p-Edge/Vertex-Connected Vertex Cover: Parameterized and Approximation Algorithms
Carl Einarson, Gregory Gutin, Bart M. P. Jansen, Diptapriyo Majumdar, Magnus Wahlstrom
aa r X i v : . [ c s . D S ] S e p p -Edge/Vertex-Connected Vertex Cover:Parameterized and Approximation Algorithms Carl Einarson , Gregory Gutin , Bart M. P. Jansen , Diptapriyo Majumdar , and MagnusWahlstr¨om Royal Holloway, University of London, Egham, United Kingdom, [email protected], { g.gutin|diptapriyo.majumdar|magnus.wahlstrom } @rhul.ac.uk Eindhoven University of Technology, Netherlands, [email protected]
Abstract
We introduce and study two natural generalizations of the Connected VertexCover (VC) problem: the p -Edge-Connected and p -Vertex-Connected VC problem(where p ≥ ⊆ coNP/poly, which ishighly unlikely. We prove however that both problems admit time efficient polyno-mial sized approximate kernelization schemes. We obtain an O (2 O ( pk ) n O (1) )-timealgorithm for the p -Edge-Connected VC and an O (2 O ( k ) n O (1) )-time algorithm forthe p -Vertex-Connected VC. Finally, we describe a 2( p +1)-approximation algorithmfor the p -Edge-Connected VC. The proofs for the new VC problems require moresophisticated arguments than for Connected VC. In particular, for the approxima-tion algorithm we use Gomory-Hu trees and for the approximate kernels a result onsmall-size spanning p -vertex/edge-connected subgraph of a p -vertex/edge-connectedgraph obtained independently by Nishizeki and Poljak (1994) and Nagamochi andIbaraki (1992). For a graph G = ( V, E ), a set C ⊆ V is a vertex cover if for every edge uv ∈ E atleast one of the vertices u, v belongs to C . The well-known classical Vertex Cover problem is the problem of deciding whether a graph G has a vertex cover of size at most k. This problem is NP-complete and thus was studied from parameterized (it is usuallyparameterized by k ) and approximation algorithms view points. Vertex Cover andits generalizations have been important in developing basic and advanced methods andapproaches for parameterized and approximation algorithms. Thus, in particular it wasnamed the
Drosophila of fixed-parameter algorithmics [17].It is well-known that
Vertex Cover is fixed-parameter tractable and admits a kernelwith 2 k vertices [2,8]. However, a connectivity variation of Vertex Cover , ConnectedVertex Cover was shown to have no polynomial kernel unless NP ⊆ coNP/poly [4] (in Connected Vertex Cover , the vertex cover has to be connected). This immediatelyrules out the possibility of admitting a polynomial kernel for several problems that gen-eralize
Connected Vertex Cover . In their pioneering work, Lokshtanov et al. [13]1onsidered several problems that are known not to admit polynomial kernels (unless NP ⊆ coNP/poly) and presented an α -approximate polynomial kernel for these problems forevery fixed α >
1. In particular, they proved that
Connected Vertex Cover admitsan α -approximate polynomial kernel for every fixed α .In this paper, we introduce the following two natural generalization of ConnectedVertex Cover , the p -Edge-Connected Vertex Cover and p -Connected Ver-tex Cover problems as follows. For both problem, p is a fixed positive integer. Input:
An undirected graph G , and an integer k Parameter: k Problem:
Does G have a set of at most k vertices that is a vertex cover and inducesa p -edge-connected subgraph? p -Edge-Connected Vertex Cover( p -Edge-CVC) Input:
An undirected graph G , and an integer k Parameter: k Problem:
Does G have a set of at most k vertices that is a vertex cover and inducesa p -connected subgraph? p -Connected Vertex Cover( p -CVC) To the best of our knowledge, none of these problems were studied in the fields ofparameterized and approximation algorithms before. These problems may be of interestin the following security setting. We have a network whose links (edges) have to beprotected by monitors positioned in its nodes against an attacker targeting the links. Toallow us to overview all the links and inform all other monitors about the attack, themonitors should form a connected vertex cover C . Higher vertex and edge connectivitiesof C provide a higher degree of resiliency in the monitoring system when monitors/linkscan be disabled.Graph theoretic problems with 2-vertex/edge-connectivity constraint has been studiedbefore. Li et al. [11] proved structural results for 2 -Edge-Connected DominatingSet and 2 -Node Connected Dominating Set . Recently, Nutov [19] obtained anapproximation algorithm with expected ratio ˜ O (log n ) for Biconnected Domination .Note that no study has investigated p -vertex/edge-connectivity constraint for p > Our results.
After showing that both p -Edge-CVC and p -CVC do not admit polynomialkernels unless NP ⊆ coNP/poly, we prove that there are (1 + ε )-approximate polynomialkernels for both problems for every 0 < ε < ε )-approximate kernels are as follows. Theorem 1.
For every < ε < and every fixed p ≥ , p -Edge-Connected VertexCover admits a (1 + ε ) -approximate kernel with O ((1 + ε ) k O ( p/ε ) ) vertices. Theorem 2.
For every < ε < and every fixed p ≥ , p -Connected Vertex Cover admits a (1 + ε ) -approximate kernel with O ((1 + ε ) p k O ( p /ε ) )) vertices. V ( G ) = H ⊎ I ⊎ R ,where H is a vertex set which has to be in every p -vertex/edge-connected vertex coverof G , I = { v ∈ V ( G ) \ H : N G ( v ) ⊆ H } and R = V ( G ) \ ( H ∪ I ). The sizes of H and R are bounded by k and 2 k , respectively, but the size of I can be unbounded in k . Weshow how to select a subset L of I of size bounded in k such that if we replace I in G by L and add a pendant vertex to every vertex of H , then we obtain a new graph G ′ withthe number of vertices bounded by a function of k and such that the minimum size of p -vertex/edge-connected vertex cover in G ′ is at most 1 + ε times the minimum size of p -vertex/edge-connected vertex cover in G. Note that a key ingredient in the proves of the two above theorems is the followingresult of Nishizeki and Poljak [18] and Nagamochi and Ibaraki [16] on a small-size span-ning p -vertex/edge-connected subgraph of a p -vertex/edge-connected graph (the problemof finding such a subgraph with minimum number of edges in NP -complete). Theorem 3.
Let G be a p -vertex/edge-connected graph. Then, there exists a polynomialtime algorithm that computes a p -vertex/edge-connected spanning subgraph H of G suchthat H has at most p | V ( G ) | edges. It is not hard to obtain simple FPT algorithms for both p -Edge-Connected Ver-tex Cover and p -Connected Vertex Cover . But, the running time of such al-gorithms is O ∗ (2 O ( k ) ) . For the sake of completeness, we give a short proof for theexistence of such an algorithm for p -Connected Vertex Cover . However, for p -Edge-Connected Vertex Cover we can do better: our next main result is a singleexponential fixed-parameter algorithm for p -Edge-Connected Vertex Cover usingdynamic programming on matroids and some specific characteristics of p -edge-connectedsubgraphs. Theorem 4.
For every fixed p ≥ , p -Edge-Connected Vertex Cover can be solvedin O (2 O ( pk ) n O (1) ) deterministic time and space. Our algorithm for p -Edge-CVC is as follows. First, we enumerate all minimal vertexcovers of G of size at most k . The number of such vertex covers is at most 2 k , and they canbe enumerated in O ∗ (2 k ) time, and space (see [15]). Then, for every minimal vertex cover H of G , we use representative sets to check if it can be extended to a p -edge-connectedvertex cover S ⊇ H of size at most k .Unfortunately, our approach to prove Theorem 4 does not work for p -ConnectedVertex Cover .Our last main result is the following: Theorem 5.
For every fixed p ≥ , p -Edge-Connected Vertex Cover admits a p + 1) -factor approximation algorithm. The proof of this theorem uses the notion of p -blocks which can be obtained fromGomory-Hu tree. Unfortunately, our approach for proving it is not applicable to p -Connected Vertex Cover and the existence of a fixed-factor approximation algo-rithm for p -Connected Vertex Cover is an open problem (recall that p is a fixedpositive integer). The O ∗ notation suppresses the polynomial factors. p -Connected Vertex Cover nor p -Edge-ConnectedVertex Cover has a polynomial kernel and we also prove our first two main results,Theorem 1, and Theorem 2. In Section 4, we show our next main result, Theorem 4. InSection 5, we give our constant factor approximation algorithm. We conclude the paperwith Section 6, where we discuss some open problems on the topic. Related Work on Connected Hitting Set problems.
Ghadikolaei et al. [10] studieda variant of
Connected Vertex Cover problem where a minimal connected vertexcover has to contain a fixed set of vertices. Ramanujan [21] proved that for every α > Connected Feedback Vertex Set has an α -approximate polynomial kernel. Eibenet al. [6] obtained a similar result for the Connected H -Hitting Set problem, where H is a fixed set of graphs. Eiben et al. [7] also obtained an approximate kernelization forthe Connected Dominating Set problem.
