Finding small stabilizers for unstable graphs

Bock, Adrian; Chandrasekaran, Karthekeyan; Könemann, Jochen; Peis, Britta; Sanità, Laura

doi:10.1007/s10107-014-0854-1

Cited by 22 publications

(11 citation statements)

References 22 publications

(29 reference statements)

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“…Matching games are not simple games. Yet their core constraints are readily seen to simplify to p ≥ 0 and p i + p j ≥ 1 for all ij ∈ E. Classical solution concepts, such as the core and core-related ones like least core, nucleolus or nucleon are well studied for matching games, see, for example, [3,4,10,18,19,27]. However, the problems encountered there differ with respect to the objective function.…”

Section: Related Workmentioning

confidence: 99%

“…Simple games form a classical class of games, which are well studied; see also the book of Taylor and Zwicker [29]. 4 The notion of being simple means that every coalition either has some equal amount of power or no power at all. Formally, a cooperative game (N, v) is simple if v is a monotone 0-1 function with v(∅) = 0 and v(N ) = 1, so v(S) ∈ {0, 1} for all S ⊆ N and v(S) ≤ v(T ) whenever S ⊆ T .…”

Section: Introductionmentioning

confidence: 99%

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Simple Games Versus Weighted Voting Games

Hof

Kern

Kurz

et al. 2018

Algorithmic Game Theory

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A simple game (N, v) is given by a set N of n players and a partition of 2 N into a set L of losing coalitions L with value v(L) = 0 that is closed under taking subsets and a set W of winning coalitions W with v(W ) = 1. Simple games with α = min p≥0 maxW ∈W,L∈L p(L) p(W ) < 1 are known as weighted voting games. Freixas and Kurz (IJGT, 2014) conjectured that α ≤ 1 4 n for every simple game (N, v). We confirm this conjecture for two complementary cases, namely when all minimal winning coalitions have size 3 and when no minimal winning coalition has size 3. As a general bound we prove that α ≤ 2 7 n for every simple game (N, v). For complete simple games, Freixas and Kurz conjectured that α = O( √ n). We prove this conjecture up to a ln n factor. We also prove that for graphic simple games, that is, simple games in which every minimal winning coalition has size 2, computing α is NP-hard, but polynomial-time solvable if the underlying graph is bipartite. Moreover, we show that for every graphic simple game, deciding if α < a is polynomial-time solvable for every fixed a > 0.

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Simple Games Versus Weighted Voting Games

Hof

Kern

Kurz

et al. 2018

Algorithmic Game Theory

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“…the optimum value of the maximum matching linear program). It is known that finding a minimum number of vertices whose removal results in a stable graph is solvable in polynomial time, but the related edge version is NP-Complete [3]. Is the problem of deleting at most k edges to obtain a stable graph fixed-parameter tractable?…”

Section: Resultsmentioning

confidence: 99%

Tractability of König edge deletion problems

Majumdar

Neogi

Raman

et al. 2019

Theoretical Computer Science

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A graph is said to be a König graph if the size of its maximum matching is equal to the size of its minimum vertex cover. The König Edge Deletion problem asks if in a given graph there exists a set of at most k edges whose deletion results in a König graph. While the vertex version of the problem (König vertex deletion) has been shown to be fixed-parameter tractable more than a decade ago, the fixed-parameter-tractability of the König Edge Deletion problem has been open since then, and has been conjectured to be W[1]-hard in several papers. In this paper, we settle the conjecture by proving it W[1]-hard. We prove that a variant of this problem, where we are given a graph G and a maximum matching M and we want a k-sized König edge deletion set that is disjoint from M, is fixed-parameter-tractable.2. Let U be the set of unsaturated vertices of M, then S ∩ U = ∅. From the above observation it follows that for every minimum vertex cover S and maximum matching M of a König graph G, M lies across the edges (S, V(G) \ S) and M saturates S. Moreover, if there exists a maximum matching M and minimum vertex cover S such that M saturates S and M lies across (S, V(G) \ S), then |M| = |S| and hence G must be König. So we have the following characterization of König graphs. Lemma 2.1. [7, 26] A graph G is König if and only if for every minimum vertex cover S of G, there exists a matching across (S, V(G) \ S) that saturates S.As for any vertex cover S and a matching M of G, |S| ≥ |M|, we also have the following equivalent characterization using the fact that if there is a matching M and a vertex cover S such that |M| = |S|, then M must be a maximum matching and S must be a minimum vertex cover. E) is König if and only if for some vertex cover S of G, there exists a matching across (S, V(G) \ S) that saturates S.In passing, we observe the following interesting property of König graphs that we couldn't find in literature. Though we don't use it in the rest of our paper, we feel that this can be of independent interest. Lemma 2.3. A graph G is König if and only if vc f (G) = vc(G), that is, its vertex cover number is equal to its fractional vertex cover number.Proof. If G is König, then vc f (G) = vc(G) as observed in Subsection 2.2. To prove the converse, find the optimum solution to the vertex cover linear program (that is, the fractional vertex cover) of G that satisfies the properties in Theorem 1.By definition, S 0 induces an independent set. As vc f (G) = vc(G), by the second property of the theorem, S 1/2 = ∅ as there exists an optimum solution to the vertex cover linear program which is integral. Hence (S 1 , S 0 ) is a partition of the vertex set and S 1 is a vertex cover. By the third property of the theorem, there exists a matching across (S 1 , S 0 ) that saturates S 1 . Hence the lemma follows from Lemma 2.2. 4

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“…Matching games are not simple games. Yet their core constraints are readily seen to simplify to p 0 and p i + p j 1 for all ij ∈ E. Classical solution concepts, such as the core and core-related ones like least core, nucleolus or nucleon are well studied for matching games, see, for example, [4,5,12,22,23,31].…”

Section: Introductionmentioning

confidence: 99%

Simple Games Versus Weighted Voting Games: Bounding the Critical Threshold Value

et al. 2018

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A simple game (N, v) is given by a set N of n players and a partition of 2 N into a set L of losing coalitions L with value v(L) = 0 that is closed under taking subsets and a set W of winning coalitions W with v(W ) = 1. Simple games with α = min p 0 maxW ∈W,L∈L p(L) p(W ) < 1 are exactly the weighted voting games. We show that α 1 4 n for every simple game (N, v), confirming the conjecture of Freixas and Kurz (IJGT, 2014). For complete simple games, Freixas and Kurz conjectured that α = O( √ n). We prove this conjecture up to a ln n factor. We also prove that for graphic simple games, that is, simple games in which every minimal winning coalition has size 2, computing α is NP-hard, but polynomial-time solvable if the underlying graph is bipartite. Moreover, we show that for every graphic simple game, deciding if α < a is polynomial-time solvable for every fixed a > 0.⋆ A partial answer to the conjecture of Freixas and Kurz appeared, together with some results of this paper, in an extended abstract published in the proceedings of SAGT 2018 [20].

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Finding small stabilizers for unstable graphs

Cited by 22 publications

References 22 publications

Simple Games Versus Weighted Voting Games

Simple Games Versus Weighted Voting Games

Tractability of König edge deletion problems

Simple Games Versus Weighted Voting Games: Bounding the Critical Threshold Value

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