397 research outputs found

    On some interconnections between combinatorial optimization and extremal graph theory

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    The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions

    A linear time algorithm for a variant of the max cut problem in series parallel graphs

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    Given a graph G=(V,E)G=(V, E), a connected sides cut (U,V\U)(U, V\backslash U) or δ(U)\delta (U) is the set of edges of E linking all vertices of U to all vertices of V\UV\backslash U such that the induced subgraphs G[U]G[U] and G[V\U]G[V\backslash U] are connected. Given a positive weight function ww defined on EE, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut Ω\Omega such that w(Ω)w(\Omega) is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.Comment: 6 page

    The generalized minimum spanning tree polytope and related polytopes

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    The Generalized Minimum Spanning Tree problem denoted by GMST is a variant of the classical Minimum Spanning Tree problem in which nodes are partitioned into clusters and the problem calls for a minimum cost tree spanning at least one node from each cluster. A different version of the problem, called E-GMST arises when exactly one node from each cluster has to be visited. Both GMST problem and E-GMST problem are NP-hard problems. In this paper, we model GMST problem and E-GMST problem as integer linear programs and study the facial structure of the corresponding polytopes

    Discrete Particle Swarm Optimization for the minimum labelling Steiner tree problem

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    Particle Swarm Optimization is an evolutionary method inspired by the social behaviour of individuals inside swarms in nature. Solutions of the problem are modelled as members of the swarm which fly in the solution space. The evolution is obtained from the continuous movement of the particles that constitute the swarm submitted to the effect of the inertia and the attraction of the members who lead the swarm. This work focuses on a recent Discrete Particle Swarm Optimization for combinatorial optimization, called Jumping Particle Swarm Optimization. Its effectiveness is illustrated on the minimum labelling Steiner tree problem: given an undirected labelled connected graph, the aim is to find a spanning tree covering a given subset of nodes, whose edges have the smallest number of distinct labels