Sets
For r ∈ N , we use [ r ] to denote the set { , , . . . , r } . Let U be a set of elements,and F a family of subsets of U. (A family may have multiple copies of the same subset.)Then F is said to be a laminar family , if for every X, Y ∈ F , either X ⊆ Y , or Y ⊆ X ,or X ∩ Y = ∅ . Graph Theory
Let G be a graph. A graph is p -connected if there are at least p vertex-disjoint paths between every pair of vertices. A graph is p -edge-connected if there are atleast p edge-disjoint paths between every pair of vertices. For S ⊆ V ( G ), G [ S ] denotesthe subgraph of G induced by S . Given a connected graph G = ( V, E ), a set of vertices S ⊆ V ( G ) is called a separator of G if G − S is a disconnected graph. For two disjointvertex sets A and B of G, an ( A, B )- cut is a set F of edges such that G − F is disconnectedand no vertex of A belongs to the same connectivity component of G − F which containsthe vertices of B. Parameterized Algorithms and Kernels A parameterized problem Π is a subset ofΣ ∗ × N for some finite alphabet Σ. An instance of a parameterized problem is a pair ( x, k )where k is called the parameter and x is the input . We assume that k is given in unary and without loss of generality k ≤ | x | , | x | is the length of the input. Definition 1 (Fixed-Parameter Tractability) . A parameterized problem Π ⊆ Σ ∗ × N issaid to be fixed-parameter tractable (or FPT ) if there exists an algorithm for solving theproblem Π that on input ( x, k ) , runs in f ( k ) | x | c time where f : N → N is a computablefunction and c is a constant. Definition 2 (Kenelization) . Let Π ⊆ Σ ∗ × N be a parameterized problem. Kernelizationis a polynomial time procedure that replaces the input instance ( x, k ) by a reduced instance ( x ′ , k ′ ) such that (1) | x ′ | + k ′ ≤ g ( k ) for some function g depending only on k , and (2) ( x, k ) ∈ Π if and only if ( x ′ , k ′ ) ∈ Π . It is well-known [3] that a decidable parameterized problem is FPT if and only if ithas a polynomial kernel. 4ernelization can be viewed as a theoretical foundation of computational preprocess-ing and thus we are interested in investigating when a parameterized problem admitsa kernel of small size. We are especially interested in polynomial kernels for which thebound g ( k ) is a polynomial and thus in classifying which parameterized problems admitpolynomial kernels or not. Over the last decade, the area of kernelization has developed ofa large number of approaches to design polynomial kernels as well as a number of tools forproving lower bounds based on assumptions from complexity theory. The lower boundsrule out the existence of polynomial kernels for many parameterized problems. We re-fer the reader to the textbooks [3, 5] for an introduction to the field of parameterizedalgorithms and to [8] for a recent introduction to kernelization. This section is partitioned into five parts. In Section 3.1, we prove that both p - ConnectedVertex Cover and p -Edge-Connected Vertex Cover do not admit a plynomialkernel unless NP ⊆ coNP/poly. In Section 3.2, we give terminology and notation onapproximate kernels. In Section 3.3, we provide material which is very similar for both p -vertex-connectivity and p -edge-connectivity and used in the next two sections. In Sec-tions 3.4 and 3.5, we obtain an approximate kernel for p -Edge-Connected VertexCover and p - Connected Vertex Cover , respectively. In these sections, we assumethat p ≥ ε < . We will first show that p - Connected Vertex Cover admits no polynomial kernelunless NP ⊆ coNP/poly using the fact that Connected Vertex Cover admits nopolynomial kernel unless NP ⊆ coNP/poly [4]. Theorem 6. p -Connected Vertex Cover admits no polynomial kernel unless NP ⊆ coNP/poly.Proof. Let (
G, k ) be an instance of
Connected Vertex Cover . We construct aninstance of p - Connected Vertex Cover as follows. We add p − v , . . . , v p − to ( G, k ) and edges { uv i | i ∈ [ p − , u ∈ V ( G ) } obtaining a new graph G ′ . Observe that G ′ can be computed in polynomial time. Furthermore, ( G, k ) is a yes-instance of
Connected Vertex Cover if and only if ( G ′ , k + p −
1) is a yes-instanceof p - Connected Vertex Cover . Hence, p - Connected Vertex Cover does notadmit a polynomial kernel unless NP ⊆ coNP/poly.The reduction in the next theorem is from Red Blue Dominating Set . In theproblem, given a bipartite graph G with partite sets B and R , we are to decide whetherthere is B ′ ⊆ B such that | B ′ | ≤ k and N G ( B ′ ) = R. Dom et al. [4] proved that
RedBlue Dominating Set parameterized by k + | R | does not admit a polynomial kernelunless NP ⊆ coNP/poly. Theorem 7. p -Edge-Connected Vertex Cover has no polynomial kernel unlessNP ⊆ coNP/poly. roof. Let ( G = ( R ⊎ B, E ) , k + | R | ) be an instance of Red Blue Dominating Set andlet R = { r , r , . . . , r t } . Without loss of generality, we may assume that for every v ∈ B ,there exists u ∈ R such that uv ∈ E ( G ) (otherwise we can just delete v ). Similarly, wemay assume that for every v ∈ R , there exists u ∈ B such that uv ∈ E ( G ) (otherwise,there is no feasible solution). We will also assume that t ≥ p , k ≥ p and p ≥
2. Constructa new graph H from G as follows. • Add a complete graph K p with vertex set A = { a , . . . , a p } such that V ( G ) ∩ A = ∅ ,and edges a i b for every i ∈ [ p ] and b ∈ B. • Replace every vertex r j of R by a complete graph K p with vertices { r j , . . . , r jp } suchthat if br j ∈ E ( G ) then br ji ∈ E ( H ) for every i ∈ [ p ] . Thus, R is replaced by ˆ R ofsize pt. • Attach a pendant vertex to every vertex in ˆ R ∪ A . • Set k ′ = k + p ( t + 1). Note that since p is a fixed constant, we have that k ′ is O ( k + t ).To complete the proof, it suffices to prove that ( G, k + t ) is a yes-instance of RedBlue Dominating Set if and only if (
H, k ′ ) is a yes-instance of p -Edge-ConnectedVertex Cover .( ⇐ ) Let S be a p -edge-connected vertex cover of H such that | S | ≤ k ′ . Observe thatˆ R ∪ A ⊆ S since all of the vertices in ˆ R ∪ A have pendant neighbors. Since every vertexin ˆ R have just p − R and G [ S ] is p -edge-connected, S must contain at leastone neighbor in B of every vertex of ˆ R. Since | ˆ R ∪ A | = p ( t + 1), k ′ = | ˆ R ∪ A | + k and sothere must be a subset B ′ of B of size at most k such that N G ( B ′ ) = R. Hence, (
G, k ) isa yes-instance of
Red Blue Dominating Set .( ⇒ ) Let B ∗ ⊆ B with | B ∗ | = k such that N G ( B ∗ ) = R . Let B ∗ = { b , . . . , b k } , recallthat k ≥ p. We claim that A ∪ B ∗ ∪ ˆ R is a p -edge-connected vertex cover of H . Clearly,it is a vertex cover, so it remains to prove that H [ A ∪ B ∗ ∪ ˆ R ] is p -edge-connected. Let u, v be vertices of A ∪ B ∗ ∪ ˆ R . It suffices to prove that there are p edge-disjoint pathsbetween u and v for every choice of u and v . Subject to symmetry, it suffices to considersix cases: u, v ∈ A ; u, v ∈ B ∗ ; u, v ∈ ˆ R ; u ∈ A, v ∈ B ∗ ; u ∈ A, v ∈ ˆ R ; u ∈ B ∗ , v ∈ ˆ R. Below we will consider these cases one by one. u, v ∈ A . Without loss of generality, let u = a and v = a . Then a a , a a q a , 3 ≤ q ≤ p and a b a are p edge-disjoint paths between u and v.u, v ∈ B ∗ . Without loss of generality, let u = b and v = b . Then b a i b , i ∈ [ p ] are p edge-disjoint paths between u and v.u, v ∈ ˆ R. We will first consider the subcase when u, v are from the same clique in ˆ R .Without loss of generality, let u = r , v = r and b r ∈ E ( G ) . Then r r , r r q r , ≤ q ≤ p and r b r are p edge-disjoint paths between u and v. u, v are from different cliques in ˆ R . Without loss ofgenerality, let u = r , v = r . For every i ∈ [ p ] , let P i = b j if r and r are adjacentto a common vertex b j in G and P i = b i ′ a i b i ′′ otherwise, where b i ′ r , b i ′′ r ∈ E ( G ) . Then r P r , r r i P i r i r , 2 ≤ i ≤ p are p edge-disjoint paths between u and v.u ∈ A, v ∈ B ∗ . There are p edge-disjoint paths between u and v since A ∪ { v } forms aclique with p + 1 vertices. u ∈ A, v ∈ ˆ R . Without loss of generality, let u = a , v = r and b r ′ ∈ E ( G ) . Then a b r , a a i b r i r (2 ≤ i ≤ p ) are p edge-disjoint paths between u and v.u ∈ B ∗ , v ∈ ˆ R . Without loss of generality, let u = b and v = r . We will first considerthe subcase when b r ′ ∈ E ( G ) . Then Q = b r , Q i = b r i r (2 ≤ i ≤ p ) are p edge-disjoint paths between u and v. Now consider the subcase when b j r ′ ∈ E ( G )for some j > . Then b a i b j Q ′ i ( i ∈ [ p ]) are p edge-disjoint paths between u and v, where Q ′ i = Q i − b and Q i is defined in the previous subcase.This completes the proof. Informally, an α -approximate kernelization is a polynomial time algorithm that, givenan instance ( I, k ) of a parameterized problem, outputs an instance ( I ′ , k ′ ) (called an α -approximate kernel ) such that | I ′ | + k ′ ≤ g ( k ) for some computable function g , andany c -approximate solution to the instance ( I ′ , k ′ ) can be turned into a ( cα )-factor ap-proximate solution of ( I, k ) in polynomial time. α -approximate kernelization is a newarea of parameterized complexity and it is important to understand which parameterizedproblems admit α -approximate kernels and which are not. Now we will mainly followLokshtanov et al. [13] to formally introduce approximate kernelization. (Several otherrecent papers have dealt with approximate kernelization, e.g., [6, 7, 13, 21].) Definition 3. A parameterized optimization (maximization or minimization) problem isa computable function Π : Σ ∗ × N × Σ ∗ → R ∪ {±∞} . The instances of a parameterized optimization problem Π are pairs ( x, k ) ∈ Σ ∗ × N , and a solution to ( x, k ) is simply a string s ∈ Σ ∗ such that | s | ≤ | x | + k . The value of the solution s is Π( x, k, s ). The problems we deal with in this paper are allminimization problems. Therefore, we state the definitions only in terms of minimizationproblems when the definition of maximization problems are analogous. Consider p -Edge-Connected Vertex Cover parameterized by solution size. This is a minimizationproblem where the optimization function pECV C : Σ ∗ × N × Σ ∗ → R ∪ {∞} is definedas follows. pECV C ( G, k, S ) = (cid:26) ∞ if S is not a p -edge-connected vertex cover of G, min {| S | , k + 1 } otherwise.For p -Connected Vertex Cover , pCV C ( G, k, S ) can be defined similarly.7 efinition 4.
For a parameterized minimization problem Π , the optimum value of aninstance ( x, k ) ∈ Σ ∗ × N is OPT Π ( x, k ) = min s ∈ Σ ∗ , | s |≤| x | + k Π( x, k, s ) . Naturally for the case of p -Edge-Connected Vertex Cover , we denoteOPT( G, k ) = min S ⊆ V ( G ) pECV C ( G, k, S ) . Similarly, for p -Connected Vertex Cover . Definition 5.
Let α ≥ be a real number and Π a parameterized minimization prob-lem. An α -approximate polynomial time preprocessing algorithm A for Π is a pair ofpolynomial time algorithms as follows. The first one is called the reduction algorithm that computes a map R A : Σ ∗ × N → Σ ∗ × N . Given an input instance ( x, k ) of Π , thereduction algorithm outputs another instance ( x ′ , k ′ ) = R A ( x, k ) .The second algorithm is called the solution lifting algorithm . This algorithm takes aninput instance ( x, k ) , the output instance ( x ′ , k ′ ) , and a solution s ′ to the output instance ( x ′ , k ′ ) . The solution lifting algorithm works in polynomial time in | x | , k, | x ′ | , k ′ , and s ′ ,and outputs a solution s to ( x, k ) such that the following holds. Π( x, k, s )OPT( x, k ) ≤ α · Π( x ′ , k ′ , s ′ )OPT( x ′ , k ′ ) The size of the reduction algorithm A is a function size A : N → N defined as size A ( k ) =sup {| x ′ | + k ′ : ( x ′ , k ′ ) = R A ( x, k ) , x ∈ Σ ∗ } . Definition 6.
Let α ≥ be a real number. An α -approximate kernelization (and ( x ′ , k ′ ) is an α -approximate kernel ) for a parameterized optimization problem Π , is an α -approximate polynomial time preprocessing algorithm A for Π such that size A is upperbounded by a computable function g : N → N . We say that ( x ′ , k ′ ) is an α -approximatepolynomial kernel if g is a polynomial function. Definition 7. A polynomial size approximate kernelization scheme (PSAKS) for a pa-rameterized optimization problem Π is a family of α -approximate polynomial kernelizationalgorithms, one such algorithm for every α > . Definition 8.
