We classify generic systems of polynomial equations with a single solution, or, equivalently, collections of lattice polytopes of minimal positive mixed volume. As a byproduct, this classification provides an algorith...
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We classify generic systems of polynomial equations with a single solution, or, equivalently, collections of lattice polytopes of minimal positive mixed volume. As a byproduct, this classification provides an algorithm to evaluate the single solution of such a system. (C) 2014 Elsevier Ltd. All rights reserved.
Given a connected graph G = (V, E), a rainbow disconnection k-coloring of G is a k-edge coloring of G such that for each pair of vertices u, v is an element of V, there is a rainbow (edge) cut of u, v, which is a subs...
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Given a connected graph G = (V, E), a rainbow disconnection k-coloring of G is a k-edge coloring of G such that for each pair of vertices u, v is an element of V, there is a rainbow (edge) cut of u, v, which is a subset RC(u, v)subset of E such that u, v are disconnected in G\RC(u, v) and that all edges in RC(u, v) have different colors. Theminimum integer k such that G admits a rainbow disconnection k-coloring is the rainbow disconnection number of G, denoted by rd(G). It has been shown that rd( G) is closely related to the maximum number lambda(+) ( G) of edge disjoint paths among all pairs of vertices in G. In this paper, we propose a polynomial-time algorithm for finding a rainbow disconnection rd(G)-coloring of a 2-tree, a graph G formed by starting with a triangle and then repeatedly adding vertices in such a way that each added vertex is adjacent to two ends of an edge. Moreover, the algorithm shows that rd(G) = lambda(+) (G) if G is a 2-tree. This justifies a conjecture of Bai et al. [MR4385957] in 2-trees, which states that rd( G) is either lambda(+) (G) or lambda(+) (G) + 1 for each graph G.
It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradient-like systems can be reduced to a combinatorial ...
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It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradient-like systems can be reduced to a combinatorial problem of distinguishing graphs up to isomorphism. The isomorphism problem of general graphs obviously can be solved by a standard enumeration algorithm. However, an efficient algorithm (i. e., polynomial in the number of vertices) has not yet been developed for it, and the problem has not been proved to be intractable (i. e., NPcomplete). We give polynomial-time algorithms for recognition of the corresponding graphs for two gradient-like systems. Moreover, we present efficient algorithms for determining the orientability and the genus of the ambient surface. This result, in particular, sheds light on the classification of configurations that arise from simple, point-source potential-field models in efforts to determine the nature of the quiet-Sun magnetic field.
We consider the problem of finding a popular matching in the Weighted Capacitated House Allocation problem (WCHA). An instance of WCHA involves a set of agents and a set of houses. Each agent has a positive weight ind...
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We consider the problem of finding a popular matching in the Weighted Capacitated House Allocation problem (WCHA). An instance of WCHA involves a set of agents and a set of houses. Each agent has a positive weight indicating his priority, and a preference list in which a subset of houses are ranked in strict order. Each house has a capacity that indicates the maximum number of agents who could be matched to it. A matching M of agents to houses is popular if there is no other matching M' such that the total weight of the agents who prefer their allocation in M to that in M exceeds the total weight of the agents who prefer their allocation in M to that in M'. Here, we give an O(root Cn1 + m) algorithm to determine if an instance of WCHA admits a popular matching, and if so, to find a largest such matching, where C is the total capacity of the houses, n(1) is the number of agents, and m is the total length of the agents' preference lists. (C) 2009 Elsevier B.V. All rights reserved.
For a given graph G and p pairs (s(i), t(i)), 1 less than or equal to i less than or equal to p, of vertices in G, the edge-disjoint paths problem is to find p pairwise edge-disjoint paths P-i, 1 less than or equal to...
