A 2-Approximation Algorithm for Flexible Graph Connectivity
Sylvia Boyd, Joseph Cheriyan, Arash Haddadan, Sharat Ibrahimpur
aa r X i v : . [ c s . D S ] F e b A 2-Approximation Algorithm for Flexible Graph Connectivity
Sylvia Boyd ∗ Joseph Cheriyan † Arash Haddadan ‡ Sharat Ibrahimpur † Abstract
We present a 2-approximation algorithm for the Flexible Graph Connectivity problem [AHM20] via a reduction to theminimum cost r -out 2-arborescence problem. In this paper, we consider the Flexible Graph Connectivity (FGC) problem which was introduced by Adjiashvili, Hommelsheimand M¨uhlenthaler [AHM20]. In an instance of FGC, we have an undirected connected graph G = ( V, E ), a partition of E into unsafe edges U and safe edges S , and nonnegative costs { c e } e ∈ E on the edges. The graph G may have multiedges, butno self-loops. A subset F ⊆ E of edges is feasible for FGC if for any unsafe edge e ∈ F ∩ U , the subgraph ( V, F \ { e } ) isconnected. We seek a (feasible) solution F minimizing c ( F ) = P e ∈ F c e . The motivation for studying FGC is two-fold. First,FGC generalizes many well-studied survivable network design problems. Most notably, the minimum-cost 2-edge connectedspanning subgraph (2ECSS) problem corresponds to an instance of FGC where all edges are unsafe. Second, FGC capturesa 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 2 . Theorem 1.
There is a -approximation algorithm for FGC. Adjiashvili et al. [AHM20] also consider the following generalization of FGC. Let k ≥ F ⊆ E ofedges is feasible for the k -FGC problem if for any edge-set X ⊆ F ∩ U with | X | ≤ k , the subgraph ( V, F \ X ) is connected. Thegoal in k -FGC is to find a solution of minimum cost. The usual FGC corresponds to 1-FGC. The following result generalizesTheorem 1. Theorem 2.
There is a ( k + 1) -approximation algorithm for k -FGC. Our proof of Theorem 2 is based on a reduction from k -FGC to the minimum-cost ( k +1)-arborescence problem (see [Sch03],Chapters 52 and 53). We lose a factor of k + 1 in this reduction. Fix some k -FGC solution F and designate a vertex r ∈ V as the root vertex. For an edge e = uv , we call the arc-set { ( u, v ) , ( v, u ) } as a bidirected pair arising from e . The key idea inour proof is that there exists an arc-set T that contains k + 1 arc-disjoint r → v dipaths for each v ∈ V \ { r } while satisfyingthe following two conditions: (i) for an unsafe edge e = uv ∈ F , T uses at most 2 arcs from a bidirected pair arising from e ;and (ii) for a safe edge e = uv ∈ F , T uses at most k + 1 arcs from the disjoint union of k + 1 bidirected pairs arising from e . This argument is formalized in Lemma 7. Complementing this step, we show that any arc-set T (consisting of appropriateorientations of edges in E ) that contains k + 1 arc-disjoint r → v dipaths for every v ∈ V \ { r } can be mapped to a k -FGCsolution. k + 1)-Approximation Algorithm for k -FGC For a subset of vertices S and a subgraph H of G , we use δ H ( S ) to denote the set of edges in H that have one endpoint in S and the other in V \ S . The following characterization of k -FGC solutions is straightforward. Proposition 3. F is feasible for k -FGC ⇐⇒ ∀ ∅ ( S ( V , δ F ( S ) contains a safe edge or k + 1 unsafe edges. For the rest of the paper, we assume that the given instance of k -FGC is feasible: this can be easily checked by computinga (global) minimum-cut in G where we assign a capacity of k + 1 to safe edges and a capacity of 1 to unsafe edges. Let D = ( W, A ) be a digraph and { c ′ a } a ∈ A be nonnegative costs on the arcs. We remark that D may have parallel arcs but it hasno self-loops. Let r ∈ W be a designated root vertex. For a subgraph H of D and a set of vertices S ⊆ W , we use δ in H ( S ) todenote the set of arcs such that the head of the arc is in S and the tail of the arc is in W \ S . ∗ [email protected] . School of Electrical Engineering and Computer Science, University of Ottawa. † { jcheriyan,sharat.ibrahimpur } @uwaterloo.ca . Department of Combinatorics and Optimization, University of Waterloo. ‡ [email protected] . Biocomplexity Institute and Initiative, University of Virginia. efinition 1 ( r -out arborescence) . An r -out arborescence ( W, T ) is a subgraph of D satisfying: (i) the undirected version of T is acyclic; and (ii) for every v ∈ W \ { r } , there is an r → v dipath in ( W, T ) . Definition 2 ( r -out k -arborescence) . For a positive integer k , a subgraph ( W, T ) is an r -out k -arborescence if T can bepartitioned into k arc-disjoint r -out arborescences. Theorem 4 ([Sch03], Chapter 53.8) . Let D = ( W, A ) be a digraph and let k be a positive integer. For r ∈ W , the digraph D contains an r -out k -arborescence if and only if | δ in D ( S ) | ≥ k for every nonempty S ⊆ V \ { r } . Claim 5.