A PSAKS is said to be time efficient if both the reduction algorithm andthe solution lifting algorithm run in time f ( α ) | x | c for some function f and a constant c independent of | x | , k, and α . We will use the following useful lemma whose vertex-connectivity part is proved in [23](Lemma 4.2.2). We were unable to find a proof of the edge-connectivity part of the lemmain the literature and thus provide a short proof here.
Lemma 1. If G is a p -vertex-connected ( p -edge-connected, respectively) graph and G ′ isobtained from G by adding a new vertex y adjacent with at least p vertices of G , then G ′ is p -vertex-connected ( p -edge-connected, respectively). roof. Let C be a cut in G ′ . If C = ( y, V ( G )) then by construction | C | ≥ p . Otherwise, C − y is a cut in G and | C | ≥ p. The next lemma will help us to identify whether a graph G = ( V, E ) has a p -vertex/edge-connected vertex cover or not. Lemma 2.
Let G = ( V, E ) be a graph, L the set of vertices of G with degree at most p − and S = V ( G − L ) . Then G has a p -vertex/edge-connected vertex cover if and onlyif S is a p -vertex/edge-connected vertex cover of G .Proof. The backward direction ( ⇐ ) of the proof is trivial.We will prove the forward direction ( ⇒ ). Let S ∗ be a p -vertex/edge-connected vertexcover of G . If S ∗ = V ( G − L ), then we are done. Hence, we may assume that S ∗ = V ( G − L ) . Observe that S ∗ ∩ L = ∅ as otherwise for every u ∈ S ∗ ∩ L deleting the vertices adjacentto u (the edges incident to u ) will make G [ S ∗ ] disconnected. Let A = V ( G ) \ ( S ∗ ∪ L ) andnote that A is an independent set as S ∗ is a vertex cover. Hence, by Lemma 1 adding A to S ∗ will result in a p -vertex/edge-connected vertex cover of G . So, S = S ∗ ∪ A is a p -vertex/edge-connected vertex cover of G , which completes the proof.We now start describing a (1 + ε )-approximate kernel for both p -Edge-ConnectedVertex Cover and p - Connected Vertex Cover . Let (
G, k ) be an input instance.Using Lemma 2, we can check in polynomial time if a graph G has a p -vertex/edge-connected vertex cover. If ( G, k ) does not have a p -vertex/edge-connected vertex cover,then we just return (2 K ,
1) as the output instance. Otherwise, (
G, k ) has a p -vertex/edge-connected vertex cover. Observe that any vertex with degree at least k +1 must be presentin any feasible solution of size at most k . Similarly, if v is a vertex with degree at most p −
1, then N G ( v ) must be part of any feasible solution. We call these vertices necessaryvertices and initialize H := ∅ . Based on this discussion, we apply the following reductionrule which is correct. Reduction Rule 1.
Remove all isolated vertices from G . Add every vertex of degree atleast k + 1 to H . For every vertex v with degree at most p − , add N ( v ) to H . Afterwards we partition V ( G ) = H ⊎ I ⊎ R, where • I = { u ∈ V ( G ) | N G ( u ) ⊆ H } , and • R = V ( G ) \ ( H ∪ I ).Observe that any vertex in R has at least one neighbor in R , and possibly someneighbor in H . By definition, I is an independent set. Also, any vertex v ∈ R has degreeat most k in G . If | H | > k or G [ R ] has more than k edges, then ( G, k ) cannot have anyfeasible solution with at most k vertices. In this case, we return ( K p ,
1) as the outputinstance. Otherwise, | H | ≤ k , and G [ R ] has at most k edges. Hence, | R | ≤ k . p -Edge-Connected Vertex Cover Set d = ⌈ p/ε ⌉ and first run Algorithm 1 on ( G, k ) to mark some vertices in I . Note thatfor every subset of H of size at most 2 d + 2 p + 2, we mark at most ⌈ (1 + ε ) k ⌉ verticesof I . For the rest of this section, we will be using ‘a vertex v ∈ L ’ or ‘a marked vertex v ’to mean the same thing. 9 lgorithm 1 (Marking vertices in I ) input : ( G, k ) and k ∈ N output : ( G ′ , k ′ ) , where G ′ satisfies some additional properties L ← ∅ ; I p ← { u ∈ I | deg G ( u ) ≥ p } ; for every set S ∈ (cid:0) H ≤ d + p +1) (cid:1) doif | ( T u ∈ S N G ( u )) ∩ I p | ≥ (1 + ε ) k then Mark ⌈ (1 + ε ) k ⌉ vertices in ( T u ∈ S N G ( u )) ∩ I p ; Add these marked vertices to L ; endelse Mark all vertices in ( T u ∈ S N G ( u )) ∩ I p ; L ← L ∪ (( T u ∈ S N G ( u )) ∩ I p ); endendfor every vertex x ∈ H do add a new pendant vertex x ′ to I such that xx ′ is an edge ; L ← L ∪ { x ′ } ; Mark x ′ ; end k ′ ← k (1 + ε ); G ′ ← G [ H ∪ R ∪ L ]; Output ( G ′ , k ′ ) as the reduced instance ;Recall that d = ⌈ p/ε ⌉ and for every subset of H of size at most 2 d + 2 p + 2, we markat most ⌈ (1 + ε ) k ⌉ vertices of I . Therefore, | L | ≤ ⌈ (1 + ε ) k ⌉ k ⌈ p/ε ⌉ +2 p +2 . Also, G [ R ]has k edges. Therefore, | R | ≤ k . Hence, | V ( G ′ ) | ≤ ⌈ (1 + ε ) k ⌉ k ⌈ p/ε ⌉ +2 p +2 + 2 k + 2 k .Let us recall the following theorem of Nishizeki and Poljak [18]. Theorem 3.
Let G be a p -vertex/edge-connected graph. Then, there exists a polynomialtime algorithm that computes a p -vertex/edge-connected spanning subgraph H of G suchthat H has at most p | V ( G ) | edges. We apply Theorem 3 to prove the following lemma, which we will use to prove ourmain result of this subsection.
Lemma 3.
Let ( G, k ) be the input instance of p -Edge-Connected Vertex Cover and ( G ′ , k ′ ) be the output instance after applying Algorithm 1. Then, the following state-ments hold true.1. If OPT(
G, k ) ≤ k , then OPT( G ′ , k ′ ) ≤ (1 + ε )OPT( G, k ) .2. If A is a p -edge-connected vertex cover of ( G ′ , k ′ ) , then A is also a p -edge-connectedvertex cover of ( G, k ) .Proof. We prove the statements in the given order.10. Let X be an optimal p -edge-connected vertex cover of ( G, k ) and | X | ≤ k . Since | X | ≤ k , H ⊆ S . If X ⊆ V ( G ′ ), then X is also a p -edge-connected vertex cover of G ′ . Hence, OPT( G ′ , k ′ ) ≤ | X | = OPT( G, k ). So, we may assume that there existsa vertex u ∈ X such that u / ∈ V ( G ′ ). Recall that by construction, L is the set of allmarked vertices from I and the additional marked pendant vertices. This ensuresthat H is contained in any feasible solution of ( G ′ , k ′ ).By Theorem 3, G [ X ] has a p -edge-connected spanning subgraph F having at most p | X | edges. We initialize X ′ := X and replace one by one all vertices of X which arenot in G ′ by vertices of G ′ forming a new set X ′ , which will be proved to be p -edge-connected and of size at most (1 + ε ) | X | . (Note that all replaced vertices form anindependent set and the order of replacing them is immaterial.) Let u ∈ X \ V ( G ′ ), B = N F ( u ) and consider the step when u is replaced by vertices in G ′ . We will usethe fact that | X ′ | ≤ (1 + ε ) | X | , the bound which will be shown in the last part ofthe proof. Let us consider two cases. Case | B | ≤ d . Since | X ′ | ≤ (1 + ε ) k , and we have marked ⌈ (1 + ε ) k ⌉ verticeswhose neighborhood contains B, (in other words, u is unmarked, u ∈ ∩ w ∈ B N G ( w ) , | ∩ w ∈ B N G ( w ) ∩ L | = ⌈ (1 + ε ) k ⌉ ) there is a marked vertex v / ∈ X ′ such that B ⊆ N ( v ) . (or in other words, v ∈ ∩ w ∈ B N G ( w )). We set X ′ := ( X ′ \ { u } ) ∪ { v } . Inthis case, we replace one unmarked vertex by one marked vertex in X ′ . Case | B | > d . We partition B into r sets B , B , . . . , B r such that | B | = · · · = | B r − | = d and d ≤ | B r | < d. Recall that d = ⌈ p/ε ⌉ and ε < . Hence d ≥ p + 1.For every i ∈ [ r ], let U i = B i ∪ A i, ∪ A i, where A i, is an arbitrary set of p +1 verticesfrom B i − , and A i, is an arbitrary set of p + 1 vertices from B i +1 . Here i is takenmodulo r , i.e., r + 1 = 1 and 0 = r. Recall that for all i ∈ [ r ], U i ⊆ N F ( u ) ⊆ N G ( u ).Since u is unmarked, | ∩ w ∈ U i N G ( w ) ∩ L | = ⌈ (1 + ε ) k ⌉ . However, X ′ covers at most ⌈ (1 + ε ) k ⌉ marked vertices from r S i =1 ( ∩ w ∈ U i N G ( w ) ∩ I ). Hence, we have r distinctmarked vertices v , . . . , v r / ∈ X ′ such that for all i ∈ [ r ], v i is marked, U i ⊆ N G ( v i ) (orequivalently v i ∈ ∩ w ∈ U i N G ( w )), but v i / ∈ X ′ . We replace u by v , . . . , v r by setting X ′ := ( X ′ \ { u } ) ∪ { v , . . . , v r } . So, in this case, we replace one unmarked vertex by r distinct marked vertices. Observe that r ≤ ⌈| B | /d ⌉ = ⌈ deg F ( u ) /d ⌉ . Also, observethat by construction the sets of marked vertices replacing two unmarked verticesdo not intersect.We claim that G ′ [ X ′ ] is a p -edge-connected subgraph. We prove this by contradic-tion. In particular we prove that “If G ′ [ X ′ ] is not a p -edge-connected subgraph,then F is not a p -edge-connected subgraph.”Let us consider a minimum cut ( X ′ , X ′ ) of G ′ [ X ′ ] with at most p − X , X ) of F . We initialize X := X ′ , and X := X ′ . Weprocess all the unmarked vertices from X \ X ′ one by one as follows. Consideran arbitrary unmarked vertex u ∈ X \ X ′ . Either u is replaced by exactly onemarked vertex, or a u is replaced by r marked vertices v , . . . , v r ∈ I (but notboth). Suppose that u is replaced by exactly one marked vertex u ∗ ∈ X ′ . Then, itmust have been because | N F ( u ) | ≤ d and N F ( u ) ⊆ N G ′ ( u ∗ ). If u ∗ ∈ X ′ , we set11 := X ∪ { u } \ { u ∗ } . Otherwise, u ∗ ∈ X ′ . Then, we set X := X ∪ { u } \ { u ∗ } .Observe that N F ( u ) ⊆ N G ′ ( u ∗ ). Therefore, this step does not increase the number ofedges of the form x x with x ∈ X , x ∈ X . Otherwise, u is replaced by r distinctmarked vertices v , . . . , v r . First, observe by construction that for all i < j , there are( p + 1) edge-disjoint paths from v i to v j passing through v i +1 , . . . , v j − . If j = i + 1,then it is trivial. Because, U i ⊆ N G ′ ( v i ) , U j ⊆ N G ′ ( v j ). Since, | U i ∩ U j | ≥ p + 1, v i and v j have at least p + 1 common neighbors. Similarly, v and v r has at least p + 1common neighbors. Otherwise, j ≥ i + 2 consider the sets A i, , . . . , A j − , . Thesesets are all pairwise disjoint. Observe that for all i ∈ [ r ], we have A i, ∈ (cid:0) B i +1 p +1 (cid:1) and A i, ⊆ N ( v i ) ∩ N ( v i +1 ). Similarly, A j − , ⊆ N ( v j − ) ∩ N ( v j ). Therefore, there are atleast p edge-disjoint paths from v i to v j . Since ( X ′ , X ′ ) is a minimum cut of G ′ [ X ′ ],either { v , . . . , v r } ⊆ X ′ or { v , . . . , v r } ⊆ X ′ (but not both). If { v , . . . , v r } ⊆ X ′ ,then we set X := ( X ∪ { u } ) \ { v , . . . , v r } . Otherwise { v , . . . , v r } ⊆ X ′ . Then,we set X := ( X ∪ { u } ) \ { v , . . . , v r } . Observe that N F ( u ) ⊆ N G ′ ( { v , . . . , v r } ).Therefore, this step also does not increase the number of edges of the form x x such that x ∈ X , x ∈ X .Note that the above process removes all vertices of X ′ \ X and adds all vertices of X \ X ′ in sequence to compute a cut ( X , X ). Moreover, by construction, we have | E ( X , X ) | ≤ | E ( X ′ , X ′ ) | ≤ p −