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For a given graph G and p pairs (s(i), t(i)), 1 less than or equal to i less than or equal to p, of vertices in G, the edge-disjoint paths problem is to find p pairwise edge-disjoint paths P-i, 1 less than or equal to i less than or equal to p, connecting s(i) and t(i). Many combinatorial problems can be efficiently solved for partial k-trees (graphs of treewidth bounded by a fixed integer k), but the edge-disjoint paths problem is NP-complete even for partial 3-trees. This paper gives two algorithms for the edge-disjoint paths problem on partial k-trees. The first one serves the problem for any partial k-tree G and runs in polynomialtime if p = O(log n) and in linear time if p = O(1), where n is the number of vertices in G. The second one solves the problem under some restriction on the location of terminal pairs even if p greater than or equal to log n.
The independence gap of a graph was introduced by Ekim et al. in 2018 as a measure of how far a graph is from being well-covered. It is defined as the difference between the maximum and minimum size of a maximal indep...
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The independence gap of a graph was introduced by Ekim et al. in 2018 as a measure of how far a graph is from being well-covered. It is defined as the difference between the maximum and minimum size of a maximal independent set. We investigate the independence gap of a graph from structural and algorithmic points of view, with a focus on classes of perfect graphs. Generalizing results on well-covered graphs due to Dean and Zito (1994) and Hujdurovic et al. (2018), we express the independence gap of a perfect graph in terms of clique partitions and use this characterization to develop a polynomial-time algorithm for recognizing graphs of constant independence gap in any class of perfect graphs of bounded clique number. Next, we introduce a hereditary variant of the parameter, which we call hereditary independence gap and which measures the maximum independence gap over all induced subgraphs of the graph. We show that determining whether a given graph has hereditary independence gap at most k is polynomial-time solvable if k is fixed and co-NP-complete if k is part of input. We also investigate the complexity of the independent set and independent domination problems in graph classes related to independence gap. In particular, we show that the two problems are NP-complete in the class of graphs of independence gap at most one and that the weighted independent set problem is polynomial-time solvable in any class of graphs with bounded hereditary independence gap. Combined with some known results on claw-free graphs, our results imply that the independent domination problem is solvable in polynomialtime in the class of {claw, 2P(3)}-free graphs. (C) 2020 Elsevier B.V. All rights reserved.
A cycle cover of a graph is a spanning subgraph whose connected components are simple cycles. Given a complete weighted directed graph, consider the intractable problem of finding a maximum-weight cycle cover which sa...
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A cycle cover of a graph is a spanning subgraph whose connected components are simple cycles. Given a complete weighted directed graph, consider the intractable problem of finding a maximum-weight cycle cover which satisfies an upper bound on the number of cycles and a lower bound on the number of edges in each cycle. We suggest a polynomial-time algorithm for solving this problem in the geometric case where the vertices of the graph are points in a multidimensional real space and the distances between them are induced by a positively homogeneous function whose unit ball is an arbitrary convex polytope with a fixed number of facets. The obtained result extends the ideas underlying the well-known algorithm for the polyhedral Max TSP.
A cycle base of a permutation group is defined to be a maximal set of its pairwise non-conjugate regular cyclic subgroups. It is proved in this paper that a cycle base of a permutation group of degree n can be constru...
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A cycle base of a permutation group is defined to be a maximal set of its pairwise non-conjugate regular cyclic subgroups. It is proved in this paper that a cycle base of a permutation group of degree n can be constructed in polynomialtime in n. (C) 2018 Elsevier Inc. All rights reserved.
We obtain a polynomial-time algorithm to compute the supremum of the max-min powers of a map from the Cartesian product of a set to a bounded subset of the real numbers. It is also shown that the value at a pair (x, y...
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We obtain a polynomial-time algorithm to compute the supremum of the max-min powers of a map from the Cartesian product of a set to a bounded subset of the real numbers. It is also shown that the value at a pair (x, y) is equal to the maximum of the first k powers of the map where k is the diameter between x and y in a graph corresponding to the map. A characterization is obtained of the metric introduced by Rosenfeld (1975) for a fuzzy graph.
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