286,791 research outputs found

    Supermodularity in Unweighted Graph Optimization I: Branchings and Matchings

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    The main result of this paper is motivated by the following two apparently unrelated graph optimization problems: (A) As an extension of Edmonds' disjoint branchings theorem, characterize digraphs comprising k disjoint branchings B-i each having a specified number mu(i) of arcs. (B) As an extension of Ryser's maximum term rank formula, determine the largest possible matching number of simple bipartite graphs complying with degree-constraints. The solutions to these problems and to their generalizations will be obtained from a new min-max theorem on covering a supermodular function by a simple degree-constrained bipartite graph. A specific feature of the result is that its minimum cost extension is already NP-hard. Therefore classic polyhedral tools themselves definitely cannot be sufficient for solving the problem, even though they make some good service in our approach

    Parameterized Rural Postman Problem

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    The Directed Rural Postman Problem (DRPP) can be formulated as follows: given a strongly connected directed multigraph D=(V,A)D=(V,A) with nonnegative integral weights on the arcs, a subset RR of AA and a nonnegative integer ℓ\ell, decide whether DD has a closed directed walk containing every arc of RR and of total weight at most ℓ\ell. Let kk be the number of weakly connected components in the the subgraph of DD induced by RR. Sorge et al. (2012) ask whether the DRPP is fixed-parameter tractable (FPT) when parameterized by kk, i.e., whether there is an algorithm of running time O∗(f(k))O^*(f(k)) where ff is a function of kk only and the O∗O^* notation suppresses polynomial factors. Sorge et al. (2012) note that this question is of significant practical relevance and has been open for more than thirty years. Using an algebraic approach, we prove that DRPP has a randomized algorithm of running time O∗(2k)O^*(2^k) when ℓ\ell is bounded by a polynomial in the number of vertices in DD. We also show that the same result holds for the undirected version of DRPP, where DD is a connected undirected multigraph

    Coloring Sums of Extensions of Certain Graphs

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    Recall that the minimum number of colors that allow a proper coloring of graph GG is called the chromatic number of GG and denoted by χ(G).\chi(G). In this paper the concepts of χ\chi'-chromatic sum and χ+\chi^+-chromatic sum are introduced. The extended graph GxG^x of a graph GG was recently introduced for certain regular graphs. We further the concepts of χ\chi'-chromatic sum and χ+\chi^+-chromatic sum to extended paths and cycles. The paper concludes with \emph{patterned structured} graphs.Comment: 12 page

    Algorithm and Complexity for a Network Assortativity Measure

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    We show that finding a graph realization with the minimum Randi\'c index for a given degree sequence is solvable in polynomial time by formulating the problem as a minimum weight perfect b-matching problem. However, the realization found via this reduction is not guaranteed to be connected. Approximating the minimum weight b-matching problem subject to a connectivity constraint is shown to be NP-Hard. For instances in which the optimal solution to the minimum Randi\'c index problem is not connected, we describe a heuristic to connect the graph using pairwise edge exchanges that preserves the degree sequence. In our computational experiments, the heuristic performs well and the Randi\'c index of the realization after our heuristic is within 3% of the unconstrained optimal value on average. Although we focus on minimizing the Randi\'c index, our results extend to maximizing the Randi\'c index as well. Applications of the Randi\'c index to synchronization of neuronal networks controlling respiration in mammals and to normalizing cortical thickness networks in diagnosing individuals with dementia are provided.Comment: Added additional section on application
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