1. Then, F is also not p -edge-connected graphthat is a contradiction. Therefore, G ′ [ X ′ ] induces a p -edge-connected subgraph.We have argued that G ′ [ X ′ ] is also a p -edge-connected subgraph. Recall that ineach the replacement steps, we replace an unmarked vertex u ∈ I by a collectionof marked vertices from I . But | X | ≤ k , and therefore, H ⊆ X . So, X ′ is also avertex cover of ( G ′ , k ′ ).Now we will complete the proof by showing that | X ′ | ≤ (1 + ε ) | X | . Recall that F has at most p | X | edges. At every step, our procedure replaces an unmarked vertex u ∈ X ∩ I by at most r marked vertices such that 1 ≤ r ≤ ⌈ deg F ( u ) /d ⌉ . Also recallthat d = ⌈ p/ε ⌉ . Hence, we have | X ′ | ≤ | X ∩ X ′ | + | X \ X ′ | + X u ∈ X \ X ′ (cid:18)(cid:24) deg F ( u ) d (cid:25) − (cid:19) ≤ | X | + X u ∈ X \ X ′ deg F ( u ) d = | X | + P u ∈ X \ X ′ deg F ( u ) d = | X | + 2 | E ( F ) | d ≤ | X | + 2 p | X | d ≤ (1+2 p/d ) | X | ≤ (1+ ε ) | X | . Therefore, OPT( G ′ , k ′ ) ≤ | X ′ | ≤ (1 + ε ) | X | = (1 + ε )OPT( G, k ).2. Let A be a p -edge-connected vertex cover of ( G ′ , k ′ ). By construction, for every x ∈ H , there is x ′ ∈ I such that x ′ is a pendant neighbor of x in G ′ . Since p ≥ A cannot contain any pendant vertex of G ′ . Hence, H ⊆ A . Consider an arbitraryvertex u ∈ V ( G ) \ V ( G ′ ). By construction, u ∈ I . Since H ⊆ A , and N G ( u ) ⊆ H ,we have that N G ( u ) ⊆ A . Hence, A is also a vertex cover of ( G, k ). Since G ′ [ A ]is p -edge-connected, A ⊆ V ( G ) and G [ A ] = G ′ [ A ], we have that G [ A ] is also a p -edge-connected graph. Therefore, A is a p -edge-connected vertex cover of ( G, k )as well.This completes the proof of the lemma. 12e are now ready to prove the main result of this section, Theorem 1 (we restate ithere).
Theorem 1.
For every < ε < and every fixed p ≥ , p -Edge-Connected VertexCover admits a (1 + ε ) -approximate kernel with O ((1 + ε ) k O ( p/ε ) ) vertices.Proof. The approximate kernelization algorithm has two parts. First part is a reductionalgorithm, and the second part is a solution lifting algorithm. Let (
G, k ) be an inputinstance of p -Edge-Connected Vertex Cover . • Reduction algorithm:
First we invoke Lemma 2 to check if G has a p -edge-connected vertex cover or not. If G has no p -edge-connected vertex cover, then wereturn (2 K ,
1) as the output instance. Otherwise, we first apply Reduction Rule 1.Reduction Rule 1 obtains a new instance ( G , k ) by deleting isolated vertices from G .Then it computes H and V ( G ) = H ⊎ I ⊎ R . Observe that G [ R ] = G [ R ]. If | H | > k ,or G [ R ] (equivalently G [ R ]) has more than k edges, then OPT( G, k ) = k + 1 andwe return ( K p ,
1) as the output instance. Otherwise, | H | ≤ k , and G [ R ] has atmost k edges. Since Reduction Rule 1 removes only isolated vertices, OPT( G, k ) =OPT( G , k ) and for any S ⊆ V ( G ), pECV C ( G, k, S ) = pECV C ( G , k, S ).In this situation, if | V ( G ) | ≤ ⌈ (1 + ε ) k ⌉ k ⌈ p/ε ⌉ +2 p +2 + 2 k + 2 k , we output ( G , k )itself. Otherwise, | V ( G ) | > ⌈ (1 + ε ) k ⌉ k ⌈ p/ε ⌉ +2 p +2 + 2 k + 2 k . Then, we applyAlgoirhm 1 on ( G , k ) to construct ( G ′ , k ′ ) and return ( G ′ , k ′ ) as an output instance. • Solution lifting algorithm:
Let S ∗ be the given solution to ( G ′ , k ′ ). If S ∗ isnot a p -edge-connected vertex cover of ( G ′ , k ′ ), then we return ∅ as the solution to( G, k ). Otherwise, given S ∗ as a feasible p -edge-connected vertex cover of ( G ′ , k ′ ),we return S ∗ as a solution for the instance ( G, k ). Running time:
Observe that Reduction Rule 1 can be implemented in O ( n O (1) ) time.The marking algorithm (Algorithm 1) is implemented only when n > (1 + ε ) k ⌈ p/ε ⌉ +2 p +2 +2 k +2 k . The marking algorithm can be implemented in O ( k ⌈ p/ε ⌉ +2 p +4 n O (1) ) time. Since n > ⌈ (1 + ε ) k ⌉ k ⌈ p/ε ⌉ +2 p +2 + 2 k + 2 k , the marking algorithm can be implemented in O ( n O (1) ) time. Therefore, the reduction algorithm can be implemented in O ( n O (1) ) time.By construction, the solution lifting algorithm can be implemented in O ( n O (1) ) time.Therefore, our approximate kernel is time-efficient. Analysis:
We argue now that the reduction algorithm and the solution lifting al-gorithm together provide a (1 + ε )-approximate kernel. If S ∗ is not a p -edge-connectedvertex cover of ( G ′ , k ′ ), then pECV C ( G ′ , k ′ , S ∗ ) = ∞ and pECV C ( G, k, ∅ ) ≤ ∞ . Hence, pECV C ( G,k,S ∗ ) OPT ( G,k ) ≤ pECV C ( G ′ ,k ′ ,S ∗ ) OPT ( G ′ ,k ′ ) . Otherwise, S ∗ is a p -edge-connected vertex cover in( G ′ , k ′ ). By Lemma 3, S ∗ is also a p -edge-connected vertex cover of ( G , k ). Reduc-tion Rule 1 removes only isolated vertices from ( G, k ), therefore, any feasible solution of( G , k ) is also a feasible solution of ( G, k ). Therefore, S ∗ is a p -edge-connected vertexcover of ( G, k ). By Lemma 3, if OPT( G , k ) ≤ k , then OPT( G ′ , k ′ ) ≤ (1 + ε )OPT( G , k ).But, as we have argued before, OPT( G , k ) = OPT( G, k ). Hence, if OPT(
G, k ) ≤ k , thenOPT( G ′ , k ′ ) ≤ (1 + ε )OPT( G, k ). We will use this fact to prove our theorem’s statement.We explain by case analysis that if we are given a c -approximate feasible solution to( G ′ , k ′ ), then we would be able to give a c (1 + ε )-approximate feasible solution to ( G, k ).13
Case 1: | S ∗ | > k ′ ≥ k . Then pECV C ( G ′ , k ′ , S ∗ ) = k ′ +1. Hence, pECV C ( G, k, S ∗ ) = k + 1 ≤ k ′ + 1 = pECV C ( G ′ , k ′ , S ∗ ). If OPT( G, k ) ≤ k then, as we have argued,OPT( G ′ , k ′ ) ≤ (1 + ε )OPT( G, k ). Hence, we have the following: pECV C ( G, k, S ∗ )OPT( G, k ) = k + 1OPT( G, k ) ≤ (1 + ε ) k ′ + 1OPT( G ′ , k ′ )= (1 + ε ) pECV C ( G ′ , k ′ , S ∗ )OPT( G ′ , k ′ )Otherwise, OPT( G, k ) = k + 1. Then, pECV C ( G, k, S ∗ )OPT( G, k ) = k + 1 k + 1 = 1 ≤ (1 + ε ) pECV C ( G ′ , k ′ , S ∗ )OPT( G ′ , k ′ ) • Case 2: k +1 ≤ | S ∗ | ≤ k ′ . Then pECV C ( G ′ , k ′ , S ∗ ) = | S ∗ | , and pECV C ( G, k, S ∗ ) =min { k + 1 , | S ∗ |} = k + 1. If OPT( G, k ) ≤ k , then as we have argued before,OPT( G ′ , k ′ ) ≤ (1 + ε )OPT( G, k ). Then, we have the following: pECV C ( G, k, S ∗ )OPT( G, k ) = k + 1OPT( G, k ) ≤ (1 + ε ) k + 1OPT( G ′ , k ′ ) ≤ (1 + ε ) | S ∗ | OPT( G ′ , k ′ ) = (1 + ε ) pECV C ( G ′ , k ′ , S ∗ )OPT( G ′ , k ′ )Otherwise OPT( G, k ) = k + 1. Then, pECV C ( G, k, S ∗ )OPT( G, k ) = k + 1 k + 1 = 1 ≤ (1 + ε ) pECV C ( G ′ , k ′ , S ∗ )OPT( G ′ , k ′ ) • Case 3: | S ∗ | ≤ k. Then pECV C ( G, k, S ∗ ) = | S ∗ | = pECV C ( G ′ , k ′ , S ∗ ). In thiscase, OPT( G, k ) ≤ k . So, we get the following by using the fact that OPT( G ′ , k ′ ) ≤ (1 + ε )OPT( G, k ). pECV C ( G, k, S ∗ )OPT( G, k ) = pECV C ( G ′ , k ′ , S ∗ )OPT( G, k ) ≤ (1 + ε ) pECV C ( G ′ , k ′ , S ∗ )OPT( G ′ , k ′ )By construction, ( G ′ , k ′ ) has O ((1 + ε ) k ⌈ p/ε ⌉ +2 p +4 ) vertices. Since, p is fixed, our resultimplies a time efficient PSAKS with the claimed bound. This completes the proof. p -Connected Vertex Cover We set d = ⌈ p /ε ⌉ and run Algorithm 2 to mark some vertices in I . Note that for everysubset of H of size at most 2 d + 2 p + 2, we mark at most ⌈ (1 + ε )( p + 1) k ⌉ vertices of I .Similar to Section 3.4, we use ‘a vertex v ∈ L ’ or ‘a marked vertex v ’ to mean the samething.Since, d = ⌈ p /ε ⌉ , we have | L | ≤ ⌈ (1 + ε )( p + 1) k ⌉ k ⌈ p /ε ⌉ +2 p +2 . Also, G [ R ] has k edges, and hence | R | ≤ k . Hence, | V ( G ′ ) | ≤ ⌈ (1 + ε )( p + 1) k ⌉ k ⌈ p /ε ⌉ +2 p +2 +2 k +2 k .We apply Theorem 3 to prove the following lemma. We will use this following lemmato prove the main result of this section. 14 lgorithm 2 (Marking vertices in I ) input : G = ( V, E ) and k ∈ N output : ( G ′ , k ′ ) , where G ′ satisfies some additional properties L ← ∅ ; I p ← { u ∈ I | deg G ( u ) ≥ p } ; for every set S ∈ (cid:0) H ≤ d + p +1) (cid:1) doif | ( T u ∈ S N G ( u )) ∩ I p | ≥ (1 + ε )( p + 1) k then Mark ⌈ (1 + ε )( p + 1) k ⌉ vertices in ( T u ∈ S N G ( u )) ∩ I p ; Add these marked vertices to L ; endelse Mark all vertices in ( T u ∈ S N G ( u )) ∩ I p ; L ← L ∪ (( T u ∈ S N G ( u )) ∩ I p ); endendfor every vertex x ∈ H do add a new pendant vertex x ′ to I such that xx ′ is an edge ; L ← L ∪ { x ′ } ; Mark x ′ ; end k ′ ← k (1 + ε ); G ′ ← G [ H ∪ R ∪ L ]; Output ( G ′ , k ′ ) as the reduced instance ; Lemma 4.
