We study the complexity of the problems of finding, given a graph G, a largest induced subgraph of G with all degrees odd (called an odd subgraph), and the smallest number of odd subgraphs that partition V(G). We call...
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We study the complexity of the problems of finding, given a graph G, a largest induced subgraph of G with all degrees odd (called an odd subgraph), and the smallest number of odd subgraphs that partition V(G). We call these parameters (G) and chi(odd)(G), respectively. We prove that deciding whether chi(odd)(G) <= q is polynomial-time solvable if q = 2, and NP-complete otherwise. We provide algorithms in time 2O(rw) center dot nO(1) and 2O(q center dot rw) center dot nO(1) to compute chi(odd)(G) and to decide whether chi(odd)(G) = q on n-vertex graphs of rank-width at most rw, respectively, and we prove that the dependency on rank-width is asymptotically optimal under the ETH. Finally, we give some tight bounds for these parameters on restricted graph classes or in relation to other parameters.
BALANCED STABLE MARRIAGE (BSM) is a central optimization version of the classic STABLE Marriage (SM) problem. We study BSM from the viewpoint of parameterized complexity. Informally, the input of BSM consists of n men...
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BALANCED STABLE MARRIAGE (BSM) is a central optimization version of the classic STABLE Marriage (SM) problem. We study BSM from the viewpoint of parameterized complexity. Informally, the input of BSM consists of n men, n women, and an integer k. Each person a has a (sub)set of acceptable partners, A(a), whom a ranks strictly;we use p(a)(b) to denote the position of b is an element of A(a) in a's preference list. The objective is to decide whether there exists a stable matching mu such that balance (mu) (sic) max{Sigma((m,w)is an element of mu)p(m)(w), Sigma((m,w)is an element of mu)p(w)(m)} <= k. In SM, all stable matchings match the same set of agents, A* which can be computed in polynomial time. As balance(mu) = vertical bar A*vertical bar/2 for any stable matching mu, BSM is trivially fixed-parameter tractable (FPT) with respect to k. Thus, a natural question is whether BSM is FPT with respect to k - vertical bar A*vertical bar/2. With this viewpoint in mind, we draw a line between tractability and intractability in relation to the target value. This line separates additional natural parameterizations higher/lower than ours (e.g., we automatically resolve the parameterization k - vertical bar A*vertical bar/2). The two extreme stable matchings are the man-optimal mu(M) and the woman-optimal mu(W). Let O-M = Sigma((m, w)is an element of mu M)p(m)(w), and O-W = Sigma((m,w)is an element of mu W)p(w)(m). In this work, we prove that BSM parameterized by t = k - min{O-M, O-W} admits (1) a kernel where the number of people is linear in t, and (2) a parameterized algorithm whose running time is single exponential in t. BSM parameterized by t = k - max{O-M, O-W} is W[1]-hard. (C) 2021 Published by Elsevier B.V.
Positional games have been introduced by Hales and Jewett in 1963 and have been extensively investigated in the literature since then. These games are played on a hypergraph where two players alternately select an unc...
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Positional games have been introduced by Hales and Jewett in 1963 and have been extensively investigated in the literature since then. These games are played on a hypergraph where two players alternately select an unclaimed vertex of it. In the MakerBreaker convention, if Maker manages to fully take a hyperedge, she wins, otherwise, Breaker is the winner. In the Maker-Maker convention, the first player to take a hyperedge wins, and if no one manages to do it, the game ends by a draw. In both cases, the game stops as soon as Maker has taken a hyperedge. By definition, this family of games does not handle scores and cannot represent games in which players want to maximize a quantity. In this work, we introduce scoring positional games, that consist in playing on a hypergraph until all the vertices are claimed, and by defining the score as the number of hyperedges a player has fully taken. We focus here on INCIDENCE, a scoring positional game played on a 2-uniform hypergraph, i.e. an undirected graph. In this game, two players alternately claim the vertices of a graph and score the number of edges for which they own both end vertices. In the Maker-Breaker version, Maker aims at maximizing the number of edges she owns, while Breaker aims at minimizing it. In the Maker-Maker version, both players try to take more edges than their opponent. We first give some general results on scoring positional games such that their membership in Milnor's universe and some general bounds on the score. We prove that, surprisingly, computing the score in the Maker-Breaker version of INCIDENCE is PSPACE-complete whereas in the Maker-Maker convention, the relative score can be obtained in polynomial time. In addition, for the Maker-Breaker convention, we give a formula for the score on paths by using some equivalences due to Milnor's universe. This result implies that the score on cycles can also be computed in polynomial time. (c) 2023 Elsevier B.V. All rights reserved.
Streaming is a model where an input graph is provided one edge at a time, instead of being able to inspect it at will. In this work, we take a parameterized approach by assuming a vertex cover of the graph is given, b...
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Streaming is a model where an input graph is provided one edge at a time, instead of being able to inspect it at will. In this work, we take a parameterized approach by assuming a vertex cover of the graph is given, building on work of Bishnu et al. [COCOON 2020]. We show the further potency of combining this parameter with the Adjacency List streaming model to obtain results for vertex deletion problems. This includes kernels, parameterized algorithms, and lower bounds for the problems of TI-FREE DELETION, H-FREE DELETION, and the more specific forms of CLUSTER VERTEX DELETION and ODD CYCLE TRANSVERSAL. We focus on the complexity in terms of the number of passes over the input stream, and the memory used. This leads to a pass/memory trade-off, where a different algorithm might be favourable depending on the context and instance. We also discuss implications for parameterized complexity in the non-streaming setting.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).
