm for FGC. Our main contribution is a simple -approximation algorithm for FGC. At a high level, our result is based on a straightforward extension of the -approximation algorithm of Khuller and Vishkin [KV94] for 2ECSS.

Theorem 1.

There is a -approximation algorithm for FGC.

Adjiashvili et al. [AHM20] also consider the following generalization of FGC. Let be an integer. A subset of edges is feasible for the -FGC problem if for any edge-set with , the subgraph is connected. The goal in -FGC is to find a solution of minimum cost. The usual FGC corresponds to -FGC. The following result generalizes Theorem 1.

Theorem 2.

There is a -approximation algorithm for -FGC.

Our proof of Theorem 2 is based on a reduction from -FGC to the minimum-cost -arborescence problem (see [Sch03], Chapters 52 and 53). We lose a factor of in this reduction. Fix some -FGC solution and designate a vertex as the root vertex. For an edge , we call the arc-set as a bidirected pair arising from . The key idea in our proof is that there exists an arc-set that contains arc-disjoint dipaths for each while satisfying the following two conditions: (i) for an unsafe edge , uses at most arcs from a bidirected pair arising from ; and (ii) for a safe edge , uses at most arcs from the disjoint union of bidirected pairs arising from . This argument is formalized in Lemma 7. Complementing this step, we show that any arc-set (consisting of appropriate orientations of edges in ) that contains arc-disjoint dipaths for every can be mapped to a -FGC solution.

2 A -Approximation Algorithm for -Fgc

For a subset of vertices and a subgraph of , we use to denote the set of edges in that have one endpoint in and the other in . The following characterization of -FGC solutions is straightforward.

Proposition 3.

is feasible for -FGC , contains a safe edge or unsafe edges.

For the rest of the paper, we assume that the given instance of -FGC is feasible: this can be easily checked by computing a (global) minimum-cut in where we assign a capacity of to safe edges and a capacity of to unsafe edges. Let be a digraph and be nonnegative costs on the arcs. We remark that may have parallel arcs but it has no self-loops. Let be a designated root vertex. For a subgraph of and a set of vertices , we use to denote the set of arcs such that the head of the arc is in and the tail of the arc is in .

Definition 1 (-out arborescence).

An -out arborescence is a subgraph of satisfying: (i) the undirected version of is acyclic; and (ii) for every , there is an dipath in .

Definition 2 (-out -arborescence).

For a positive integer , a subgraph is an -out -arborescence if can be partitioned into arc-disjoint -out arborescences.

Theorem 4 ([Sch03], Chapter 53.8).

Let be a digraph and let be a positive integer. For , the digraph contains an -out -arborescence if and only if for every nonempty .

Claim 5.

Let be an -out -arborescence for an integer . Let be any two vertices. Then, the number of arcs in that have one endpoint at and the other endpoint at (counting multiplicities) is .

Proof.

Since an -out -arborescence is a union of arc-disjoint -out -arborescences, it suffices to prove the result for . The claim holds for because the undirected version of is acyclic, by definition. ∎

Theorem 6 ([Sch03], Theorem 53.10).

In polynomial time, we can obtain an optimal solution to the minimum -cost -out -arborescence problem on , or conclude that there is no -out -arborescence in .

The following lemma shows how a -FGC solution can be used to obtain an -out -arborescence (in an appropriate digraph) of cost at most .

Lemma 7.

Let be a -FGC solution. Consider the digraph where the arc-set is defined as follows: for each unsafe edge , we include a bidirected pair of arcs arising from , and for each safe edge , we include bidirected pairs arising from

. Consider the natural extension of the cost vector

to where the cost of an arc is equal to the cost of the edge that gives rise to it. Then, there is an -out -arborescence in with cost at most .

Proof.

Let be a minimum-cost -out -arborescence in . First, we argue that is well-defined. By Theorem 4, it suffices to show that for any nonempty , we have . Fix some nonempty . By feasibility of , contains a safe edge or unsafe edges (see Proposition 3). If contains a safe edge with , then by our choice of , contains -arcs. Otherwise, contains unsafe edges, and for each such unsafe edge with , contains the arc . Since in both cases, is well-defined.

Finally, we use Claim 5 to show that satisfies the required cost-bound. For each unsafe edge , contains at most arcs from the bidirected pair arising from , and for each safe edge , contains at most arcs from the (disjoint) union of bidirected pairs arising from . Thus, . ∎

Lemma 7 naturally suggests a strategy for Theorem 2 via minimum-cost -arborescences.

Proof of Theorem 2.

Fix some vertex as the root vertex. Consider the digraph obtained from our FGC instance as follows: for each unsafe edge , we include a bidirected pair arising from , and for each safe edge , we include bidirected pairs arising from . For each edge , let denote the multi-set of all arcs in that arise from . For any edge and arc , we define . Let denote a minimum -cost -out -arborescence in . By Lemma 7, , where OPT denotes the optimal value for the given instance of -FGC.

We finish the proof by arguing that induces a -FGC solution with cost at most . Let . By definition of and our choice of arc-costs in , we have . It remains to show that is feasible for -FGC. Consider a nonempty set . Since is an -out -arborescence, Theorem 4 gives . If contains a safe arc (i.e., an arc that arises from a safe edge), then that safe edge belongs to . Otherwise, contains some unsafe arcs (that arise from unsafe edges). Since both orientations of an edge cannot appear in , we get that . Thus, is a feasible solution for the given instance of -FGC, and . ∎

References

  • [AHM20] David Adjiashvili, Felix Hommelsheim, and Moritz Mühlenthaler. Flexible graph connectivity. In D. Bienstock and G. Zambelli, editors,

    21st Integer Programming and Combinatorial Optimization, IPCO 2020, London, UK, June 8-10, 2020, Proceedings

    , LNCS, 12125:13–26. Springer, 2020.
  • [KV94] Samir Khuller and Uzi Vishkin. Biconnectivity approximations and graph carvings. J. ACM, 41(2):214–235, 1994.
  • [Sch03] A. Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics, Volume 24. Springer-Verlag, Berlin Heidelberg, 2003.