Let ( G, k ) be the input instance of p - Connected Vertex Cover , and let ( G ′ , k ′ ) be the output instance after applying Algorithm 2. Then, the following statementshold true.1. If A is a p -connected vertex cover of ( G ′ , k ′ ) , then A is also a p -connected vertexcover of ( G, k ) .2. If OPT(
G, k ) ≤ k , then OPT( G ′ , k ′ ) ≤ (1 + ε )OPT( G, k ) .Proof. We prove the statements in the given order.1. We can prove this statement using arguments similar to the proof of Lemma 3(2).2. Let X be an optimal p -connected vertex cover of ( G, k ) such that | X | ≤ k . Since, | X | ≤ k , H ⊆ S . If X ⊆ V ( G ′ ), then OPT( G ′ , k ′ ) ≤ | X | = OPT( G, k ). So, wemay assume that there exists a vertex u ∈ X such that u / ∈ V ( G ′ ). Recall that byconstruction, L is the set of all marked vertices from I , and the additional markedpendant vertices. By Theorem 3, G [ X ] has a p -connected spanning subgraph F with at most p | X | edges. We initialize, X ′ := X and replace one by one all verticesof X which are not in G ′ by vertices of G forming a new set X ′ . We will then provethat G ′ [ X ′ ] is a p -connected vertex cover and | X ′ | ≤ (1 + ε ) | X | ≤ (1 + ε ) k . Observethat the replaced vertices form independent set, and therefore, order of replacingthem does not make any difference. Let u ∈ X \ V ( G ′ ) , B = N F ( u ) and consider the15tep when u is replaced by vertices in G ′ . We will use the fact that | X ′ | ≤ (1 + ε ) k ,the bound which will be shown in the last part of the proof. Case (i): | B | ≤ d . Since, | X ′ | ≤ (1 + ε ) k , the vertex u is unmarked, u ∈∩ x ∈ B N G ( x ) ∩ I p , it holds true that | ∩ x ∈ B N G ( x ) ∩ L | = ⌈ (1 + ε )( p + 1) k ⌉ .So, there is a marked vertex v / ∈ X ′ such that v ∈ ∩ x ∈ B N G ( x ) ∩ L . We set X ′ := ( X \ { u } ) ∪ { v } . In this case, we replace one unmarked vertex by onemarked vertex in X ′ . Case (ii): | B | > d . We partition B = B ⊎ B ⊎ . . . B r such that | B | = · · · = | B r | = d , and d ≤ | B r | ≤ d −
1. Recall that d = ⌈ p /ε ⌉ and 0 < ε < d ≥ p + 1. For every i ∈ [ r ], let U i = B i ∪ A i, ∪ A i, where A i, is an arbitrary set of p + 1 vertices from B i − , and A i, is an arbitrary setof p + 1 vertices from B i +1 . Here i is taken modulo r . Recall that for all i ∈ [ r ], U i ⊆ N F ( u ) ⊆ N G ( u ). Since u is unmarked but u ∈ ∩ x ∈ U i N G ( x ), wehave that | ∩ x ∈ U i N G ( x ) ∩ L | = ⌈ (1 + ε )( p + 1) k ⌉ . But, X ′ covers at most(1 + ε ) k of these marked vertices from r S i =1 ( ∩ x ∈ U i N G ( x ) ∩ L ). Hence, we have pr many distinct marked vertices v , , . . . , v ,p , v , , . . . , v ,p , . . . , v r, , . . . , v r,p / ∈ X such that for all i ∈ [ r ], we have v i, , . . . , v i,p ∈ ∩ x ∈ U i N G ( x ) ∩ L . So, wereplace u by v , , . . . , v ,p , v , , . . . , v ,p , . . . , v r, , . . . , v r,p . Thus, we set X ′ :=( X ′ \ { u } ) ∪ { v , , . . . , v ,p , v , , . . . , v ,p , . . . , v r, , . . . , v r,p } . So, in this case, wehave replaced one unmarked vertex by rp marked vertices. Observe that r = ⌊| B | /d ⌋ = ⌊ deg F ( u ) /d ⌋ .Also observe that the sets of marked vertices replacing two distinct unmarked ver-tices do not intersect. We prove that G ′ [ X ′ ] is a p -connected subgraph. We provethis by contradiction. In particular, we prove that “if G ′ [ X ′ ] is not a p -connectedsubgraph, then F is also not a p -connected subgraph”. Consider a minimum vertexseparator T ′ of G ′ [ X ′ ] such that | T ′ | ≤ p −
1. Let X ′ = X ′ ⊎ X ′ ⊎ T ′ such that T ′ separates X ′ from X ′ in G ′ [ X ′ ]. We will compute a partition X = X ⊎ X ⊎ T of F such that T is separates X from X in F . We initialize X := X ′ , X := X ′ ,and T := T ′ . We process all the unmarked vertices from X \ X ′ one by oneas follows. Consider an arbitrary unmarked vertex u ∈ X \ X ′ . Either u isreplaced by exactly one marked vertex or u is replaced by pr marked vertices v , , . . . , v ,p , . . . , v r, , . . . , v r,p ∈ L , but not both. Suppose that u is replaced by pr marked vertices v , , . . . , v ,p , . . . , v r, , . . . , v r,p ∈ L . For every i ∈ [ r ], we de-fine D i = { v i, , . . . , v i,p } . Furthermore, let us denote D u = D ∪ . . . ∪ D r . Let x, y ∈ D i . Observe that U i ⊆ N G ′ ( x ) ∩ N G ′ ( y ) and | U i | > p . Hence, there are atleast p vertex-disjoint paths between two vertices of D i . Hence, for every i ∈ [ r ],either D i ⊆ X ′ ∪ T ′ or D i ⊆ X ′ ∪ T ′ . Otherwise, if for two distinct x, y ∈ D i , if x ∈ X ′ , y ∈ X ′ , then | T ′ | ≥ p which is a contradiction.Also, let us argue that for i < j , there are at least p vertex-disjoint paths from x ∈ D i to y ∈ D j . If j = i + 1, then it is trivial because A i, ⊆ N G ′ ( x ) ∩ N G ′ ( y ) and | A i, | = p + 1. Similarly, there are p vertex-disjoint paths between x ∈ D and y ∈ D r . Con-sider when j ≥ i +2. Then, consider the sets A i, , D i +1 , A i +1 , , D i +2 , . . . , A j − , , D j − , A j − , .These sets are all pairwise disjoint. And observe that for every z ∈ A j ′ , , for ev-ery w ∈ D j ′ +1 , zw ∈ E ( G ′ ). Similarly, for every z ∈ A j ′ , , for every w ∈ D j ′ ,16 w ∈ E ( G ′ ). Hence, there are at least p vertex-disjoint paths from x ∈ D i to y ∈ D j for all i < j . Therefore, for x ∈ D i , y ∈ D j , it cannot happen that x ∈ X ′ and y ∈ X ′ . Based on the above discussion, for every i, j ∈ [ r ] with i = j , ei-ther D i ∪ D j ⊆ X ′ ∪ T ′ or D i ∪ D j ⊆ X ′ ∪ T ′ , but not both. Therefore, either D ∪ . . . ∪ D r ⊆ X ′ ∪ T ′ or D ∪ . . . ∪ D r ⊆ X ′ ∪ T ′ . We replace these vertices of D u by u as follows. There are four situation that can arise. • Case 1 : If D u ⊆ X ∪ T and T ∩ D u = ∅ , then set T := ( T ∪ { u } ) \ D u . Also,we set X := X \ D u . • Case 2 : If D u ⊆ X ∪ T , and T ∩ D u = ∅ . Then we set T := ( T ∪ { u } ) \ D u .Also, we set X := X \ D u . • Case 3: If D u ⊆ X , then set X := ( X ∪ { u } ) \ D u . • Case 4: If D u ⊆ X , then set X := ( X ∪ { u } ) \ D u .Observe that the above case analysis is exhaustive. Since, N F ( u ) ⊆ N G ′ ( D u ), thisstep of replacement (even when u is added to X or to X ) cannot create any edgebetween X and X in the graph F .On the other hand, when u is replaced by exactly one marked vertex u ∗ ∈ X ′ , thenit holds true that N F ( u ) ⊆ N G ′ ( u ∗ ). Then, we do the following. If u ∗ ∈ T , thenset T := T ∪ { u } \ { u ∗ } . If u ∗ ∈ X , then set X := X ∪ { u } \ { u ∗ } . If u ∗ ∈ X ,then set X := X \ { u ∗ } ∪ { u } . Since N F ( u ) ⊆ N G ′ ( u ∗ ), this procedure also cannotcreate any edge between X and X in the graph F .Moreover, by construction, | T | ≤ | T ′ | and as argued, none of the replacement stepsin the construction can create an edge from X to X . Therefore, T is a set ofvertices that separates X from X . But, then F is also not p -connected. This is acontradiction. Hence, G ′ [ X ′ ] is p -connected.We have shown that G ′ [ X ′ ] is a p -connected subgraph. Since the removed verticesof X are from an independent set, X ′ is a vertex cover as well. Hence, X ′ is a p -connected vertex cover.Recall that F has at most p | X | edges. At every step, one unmarked vertex isreplaced by at most pr marked vertices. Recall that d = ⌈ p /ε ⌉ and 1 ≤ r = ⌊ deg F ( u ) /d ⌋ . Hence, we have | X ′ | ≤ | X ∩ X ′ | + | X \ X ′ | + X u ∈ X \ X ′ p (cid:22) deg F ( u ) d (cid:23) − ! ≤ | X | + X u ∈ X \ X ′ p · deg F ( u ) d = | X | + p P x ∈ X \ X ′ deg F ( u ) d ≤ | X | + 2 p · | E ( F ) | d ≤ | X | + 2 p | X | d ≤ (1 + ε ) | X | Therefore, OPT( G ′ , k ′ ) ≤ | X ′ | ≤ (1 + ε ) | X | = (1 + ε )OPT( G, k ).This completes the proof of the lemma. 17ow, we are ready to prove the main result of this section, i.e. Theorem 2 (we restateit here).
