753 research outputs found

    On the Threshold of Intractability

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    We study the computational complexity of the graph modification problems Threshold Editing and Chain Editing, adding and deleting as few edges as possible to transform the input into a threshold (or chain) graph. In this article, we show that both problems are NP-complete, resolving a conjecture by Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1):109--128, 2001). On the positive side, we show the problem admits a quadratic vertex kernel. Furthermore, we give a subexponential time parameterized algorithm solving Threshold Editing in 2O(klogk)+poly(n)2^{O(\surd k \log k)} + \text{poly}(n) time, making it one of relatively few natural problems in this complexity class on general graphs. These results are of broader interest to the field of social network analysis, where recent work of Brandes (ISAAC, 2014) posits that the minimum edit distance to a threshold graph gives a good measure of consistency for node centralities. Finally, we show that all our positive results extend to the related problem of Chain Editing, as well as the completion and deletion variants of both problems

    An FPT Algorithm for Elimination Distance to Bounded Degree Graphs

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    In the literature on parameterized graph problems, there has been an increased effort in recent years aimed at exploring novel notions of graph edit-distance that are more powerful than the size of a modulator to a specific graph class. In this line of research, Bulian and Dawar [Algorithmica, 2016] introduced the notion of elimination distance and showed that deciding whether a given graph has elimination distance at most k to any minor-closed class of graphs is fixed-parameter tractable parameterized by k [Algorithmica, 2017]. They showed that Graph Isomorphism parameterized by the elimination distance to bounded degree graphs is fixed-parameter tractable and asked whether determining the elimination distance to the class of bounded degree graphs is fixed-parameter tractable. Recently, Lindermayr et al. [MFCS 2020] obtained a fixed-parameter algorithm for this problem in the special case where the input is restricted to K?-minor free graphs. In this paper, we answer the question of Bulian and Dawar in the affirmative for general graphs. In fact, we give a more general result capturing elimination distance to any graph class characterized by a finite set of graphs as forbidden induced subgraphs

    On Selected Subclasses of Matroids

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    Matroids were introduced by Whitney to provide an abstract notion of independence. In this work, after giving a brief survey of matroid theory, we describe structural results for various classes of matroids. A connected matroid MM is unbreakable if, for each of its flats FF, the matroid M/FM/F is connected%or, equivalently, if MM^* has no two skew circuits. . Pfeil showed that a simple graphic matroid M(G)M(G) is unbreakable exactly when GG is either a cycle or a complete graph. We extend this result to describe which graphs are the underlying graphs of unbreakable frame matroids. A laminar family is a collection \A of subsets of a set EE such that, for any two intersecting sets, one is contained in the other. For a capacity function cc on \A, let \I be %the set \{I:|I\cap A| \leq c(A)\text{ for all A\in\A}\}. Then \I is the collection of independent sets of a (laminar) matroid on EE. We characterize the class of laminar matroids by their excluded minors and present a way to construct all laminar matroids using basic operations. %Nested matroids were introduced by Crapo in 1965 and have appeared frequently in the literature since then. A flat of a matroid MM is Hamiltonian if it has a spanning circuit. A matroid MM is nested if its Hamiltonian flats form a chain under inclusion; MM is laminar if, for every 11-element independent set XX, the Hamiltonian flats of MM containing XX form a chain under inclusion. We generalize these notions to define the classes of kk-closure-laminar and kk-laminar matroids. The second class is always minor-closed, and the first is if and only if k3k \le 3. We give excluded-minor characterizations of the classes of 2-laminar and 2-closure-laminar matroids

    All Abelian Quotient C.I.-Singularities Admit Projective Crepant Resolutions in All Dimensions

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    For Gorenstein quotient spaces Cd/GC^d/G, a direct generalization of the classical McKay correspondence in dimensions d4d\geq 4 would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces which would satisfy the above property. We prove that the underlying spaces of all Gorenstein abelian quotient singularities, which are embeddable as complete intersections of hypersurfaces in an affine space, have torus-equivariant projective crepant resolutions in all dimensions. We use techniques from toric and discrete geometry.Comment: revised version of MPI-preprint 97/4, 35 pages, 13 figures, latex2e-file (preprint.tex), macro packages and eps-file

    Vertex Sparsifiers for Hyperedge Connectivity

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    Recently, Chalermsook et al. [SODA'21(arXiv:2007.07862)] introduces a notion of vertex sparsifiers for cc-edge connectivity, which has found applications in parameterized algorithms for network design and also led to exciting dynamic algorithms for cc-edge st-connectivity [Jin and Sun FOCS'21(arXiv:2004.07650)]. We study a natural extension called vertex sparsifiers for cc-hyperedge connectivity and construct a sparsifier whose size matches the state-of-the-art for normal graphs. More specifically, we show that, given a hypergraph G=(V,E)G=(V,E) with nn vertices and mm hyperedges with kk terminal vertices and a parameter cc, there exists a hypergraph HH containing only O(kc3)O(kc^{3}) hyperedges that preserves all minimum cuts (up to value cc) between all subset of terminals. This matches the best bound of O(kc3)O(kc^{3}) edges for normal graphs by [Liu'20(arXiv:2011.15101)]. Moreover, HH can be constructed in almost-linear O(p1+o(1)+n(rclogn)O(rc)logm)O(p^{1+o(1)} + n(rc\log n)^{O(rc)}\log m) time where r=maxeEer=\max_{e\in E}|e| is the rank of GG and p=eEep=\sum_{e\in E}|e| is the total size of GG, or in poly(m,n)\text{poly}(m, n) time if we slightly relax the size to O(kc3log1.5(kc))O(kc^{3}\log^{1.5}(kc)) hyperedges.Comment: submitted to ESA 202

    Structure Theorem and Isomorphism Test for Graphs with Excluded Topological Subgraphs

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    We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph HH as a minor to graphs excluding HH as a topological subgraph. We prove that for a fixed HH, every graph excluding HH as a topological subgraph has a tree decomposition where each part is either "almost embeddable" to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, we prove that such a decomposition is computable by an algorithm that is fixed-parameter tractable with parameter H|H|. We present two algorithmic applications of our structure theorem. To illustrate the mechanics of a "typical" application of the structure theorem, we show that on graphs excluding HH as a topological subgraph, Partial Dominating Set (find kk vertices whose closed neighborhood has maximum size) can be solved in time f(H,k)nO(1)f(H,k)\cdot n^{O(1)} time. More significantly, we show that on graphs excluding HH as a topological subgraph, Graph Isomorphism can be solved in time nf(H)n^{f(H)}. This result unifies and generalizes two previously known important polynomial-time solvable cases of Graph Isomorphism: bounded-degree graphs and HH-minor free graphs. The proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition
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