Let ( W, T ) be an r -out k -arborescence for an integer k ≥ . Let u, v ∈ W be any two vertices. Then, the number ofarcs in T that have one endpoint at u and the other endpoint at v (counting multiplicities) is ≤ k .Proof. Since an r -out k -arborescence is a union of k arc-disjoint r -out 1-arborescences, it suffices to prove the result for k = 1.The claim holds for k = 1 because the undirected version of T is acyclic, by definition. Theorem 6 ([Sch03], Theorem 53.10) . In polynomial time, we can obtain an optimal solution to the minimum c ′ -cost r -out k -arborescence problem on D , or conclude that there is no r -out k -arborescence in D . The following lemma shows how a k -FGC solution F can be used to obtain an r -out ( k + 1)-arborescence (in an appropriatedigraph) of cost at most ( k + 1) c ( F ). Lemma 7.
Let F be a k -FGC solution. Consider the digraph D = ( V, A ) where the arc-set A is defined as follows: for eachunsafe edge e ∈ F ∩ U , we include a bidirected pair of arcs arising from e , and for each safe edge e ∈ F ∩ S , we include k + 1 bidirected pairs arising from e . Consider the natural extension of the cost vector c to D where the cost of an arc ( u, v ) ∈ A isequal to the cost of the edge that gives rise to it. Then, there is an r -out ( k +1) -arborescence in D with cost at most ( k +1) c ( F ) .Proof. Let (
V, T ) be a minimum-cost r -out ( k + 1)-arborescence in D . First, we argue that T is well-defined. By Theorem 4, itsuffices to show that for any nonempty S ⊆ V \ { r } , we have | δ in D ( S ) | ≥ k + 1. Fix some nonempty S ⊆ V \ { r } . By feasibilityof F , δ F ( S ) contains a safe edge or k + 1 unsafe edges (see Proposition 3). If δ F ( S ) contains a safe edge e = uv with v ∈ S ,then by our choice of A , δ in D ( S ) contains k + 1 ( u, v )-arcs. Otherwise, δ F ( S ) contains k + 1 unsafe edges, and for each suchunsafe edge uv with v ∈ S , δ in D ( S ) contains the arc ( u, v ). Since | δ in D ( S ) | ≥ k + 1 in both cases, T is well-defined.Finally, we use Claim 5 to show that T satisfies the required cost-bound. For each unsafe edge e ∈ F , T contains at most2 arcs from the bidirected pair arising from e , and for each safe edge e ∈ F , T contains at most k + 1 arcs from the (disjoint)union of k + 1 bidirected pairs arising from e . Thus, c ( T ) ≤ c ( F ∩ U ) + ( k + 1) c ( F ∩ S ) ≤ ( k + 1) c ( F ).Lemma 7 naturally suggests a strategy for Theorem 2 via minimum-cost ( k + 1)-arborescences. Proof of Theorem 2.
Fix some vertex r ∈ V as the root vertex. Consider the digraph D = ( V, A ) obtained from our FGCinstance as follows: for each unsafe edge e ∈ U , we include a bidirected pair arising from e , and for each safe edge e ∈ S , weinclude k + 1 bidirected pairs arising from e . For each edge e ∈ E , let R ( e ) denote the multi-set of all arcs in D that arisefrom e . For any edge e = uv ∈ E and arc ( u, v ) ∈ R ( e ), we define c ( u,v ) := c e . Let ( V, T ) denote a minimum c -cost r -out( k + 1)-arborescence in D . By Lemma 7, c ( T ) ≤ ( k + 1) OPT , where
OPT denotes the optimal value for the given instance of k -FGC.We finish the proof by arguing that T induces a k -FGC solution F with cost at most c ( T ). Let F := { e ∈ E : R ( e ) ∩ T = ∅} .By definition of F and our choice of arc-costs in D , we have c ( F ) ≤ c ( T ). It remains to show that F is feasible for k -FGC.Consider a nonempty set S ⊆ V \ { r } . Since T is an r -out ( k + 1)-arborescence, Theorem 4 gives | δ in T ( S ) | ≥ k + 1. If δ in T ( S )contains a safe arc (i.e., an arc that arises from a safe edge), then that safe edge belongs to δ F ( S ). Otherwise, δ in T ( S ) containssome k + 1 unsafe arcs (that arise from unsafe edges). Since both orientations of an edge cannot appear in δ in D ( S ), we get that | δ F ( S ) ∩ U | ≥ k + 1. Thus, F is a feasible solution for the given instance of k -FGC, and c ( F ) ≤ ( k + 1) OPT . References [AHM20] David Adjiashvili, Felix Hommelsheim, and Moritz M¨uhlenthaler. Flexible graph connectivity. In D. Bienstock andG. Zambelli, editors, , 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.. Algorithms and Combinatorics, Volume 24.Springer-Verlag, Berlin Heidelberg, 2003.