Theorem 2.
For every < ε < and every fixed p ≥ , p -Connected Vertex Cover admits a (1 + ε ) -approximate kernel with O ((1 + ε ) p k O ( p /ε ) )) vertices.Proof. The approximate kernelization algorithm has two parts. The first part is reductionalgorithm, and the second part is solution lifting algorithm. • Reduction Algorithm:
Let (
G, k ) be an input instance for p - Connected Ver-tex Cover . First, we invoke Lemma 2 to check if G has a p -connected vertexcover or not. If G does not have any p -connected vertex cover, then we return(2 K ,
1) as the output instance. Otherwise, we apply Reduction Rule 1 and ob-tain ( G , k ). Then, we construct V ( G ) = H ⊎ I ⊎ R . If | H | > k or G [ R ]has more than k edges, then OPT( G, k ) = k + 1 and we return ( K p ,
1) asthe output instance. Otherwise, we have | H | ≤ k and G [ R ] has k edges. If | V ( G ) | ≤ ⌈ (1 + ε )( p + 1) k ⌉ k ⌈ p /ε ⌉ +2 p +2 + 2 k + 2 k , then we output ( G , k ) it-self. Otherwise, | V ( G ) | > ⌈ (1 + ε )( p + 1) k ⌉ k ⌈ p /ε ⌉ +2 p +2 + 2 k + 2 k . In thissituation, we apply Algorithm 2 on ( G , k ) and obtain ( G ′ , k ′ ). We return ( G ′ , k ′ )as the output instance. • Solution Lifting Algorithm:
Let S ∗ be a given solution to ( G ′ , k ′ ). If S ∗ isnot p -connected vertex cover of ( G ′ , k ′ ), then we return ∅ as a solution to ( G, k ).Otherwise, S ∗ is a p -connected vertex cover of ( G ′ , k ′ ) and we return S ∗ as a feasiblesolution to ( G, k ′ ).Again observe that Reduction Rule 1 only removes isolated vertices. Hence, OPT( G, k ) =OPT( G , k ) and any feasible p -connected vertex cover of ( G , k ) is also a p -connectedvertex cover of ( G, k ). Similarly, due to Lemma 4, if OPT( G , k ) ≤ k , then OPT( G ′ , k ′ ) ≤ (1 + ε )OPT( G , k ). Therefore, if OPT( G, k ) ≤ k , then OPT( G ′ , k ′ ) ≤ (1 + ε )OPT( G, k ).We use this fact and the arguments similar to that of Theorem 1 that if S ∗ is a c -approximate solution to ( G ′ , k ′ ), then S ∗ is a c (1 + ε )-approximate solution to ( G, k ).Also, by similar arguments as in Theorem 1, we can prove that this gives a time efficientPSAKS. By construction G ′ has O ((1+ ε ) p k O ( p /ε ) ) vertices. Hence, this is a time efficientPSAKS with the claimed bound. This completes the proof. In this section, we obtain FPT algorithms for the two problems. While the algorithmfor p -Connected Vertex Cover described in Section 4.3 is quite simple and runsin time O ∗ (2 O ( k ) ), the one for p -Edge-Connected Vertex Cover is based on moreadvanced techniques and is of complexity O ∗ (2 O ( pk ) ). Our algorithm for p -Edge-Connected Vertex Cover uses notions of matroid theory,which we briefly recall the basics for here. For more information, see Oxley [20].18 efinition 9 (Matroid) . Let U be a universe and I ⊆ U . Then, ( U, I ) is said to be a matroid if the following conditions are satisfied.1. ∅ ∈ I ,2. if A ∈ I , then for all A ′ ⊆ A , A ′ ∈ I , and3. if there exist A, B ∈ I with | A | < | B | , then there exists x ∈ B \ A such that A ∪ { x } ∈ I .A set A ∈ I is called an independent set . Note that all maximal independent sets of a matroid M are of the same size, calledthe rank of M and denoted by rank( M ). A maximal independent set is called a basis of M . We recall some useful standard constructions. Let U be a universe with n elementsand I = { A ⊆ U : | A | ≤ r } . Then, ( U, I ) is a matroid called a uniform matroid . Next,let G = ( V, E ) be an undirected graph and let I = { F ⊆ E ( G ) | G ′ = ( V, F ) is a forest } .Then, ( E, I ) is a matroid called a graphic matroid . Finally, let U be partitioned as U = U ∪ . . . ∪ U r and let I = { A ⊆ U : | A ∩ U i | ≤ ∀ i ∈ [ r ] } . Then, ( U, I ) is a matroidcalled a partition matroid .A matroid M is said to be representable over a field F if there is a matrix ˆ M over F and a bijection f : U → col ( ˆ M ), where col ( ˆ M ) is the set of columns of ˆ M , such that B ⊆ U is an independent set in M if and only if { f ( b ) | b ∈ B } is linearly independentover F . Clearly the rank of M is the rank of the matrix ˆ M . A matroid representable overa field M is called a linear matroid over F . A graphic matroid and partition matroid canbe represented over any field [20], while a uniform matroid ( U, I ) with | U | = n , can berepresented over any field F p for p > n [20]. Furthermore, all these representations canbe constructed in deterministic polynomial time. See also [3, 14] for expositions of theissues closer to our current needs.Given a matroid M = ( U, I ), the truncation of M to rank r is the matroid M ′ = ( U, I ′ )where a set A ⊆ U is independent in M ′ if and only if A ∈ I and | A | ≤ r . Given arepresentation of M , a representation of a truncation of M can be computed relativelyeasy in randomized polynomial time [14], but can also be computed in deterministicpolynomial time through more involved methods [12]. p -Edge-Connected VertexCover In this subsection, we provide a single exponential algorithm for p -Edge-ConnectedVertex Cover using dynamic programming and the method of representative sets . Thismethod was introduced to FPT-algorithms by Marx [14]; Fomin et al. [22] presented addi-tional applications and a faster method of computing such sets. We recall the definitions. Definition 10.
Let M = ( E, I ) be a matroid and X, Y ⊆ E . We say that X extends Y in M if X ∩ Y = ∅ and X ∪ Y ∈ I . Furthermore, let S be a family of subsets of E .A subfamily ˆ S ⊆ S is q -representative for S if the following holds: for every set Y ⊆ E with | Y | ≤ q , there is a set X ∈ S that extends Y if and only if there is a set X ∈ ˆ S thatextends Y . heorem 8 (Fomin et al. [22]) . Let M = ( E, I ) be a linear matroid of rank p + q = k over some field F and let S = { S , . . . , S t } be a family of independent sets in M , eachof cardinality p . A representative subset ˆ S ⊆ S of size | ˆ S| ≤ (cid:0) p + qp (cid:1) can be computed in O ( k O (1) (cid:0) p + qp (cid:1) ω − t ) field operations. Here, ω < . is the matrix multiplication exponent. Our algorithm is based on the following result, due to Agrawal et al. [1]. Recall thatan out-branching of a digraph is a spanning subgraph which is a tree, where every arc isoriented away from the root; or equivalently, where every vertex has at most one incomingarc.
Lemma 5 (Agrawal et al. [1]) . Let G = ( V, E ) be an undirected graph and let v r ∈ V .Define a digraph D G = ( V, A E ) by adding the arcs ( u, v ) , ( v, u ) to A E for every edge uv ∈ E . Then G is p -edge-connected if and only if D G has p pairwise arc-disjoint out-branchings rooted in v r . As previous work, we will use representative sets to facilitate a fast dynamic program-ming algorithm. We begin by recalling how out-branchings are realized using matroidtools. Let G = ( V, E ) and D G = ( V, A E ) be as above and let v r ∈ V . Let the out-partitionmatroid (with root v r ) for a digraph D with v r ∈ V ( D ) refer to the partition matroid for A ( D ) where arcs are partitioned according to their heads and where arcs ( u, v r ) are de-pendent; i.e., an arc set F is independent in the out-partition matroid if and only if v r has in-degree 0 in F and every other vertex has in-degree at most one in F . Proposition 1. F is the arc set of an out-branching rooted in v r if and only if | F | = | V ( G ) | − and F is independent in both the out-partition matroid for D G with root v r and the graphic matroid for G . Here, two arcs ( u, v ) and ( v, u ) in A ( D ) represent copies of the same underlying edge uv of the graphic matroid for G .We extend this to construct a matroid that can be used to verify the condition ofLemma 5. Lemma 6.
Let v r ∈ V ( G ) be fixed. Let M be the disjoint union of p + 1 matroids M i as follows. Matroids M , M , . . . , M p − are copies of the graphic matroid of G overa ground set of A E , where each arc represents its underlying undirected edge. Matroids M , M , . . . , M p are copies of the out-partition matroid for D G with root v r . Matroid M p +1 is the uniform matroid over A E with rank p ( k − . Let F ⊆ A E . The followingare equivalent.1. F is the arc set of p pairwise arc-disjoint out-branchings rooted in v r in D G [ S ] forsome | S | = k with v r ∈ S | V ( F ) | = k , | F | = p ( k − , v r ∈ V ( F ) , and there is an independent set I in M where every arc a ∈ F occurs in I precisely in its copies in matroids M i − , M i and M p +1 for some i ∈ [ p ] .Furthermore, a representation of M can be constructed in deterministic polynomial time.Proof. Let M and M i , i ∈ [2 p + 1] be as described. We note that graphic matroids,partition matroids (hence out-partition matroids), and uniform matroids all have deter-ministic representations [20]. Specifically, graphic matroids and partition matroids are20epresentable over any field, and uniform matroids over any sufficiently large field. Hencea representation of M can be constructed as a diagonal block matrix with 2 p + 1 blocks,where each block i is a representation of the matroid M i over some sufficiently large field F (e.g., GF( q ) for some prime q > p ( k − F meet the conditions in the first item. Since F is spanning for D G [ S ] we have | V ( F ) | = k and v r ∈ V ( F ), and furthermore | F | = p ( k −
1) since eachout-branching is spanning. Furthermore, any arc set of an out-branching is independentin both the graphic matroid and the out-partition matroid by Prop. 1. Letting F = F ∪ . . . ∪ F p where F i is the arc set of an out-branching for every i ∈ [ p ], we can thenconstruct I by letting an arc a ∈ F i be present in I in matroids M i − , M i and M p +1 .Hence all conditions in the second item are met.Now assume that F meets the conditions in the second item. Partition F = F ∪ . . . ∪ F p where F i , i ∈ [ p ] contains those arcs of F that are represented in matroids M i − and M i .By a counting argument, | F i | = k − i ∈ [ p ]. Furthermore, F i is the arc setof an out-forest (since its underlying undirected edge set is acyclic and every vertex hasin-degree at most 1 in F i ). But then F i must form a spanning tree of V ( F ) by counting,hence an out-branching of D G [ V ( F )] by Prop. 1. Furthermore v r ∈ V ( F ) and every arcinto v r is dependent in M i ; thus every out-branching F i is rooted in v r .For use in the representative set computation, we need to modify M so that the set I being described is a basis of M , not just an independent set. This is more technical,but can be done deterministically using the operation of deterministic truncation, due toLokshtanov et al. [12], as noted in Section 4.1. Lemma 7.