In the BICLUSTER EDITING (resp., FLIP CONSENSUS TREE) problem the input is a bipartite graph G = (V1, V2, E) and an integer k, and the goal is to decide whether there is a set F c V1 X V2 such that the graph (V1, V2, ...
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In the BICLUSTER EDITING (resp., FLIP CONSENSUS TREE) problem the input is a bipartite graph G = (V1, V2, E) and an integer k, and the goal is to decide whether there is a set F c V1 X V2 such that the graph (V1, V2, EAF) does not contain an induced path on four vertices (resp., an induced path on five vertices whose endpoints are in V2). In this paper we give algorithms for BICLUSTER EDITING and FLIP CONSENSUS TREE whose running times are O *(2.22k) and O(3.24k), respectively. This improves over the O *(2.636k)-time algorithm for BICLUSTER EDITING of Tsur [IPL 2021] and the O*(3.68k)-time algorithm for FLIP CONSENSUS TREE of Komusiewicz and Uhlmann [Algorithmica 2014].(c) 2023 Elsevier B.V. All rights reserved.
Graph-deletion problems involve deleting a small number of vertices so that the resulting graph belong to a given hereditary graph class. We initiate a study of a natural variation of the problem of deletion to scatte...
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Graph-deletion problems involve deleting a small number of vertices so that the resulting graph belong to a given hereditary graph class. We initiate a study of a natural variation of the problem of deletion to scattered graph classes. We want to delete at most k vertices so that each connected component of the resulting graph belongs to one of the constant number of graph classes. As our main result, we show that this problem is non-uniformly fixed-parameter tractable (FPT) when the deletion problem corresponding to each of the constant number of graph classes is known to be FPT and the properties that a graph belongs to these classes are expressible in Counting Monodic Second Order (CMSO) logic. While this is shown using some black box theorems in parameterized complexity, we give a faster FPT algorithm when each of the graph classes has a finite forbidden set. & COPY;2023 Elsevier Inc. All rights reserved.
In the CACTUS VERTEX DELETION (resp., EVEN CYCLE TRANSVERSAL) problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices whose removal from G results in a ...
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In the CACTUS VERTEX DELETION (resp., EVEN CYCLE TRANSVERSAL) problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices whose removal from G results in a graph in which every edge belongs to at most one cycle (resp., a graph without even cycles). In this paper we give deterministic O *(13.69k)-time algorithms for CACTUS VERTEX DELETION and EVEN CYCLE TRANSVERSAL.(c) 2022 Elsevier B.V. All rights reserved.
A connection tree of a graph G for a terminal set W is a tree subgraph T of G such that leaves(T) subset of W subset of V(T). A non-terminal vertex is called linker if its degree in T is exactly 2, and it is called ro...
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A connection tree of a graph G for a terminal set W is a tree subgraph T of G such that leaves(T) subset of W subset of V(T). A non-terminal vertex is called linker if its degree in T is exactly 2, and it is called router if its degree in T is at least 3. The Terminal connection problem (TCP) asks whether G admits a connection tree for W with at most l linkers and at most r routers, while the Steiner tree problem asks whether G admits a connection tree for W with at most k non-terminal vertices. We prove that, if r >= 1 is fixed, then TCP is polynomial-time solvable when restricted to split graphs. This result separates the complexity of TCP from the complexity of Steiner tree, which is known to be NP-complete on split graphs. Additionally, we prove that TCP is NP-complete on strongly chordal graphs, even if r >= 0 is fixed, whereas Steiner tree is known to be polynomial-time solvable. We also prove that, when parameterized by clique-width, TCP is W[1]-hard, whereas STeiner tree is known to be in FPT. On the other hand, agreeing with the complexity of Steiner tree, we prove that TCP is linear-time solvable when restricted to cographs (i.e. graphs of clique-width 2). Finally, we prove that, even if either l >= 0 or r >= 0 is fixed, TCP remains NP-complete on graphs of maximum degree 3.
We investigate the computational complexity of finding temporally disjoint paths and walks in temporal graphs. There, the edge set changes over discrete time steps. Temporal paths and walks use edges that appear at mo...
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We investigate the computational complexity of finding temporally disjoint paths and walks in temporal graphs. There, the edge set changes over discrete time steps. Temporal paths and walks use edges that appear at monotonically increasing time steps. Two paths (or walks) are temporally disjoint if they never visit the same vertex at the same time;otherwise, they interfere. This reflects applications in robotics, traffic routing, or finding safe pathways in dynamically changing networks. At one extreme, we show that on general graphs the problem is computationally hard. The path version is NP-hard even if we want to find only two temporally disjoint paths. The walk version is W-hard (Klobas in IJCAI 4090-4096, 2021) when parameterized by the number of walks. However, it is polynomial-time solvable for any constant number of walks. At the other extreme, restricting the input temporal graph to have a path as underlying graph, quite counter-intuitively, we find NP-hardness in general but also identify natural tractable cases.
In this paper, we study the problem of detecting maximum k-durable structures on temporal graphs, which can be used to mine and analyze more knowledge behind the temporal graphs. We first prove that this problem is NP...
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In this paper, we study the problem of detecting maximum k-durable structures on temporal graphs, which can be used to mine and analyze more knowledge behind the temporal graphs. We first prove that this problem is NP-complete and hard to approximate. Next, we propose an efficient algorithm to detect maximum k-durable structures. The algorithm accelerates the detection process by using an auxiliary graph and several well-designed pruning strategies. Massive experiments on five large temporal social networks demonstrate that our algorithm can save 2-4 orders of magnitude number of recursive invocation and is at least 30x faster than the baseline algorithm.(c) 2023 Elsevier B.V. All rights reserved.
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