We can in deterministic polynomial time compute a linear representation ofa matroid M as in Lemma 6, except truncated so that any independent set I as describedis a basis of M .Proof. The operation of truncation on a matroid is the operation of restricting its rank,i.e., defining from M = ( E, I ) a new matroid M ′ = ( E, I ′ ) where I ′ = { S ∈ I | | S | ≤ r } for some r < | E | . Hence, a matroid M as described is produced by the truncation of M from Lemma 6 to rank r = 3 p ( k − p -Edge-CVC works as follows. First, we enumerate all minimalvertex covers of G of size at most k . The number of such minimal vertex covers is at most2 k , and can be enumerated in O ∗ (2 k ) time, and space (see [15]). Then, for every minimalvertex cover H of G , we use representative sets and the above characterization to checkif it can be extended to a feasible p -edge-connected vertex cover S ⊇ H of size at most k . In detail, consider a graph G with a vertex cover H which is not p -edge-connected,where we are looking for a set S ⊃ H such that G [ S ] is p -edge-connected and | S | ≤ k .By iteration over k , we may assume that | S | < k is impossible. Let us fix v r ∈ H . Bythe above lemma, there then exists such a set S if and only if there is an independent set I in M meeting the following conditions:1. H ⊂ V ( I ) and | V ( I ) | = k | I | = 3 p ( k −
1) 21. Every arc which is represented in I is represented in precisely three matroids M i − , M i , M p +1 in M .Furthermore, the first condition can be relaxed to | V ( I ) \ H | = k − | H | (since any foreston k − S ∪ H for S ⊆ V ( G ) \ H and | S | = k − | H | must span S ∪ H ).We can address this via dynamic programming. The dynamic program is set up viaa table keeping track of | V ( I ) | and | I | , and we ensure that every time we add somearc a to a set I we add it in precisely three layers, as described. Thanks to the use ofrepresentative sets, each table entry in the dynamic programming only needs to contain2 O ( pk ) partial solutions.We provide the details of this scheme in the proof of the main result of this section,i.e. Theorem 4 (we restate it here). Theorem 4.
For every fixed p ≥ , p -Edge-Connected Vertex Cover can be solvedin O (2 O ( pk ) n O (1) ) deterministic time and space.Proof. As outlined above, we may assume that we have a vertex cover H that is not p -edge-connected and have already tested that there is no p -edge-connected vertex coverwith at most k − V ( G ) \ H as S = { v , . . . , v n ′ } . Wecreate a dynamic programming table T [( i, j, q )] with entries indexed by ( i, j, q ) for i ≤ k , j ≤ n ′ and q ≤ p ( k − T [( i, j, q )] contains independent sets I in M such that | V ( I ) \ H | = i , the largest-index vertex of S occurring in V ( I ) is v j , and | I | = q .Furthermore, for every independent set I in the table, every arc occurring in I occurs inprecisely three layers, as described above. We may then check for a solution by checkingwhether any slot T [( k − | H | , j, p ( k − M as in Lemma 7.For an arc a ∈ A E , define F a,i to be the set consisting of the copies of a in M i − , M i and M p +1 .We initialize the slots T [(0 , , q )] by a dynamic programming process within D G [ H ].Initialise T [(0 , , {∅} . Enumerate the arcs of D G [ H ] as a , . . . , a m ′ . Then, for every q = 3 , , . . . fill in the slot T [(0 , , q )] from T [(0 , , q − I ∈ T [(0 , , q − a j , and every i ∈ [ p ] such that F a j ,i extends I , add I ∪ F a j ,i to T [(0 , , q )]2. Reduce T [(0 , , q )] to a (3 p ( k − − q )-representative set in M .This is a polynomial number of steps, where every set I ∈ T [(0 , , q )] is used in a poly-nomially bounded number of new sets, hence every time we apply Theorem 8 at a level q , we do so with t ≤ ( k + p ) O (1) (cid:0) p ( k − q (cid:1) , hence up to polynomial factors each step takestime (cid:0) p ( k − q (cid:1) ω = 2 O ( pk ) .For slots T [( i, j, q )], we process vertices v j one at a time. The process is slightly morecomplex since each vertex v j can be incident to O ( k ) arcs in A E , but the principle is thesame. We process slots T [( i, j, q )] in lexicographic order by ( i, j, q ). Before we processthe sets in a slot T [( i, j, q )], we reduce them to a (3 p ( k − − q )-representative set in M .Then we proceed as follows. For every independent set I in T [( i, j, q )] and every j ′ with j < j ′ ≤ n ′ , we combine I and v j ′ as follows.1. Let d ≤ | H | be the number of arcs of A E incident with v j ′ . Create a set F forevery one out of the ( p + 1) d options of (1) adding F a,b to F for some b ∈ [ p ], or (2)not adding any set F a,b to F , for every arc a incident with v j ′ .22. For every such non-empty set F , and every independent set I of T [( i, j, q )] suchthat F extends I in M , add I ∪ F to T [( i, j ′ , q + | F | )].To roughly bound the running time, we note that the number of slots in the table ispolynomial, and for every set I in a slot, at most ( p + 1) | H | sets I ′ = I ∪ F are addedto other slots of the table. Furthermore, after the representative set reduction, every slotcontains at most 2 p ( k − sets. Thus every time we apply Theorem 8, we have | T [( i, j, q )] | < ( p + k ) O (1) pk ( p + 1) k = 2 O ( pk ) , hence the total time usage, up to a polynomial factor, is 2 O ( pk ) .It remains to prove correctness. Let I ∈ T [( i, j, q )] for some ( i, j, q ) and let S = V ( I ) \ H . We note the invariants | S | = i , max a { v a ∈ S } = j and | I | = q hold byinduction. Furthermore, by construction I is independent in M . With these observations,we proceed. First assume that there is a set I ∈ T [( k − | H | , j, p ( k − j .Then | V ( I ) \ H | = k − | H | and | I | = 3 p ( k −
1) by the invariants. Furthermore, every arc a represented in I occurs in precisely three copies by construction. Indeed, every timewe grow a set I we do so by adding a collection of sets F a,i to it. Hence every arc occursat least three times, and furthermore, since F a,i always contains a copy of a in M p +1 , wewill never add two distinct sets F a,i , F a,i ′ to the same set I . Hence Lemma 6 implies that D G [ H ∪ V ( I )] has p pairwise arc-disjoint out-branchings rooted in v r , which by Lemma 5implies that G [ H ∪ V ( I )] is p -edge-connected.On the other hand, assume that G [ H ∪ S ] is p -edge-connected for some S ⊆ V ( G ) \ H with | S | = k − | H | . By Lemma 5 there exist p pairwise edge-disjoint out-branchings in D G [ H ∪ S ] rooted in v r , hence by Lemma 6 there is an independent set I in M formedusing a set of arcs F ⊆ A E with | F | = p ( k −
1) and | I | = 3 p ( k −
1) as described. Let j be the largest index such that v j ∈ V ( I ); then I is a candidate for the table slot T [( k − | H | , j, p ( k − T [( k − | H | , j, p ( k − I is the disjoint union of sets F a,i . We partition I according tolexicographical order of ( i, j, q ) as follows. First, let F a ,i , . . . , F a t ,i t enumerate the sets F a,i contained in I for which a is contained in D G [ H ]. For r ∈ [ t ], let I ′ r = I \ r [ j =1 F a j ,i j be the subset of I which is encountered “after” F a r ,i r in the natural ordering. We showby induction that for each r ∈ [ t ], the slot T [(0 , , r )] contains a set which extends I ′ r .For r = 0 this holds trivially. Hence, assume the statement holds for T [(0 , , r )] for some r < t , and let I ∈ T [(0 , , r )] be a set which extends I ′ r . While processing (0 , , r ),the set F a r +1 ,i r +1 is considered in the loop, and clearly it extends I since F a r +1 ,i r +1 ⊆ I ′ r .Hence T [(0 , , r + 3)] contains the set I = I ∪ F a r +1 ,i r +1 before the representative setcomputation is performed. By assumption I extends I ′ r +1 . Hence by the correctness ofTheorem 8, T [(0 , , r + 3)] contains some set I that extends I ′ r +1 , as required. Hencethe claim holds up to the set T [(0 , , t )].We can now complete the proof using the same outline for entries T [( i, j i , q i )]. Enu-merate S as S = { v j , . . . , v j k −| H | } in increasing order of indices j i and for each i ∈ [ k −| H | ]let I i be the union of sets F a,i of I for which a is incident with v j i . Let I ≥ i = S k −| H | j = i I j .We show by induction in lexicographical order that T [( i, j i , q i )] for some q i contains a set23hich extends I ≥ i +1 , for each i . As a base case, the claim holds for T [(0 , , t )] as hasalready been shown. For the inductive step, the proof is precisely as above. For every i = 1 , . . . , k − | H | , let I ′ i − ∈ T [( i − , j i − , q i − )] be a set which extends I ≥ i . Then inparticular I i extends I ′ i − and is added to table T [( i, j i , q i )] in the exhaustive enumerationloop from T [( i − , j i − , q i − )]. Thus before the call to Theorem 8 there was a set in T [( i, j i , q i )] which extends I ′≥ i +1 , hence the same holds after the representative set reduc-tion. By induction, the table slot T [( k − | H | , j, p ( k − j . This completes the proof of the theorem. p -Connected Vertex Cover Theorem 9. p -Connected Vertex Cover is fixed parameter tractable with an algo-rithm running in time O ∗ (2 O ( k ) ) .Proof. Suppose that (
G, k ) is the input instance of p -Connected Vertex Cover .First, we remove all isolated vertices of G . Then we compute a set H of vertices whichhave to be in a vertex cover as follows. We put a vertex into H if the degree of this vertexis at least k + 1 or it is a neighbor to a vertex v such that deg G ( v ) ≤ p −
1. If | H | > k then ( G, k ) is a negative instance, so we may assume that | H | ≤ k. Now, we partitionthe vertices of G − H into two parts. We put a vertex u into I if N G ( u ) ⊆ H . Otherwise,we put u into R . Observe that I is an independent set. Since any vertex in R has degreeat most k , if the number of edges incident to R is more than k then ( G, k ) is a negativeinstance. Thus, | R | ≤ k . If for some u = v ∈ I , N G ( v ) = N G ( u ), then say that u and v are false twins . If avertex v has more than k +1 false twins, then we delete v . Hence, for any A ⊆ V ( G ), thereare at most k + 1 vertices in I whose neighborhood equals A . Hence, | I | = O (2 k ( k + 1)).Now it is not hard to see that H ∪ I ∪ R is a kernel and | H | + | R | + | I | = O (2 k k ). We canconsider all subsets of H ∪ I ∪ R with at most k vertices and either output a p -connectedvertex cover or that the instance is negative. This algorithm runs in time O ∗ (2 O ( k ) ) . p -Edge-Connected Vertex Cover In this section, we describe a 2( p + 1)-approximation algorithm for p -Edge-ConnectedVertex Cover . We begin by recalling the notion of a Gomory-Hu tree . Definition 11 (Gomory-Hu Tree) . Let G = ( V, E ) be a graph, and let c ( u, v ) ≥ bethe capacity of edge uv ∈ E. Denote the minimum capacity of an s - t cut by λ st for each s, t ∈ V ( G ) . Let T = ( V T , E T ) be a tree with V T = V ( G ) , and let us denote the set ofedges in the s - t path in T by P st for each s, t ∈ V T . Then T is said to be a Gomory-Hutree of G if λ st = min e ∈ P st c ( S e , T e ) for all s, t ∈ V ( G ) , where • S e and T e are sets of vertices of the two connected components of T − e such that s ∈ S e and t ∈ T e , and • c ( S e , T e ) is the capacity of the cut in G .The capacity of an edge uv of T is equal to λ uv . heorem 10. [9] Every weighted graph ( G, c ) has a Gomory-Hu tree which can beconstructed in polynomial time. For an unweighted graph G = ( V, E ) , we can introduce weights by setting c ( uv ) = 1for every uv ∈ E. Let T be a Gomory-Hu tree of G . Then, by Definition 11, for everypair of vertices u, v ∈ V ( G ), the size of a minimum edge cut between u and v in G is theminimum capacity of an edge cut between u and v in T . For i ∈ [ p ], consider the set E i ofall the edges of total capacity at most i − T . Deleting E i disconnects T into severalsubtrees. We call the vertex set of each such subtree an i -segment in G . Thus, a subsetof vertices S ⊆ V ( G ) is an i -segment in G if and only if for every u, v ∈ S , there are atleast i edge-disjoint paths between u and v in G and S is a maximal such subset. It isobvious from the construction that the i -segments of G form a partition of the vertex setof G , and can be computed in polynomial time.Now let G = ( V, E ) be an undirected graph, and X ⊆ V ( G ). For i ∈ [ p ], let an i -blockof X in G be a maximal subset X ′ ⊆ X such that for every u, v ∈ X ′ , there are at least i edge-disjoint paths between u and v in G . We can use the Gomory-Hu tree to computethe i -blocks of X in G , as follows. Lemma 8.
Let G = ( V, E ) be an undirected graph, and X ⊆ V ( G ) . The i -blocks of X in G are precisely the sets X ∩ S over all i -segments S of G .Proof. Let T be the Gomory-Hu tree of G , and let u, v ∈ X be distinct vertices. By thedefinition of a Gomory-Hu tree, λ uv ( G ) ≥ i if and only if there is no edge e on the pathfrom u to v in T with capacity c ( e ) less than i . Since this is also equivalent to u and v being in the same i -segment of G , the statement follows.Based on Lemma 8, we can compute the collection of i -blocks in polynomial timeusing Goromy-Hu tree of G .Recall that a family of subsets of a set U is called laminar if for every pair A, B ofsubsets of U one of the following holds: (i) A ∩ B = ∅ , (ii) A ⊆ B , or (iii) B ⊆ A. (A family may have several copies of the same subset.) Lemma 9.
Let G be a graph, X ⊆ V ( G ) , and p be a fixed integer. Then the collectionof all i -blocks of X for all i = 1 , . . . , p forms a laminar family.Proof. Note first that for every i , the i -segments of G form a partition of the vertex set,hence similarly, the i -blocks of X form a partition of X for every i ∈ [ p ]. Furthermore, let i, j ∈ [ p ] with 1 ≤ i < j ≤ p . It is obvious from the definition that the j -segments of G form a refinement of the i -segments of G , since they are formed from the Gomory-Hu treeby deleting an additional set of edges. Hence the j -blocks of X also form a refinement ofthe i -blocks of X in G by Lemma 8, and the statement follows.The proof of Lemma 9 leads to Algorithm 3 which sets all i -blocks of X , i ∈ [ p ], asnodes of a (laminar) tree. Lemma 10.
Let G be a graph, u ∈ V ( G ) and let A, B ⊆ V ( G − u ) such that A ∩ B = ∅ .If the size of a minimum ( A, B ) -cut in G − u is i and min {| N ( u ) ∩ A | , | N ( u ) ∩ B |} = j then the size of a minimum ( A, B ) -cut in G is at least i + j . lgorithm 3: LaminarTree ( G = ( V, E ) , X, p ) input : G = ( V, E ) , X ⊆ V ( G ) , p output : A laminar tree of X in G Initialize a tree T := ∅ ; for i = 1 , . . . , p do Compute the set of all i -blocks of X in G ; A i ← the set of all i -blocks of X in G ; end Set X as the root of T ; Make all -blocks the children of X ; for i = 2 , . . . , p dofor every Y ∈ A i do Make Y a child of W ∈ A i − in T such that Y ⊆ W ; endend Output T as the laminar tree of X in G ; Proof.
Consider a minimum (
A, B )-cut (
S, T ) in G ; A ⊆ S and B ⊆ T . Assume withoutloss of generality that u ∈ S . Suppose that the size of ( S, T ) is at most i + j −
1. Consider( S \ { u } , T ) , which is a cut in G − u. Since | N ( u ) ∩ B | ≥ j , we have that the size of thecut ( S \ { u } , T ) is at most i −
1, but this contradicts the fact that the size of a minimum(
A, B )-cut in G − u is i .We are ready to prove the main result of this section, Theorem 5 (we restate it here). Theorem 5.
For every fixed p ≥ , p -Edge-Connected Vertex Cover admits a p + 1) -factor approximation algorithm.Proof. We will show that Algorithm 4 is a 2( p + 1)-approximation algorithm.Observe that the first output “No feasible solution” is correct due to Lemma 2. Note X = N G ( L ) ∪ V ( M ) is a vertex cover of G. Suppose that T has more than one leaf andthere is no vertex u ∈ V ( G ) \ ( Y ∪ L ) such that N G ( u ) intersects two distinct p -blocksin T. Then even adding all vertices of V ( G ) \ ( Y ∪ L ) to Y will not make Y p -edge-connected vertex cover of G since there will be the same number of p -blocks of X in G [ Y ] before and after the addition because the vertices of V ( G ) \ ( Y ∪ L ) are not verticesof M and thus form an independent set. However, this is impossible as V ( G − L ) is a p -edge-connected vertex cover of G. Thus, as long as T has more than one leaf there is avertex u ∈ V ( G ) \ ( Y ∪ L ) such that N G ( u ) intersects two distinct p -blocks in T. When T has just one leaf, G has only one p -block of X in G [ Y ] . This means that forevery pair x, y of vertices in X there are p edge-disjoint paths in G [ Y ] between x and y .Let u, v ∈ Y \ X . Since | N G ( u ) | ≥ p and | N G ( v ) | ≥ p by Menger’s theorem, there are p edge-disjoint paths in G [ Y ] between N G ( u ) and N G ( v ) with distinct end-vertices andhence p edge-disjoint paths in G [ Y ] between u and v. Similarly, we can see that there are p edge-disjoint paths in G [ Y ] between u and any x ∈ X. Therefore, Y is a p -edge-connectedvertex cover of G. Let us analyze how T changes after u ∈ V ( G ) \ ( Y ∪ L ) is added to Y. First we consider T before u is added to Y . By the description of Algorithm 4, u has neighbors in two26 lgorithm 4: Approximation algorithm for p -Edge-CVC input : G = ( V, E ) output : An approximate p -edge-connected vertex cover of GL ← { v ∈ V ( G ) | deg G ( v ) < p } ; if V ( G − L ) is not a p -edge-connected vertex cover of G then Output “No feasible solution” ; end Compute a maximal matching M of G − N G [ L ]; X ← N G ( L ) ∪ V ( M ); T ← LaminarTree ( G [ X ] , X, p ); Y ← X ; while T has at least two leaves doif u ∈ V ( G ) \ ( Y ∪ L ) s.t. N G ( u ) intersects two distinct p -blocks in T then Y ← Y ∪ { u } ; T ← LaminarTree ( G [ Y ] , X, p ); endend Output Y as a solution ;distinct p -blocks X , X of X in G [ Y ]. Let X be the least common ancestor of X and X in T and let X be an i -block of X in G [ Y ]. Observe that X has two children X a and X b such that X ⊆ X a and X ⊆ X b . Note that X a and X b are ( i + 1)-blocks of X in G [ Y ] and the minimum size of a ( X a , X b )-cut is i (otherwise, X is not an i -block of X in G [ Y ]). Now consider what happens just after u is added to Y . By Lemma 10, the sizeof any ( X a , X b )-cut increases by at least one. Thus, the minimum size of a ( X a , X b )-cutbecomes i + 1 and so X a and X b become part of a new ( i + 1)-block of X in G [ Y ] . Thus,the number of the nodes of T decreases.Let us now bound the approximation factor of Algorithm 4. The number of leaves in T becomes one only when there is just one node on each level of T , i.e., T has p + 1 vertices.Initially, T may have up to p | X | + 1 nodes. Thus, at most p ( | X | −
1) nodes will be addedto X before a solution Y is obtained. Hence, | Y | ≤ | X | + p ( | X | − ≤ ( p + 1) | X | . Since M is a maximal matching of G − N G ( L ), at least one endpoint of each edge of M has to be inany vertex cover of G . Thus, OPT( G ) ≥ | N G ( L ) | + | M | , where OPT( G ) is the minimumnumber of vertices in a p -edge-connected vertex cover of G. Since | X | = 2 | M | + | N G ( L ) | , | X | ≤ G ) . Therefore, | Y | ≤ ( p + 1) | X | ≤ p + 1)OPT( G ) . We obtained time efficient polynomial sized approximate kernelization schemes (PSAKS)for both p -Edge-Connected Vertex Cover and p -Connected Vertex Cover .We also provide a O ∗ (2 O ( pk ) ) time algorithm for p -Edge-Connected Vertex Cover .But, unfortunately, the approach we use in this FPT algorithm does not work for p -vertex-connectivity. Hence, an interesting open problem would be to provide a singlyexponential FPT algorithm for p -Connected Vertex Cover .27e also obtain a polynomial time 2( p +1)-factor approximation algorithm for p -Edge-Connected Vertex Cover . Unfortunately, the main idea used in this approximationalgorithm does not work for p -Connected Vertex Cover . Although for p = 2, thereis a block decomposition of the graph. Hence, a constant factor approximation algorithmis possible to obtain for Biconnected Vertex Cover using an approach that is muchsimpler than p -Edge-Connected Vertex Cover . But, it is currently unknown howto get such a result for p -Connected Vertex Cover for any arbitrary fixed p ≥ p -Edge-Connected Vertex Cover , and for p -Connected Vertex Cover butthis is also open for Connected Vertex Cover . References [1] Akanksha Agrawal, Pranabendu Misra, Fahad Panolan, and Saket Saurabh. Fastexact algorithms for survivable network design with uniform requirements. In
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