Exact arborescences, matchings and cycles

AbstractSuppose we are given a graph in which edge has an integral weight. An ‘exact’ problem is to determine whether a desired structure exists for which the sum of the edge weights is exactly k for some prescribed k.We consider the special case of the problem in which all costs are zero or one for arborescences and show that a ‘continuity’ property is prossessed similar to that possessed by matroids. This enables us to determine in polynomial time the complete set of values of k for which a solution exists. We also give a minmax theorem for the maximum possible value of k, in terms of a packing of certain directed cuts in the graph.We also show how enumerative techniques can be used to solve the general exact problem for arborescences (implying spanning trees), perfect matchings in planar graphs and sets of disjoint cycles in a class of planar directed graphs which includes those of degree three. For these problems, we thereby obtain polynomial algorithms provided that the weights are bounded by a constant or encoded in unary

FACES OF MATCHING POLYHEDRA

Let G = (V, E, ~) be a finite loopless graph, let b=(bi:ieV) be a vector of positive integers. A feasible matching is a vector X = (x.: j e: E) J of nonnegative integers such that for each node i of G, the sum of the over the edges j of G incident with i is no greater than bi. The matching polyhedron P(G, b) is the convex hull of the set of feasible matchings. In Chapter 3 we describe a version of Edmonds' blossom algorithm which solves the problem of maximizing C • X over P (G, b) where c =. (c.: j e: E) J is an arbitrary real vector. This algorithm proves a theorem of Edmonds which gives a set of linear inequalities sufficient to define P(G, b). In Chapter 4 we prescribe the unique subset of these inequalities which are necessary to define P(G, b), that is, we characterize the facets of P(G, b). We also characterize the vertices of P(G, b), thus describing the structure possessed by the members of the minimal set X of feasible matchings of G such that for any real vector c = (c.: j e: E), c • x is maximized over P(G, b) J member of X. by a In Chapter 5 we present a generalization of the blossom algorithm which solves the problem: maximize c • x over a face F of P(G, b) for any real vector c = (c.: j e: E). J In other words, we find a feasible matching x of G which satisfies the constraints obtained by replacing an arbitrary subset of the inequalities which define P(G, b) by equations and which maximizes c • x subject to this restriction. We also describe an application of this algorithm to matching problems having a hierarchy of objective functions, so called ''multi-optimization'' problems. In Chapter 6 we show how the blossom algorithm can be combined with relatively simple initialization algorithms to give an algorithm which solves the following postoptimality problem. Given that we know a matching 0 x £ P(G, b) maximizes c · x over P(G, b), we wish to utilize 0 X which to find a feasible matching x' £ P(G, b') which maximizes c • x over P(G, b'), where b' = (b!: i £ V) ]_ vector of positive integers and arbitrary real vector. c=(c.:j£E) J is a is an In Chapter 7 we describe a computer implementation of the blossom algorithm described herein

A LINEAR PROGRAMMING RELAXATION OF THE NODE PACKING PROBLEM OR 2-BICRITICALGRAPHS AND NODE COVERS

The problem of finding a minimum cardinality set of nodes in a graph which meet every edge is of considerable theoretical as well as practical interest. Because of the difficulty of this problem, a linear relaxation of an integer programming model is sometimes used as a heuristic. In fact Nemhauser and Trotter showed that any variables which receive integer values in an optimal solution to the relaxation can retain the same values in an optimal solution to the integer program. We define 2-bicritical graphs and give several characterizations of them. One characterization is that they are precisely the graphs for which an optimal solution to the linear relaxation will have no integer valued variables. Then we show that almost all graphs are 2-bicritical, and hence the linear relaxation almost never helps.We are currently acquiring citations for the work deposited into this collection. We recognize the distribution rights of this item may have been assigned to another entity, other than the author(s) of the work.If you can provide the citation for this work or you think you own the distribution rights to this work please contact the Institutional Repository Administrator at [email protected]

The perfectly matchable subgraph polytope of an arbitrary graph

Available from Bibliothek des Instituts fuer Weltwirtschaft, ZBW, Duesternbrook Weg 120, D-24105 Kiel C 151960 / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

On cycle cones and polyhedra

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Geometric Duality and Combinatorial Optimization

Many combinatorial optimization problems have natural geometric versions, that is, versions in which the objects are points or lines in Euclidean space and the cost function is given by a planar metric. For example, a Euclidean Traveling Salesman Problem is specified by giving n points in the plane, and then requiring the construction of a tour with minimum Euclidean (L2) length passing through these points. A problem encountered in VLSI design is that of constructing minimum weight Steiner trees on a set of n points in the plane, for which the edge lengths are given by the Manhattan, or L1, metric. For many such problems there are natural ''dual'' problems. These are geometric problems, usually involving optimally packing some shape in the plane, with the property that any feasible solution to the dual problem provides a bound on the optimum solution to the original problem. Moreover, in some cases, the ''best'' such bound is tight; its value equals that of the optimum solution. We describe several such optimization problems and their geometric duals. We also discuss the solvability of these problems

Backward Error Analysis for the Travelling Salesman Problem: Generalized Convexity

We examine the notion of Backward Error Analysis for the Travelling Salesman Problem. One property that establishes the optimality of a tour for a given set of vertices is generalized convexity. Geometrically speaking, we examine the question: How far is a given set of vertices from forming a convex arrangement? There are two metrics involved in this question: The distance metric of the TSP defines the particular type of convexity that we have to consider. The second metric describes the error bounds around the tour vertices, i.e. the amount of perturbation we must apply to a given set to obtain a set that forms a convex arrangement. We consider several combinations of distance metrics and error metrics. We show that it is easy to solve the question for L1 distances and L_{infty} errors and also address the question of L1 distances with L1 errors. Our results can be generalized to polygonal norms. (A polygonal norm L_{cal P} has a centrally symmetric convex 2k-gon as its unit ball.) For Euclidean distances, we show how it can be decided in polynomial time whether a given L_{cal P} error bound around each point is sufficient to transform a point set into a convex arrangement. This is closely related to the notion of convex stabbing: Does a given family of sets allow a convex curve that intersects them all? The notion of convex stabbing was introduced by Tamir in 1987. Goodrich and Snoeyink have given a solution for the special case of a family of parallel line segments. Our result on L_{cal P} norms yields a polynomial solution for the case of a family of congruent convex polygons with a fixed number of vertices. We conclude this paper by discussing difficulties arising from convex stabbing of disks

A Note on the Traveling Preacher Problem

In this paper, we consider a problem of cost allocations for shortest roundtrips. Given a weighted graph G = (V; E), we are to nd a subset S V with a maximum weight core element for the Traveling Preacher game, i.e., a subset S V with a maximum cost allocation x(S) to the vertices, such that there is no nontrivial subset S S of vertices with a total cost allocation x(S ) exceeding the cost of a Traveling Salesman tour visiting the vertices in the subset S. This game can be considered as a variant of the so-called Traveling Salesman game, with the dierence that there is no speci ed central root node for the salesman. We show that this \Traveling Preacher Problem" can be solved in polynomial time, showing that the diculty of nding a core allocation for a combinatorial optimization problem may be caused by the existence of a special depot node, rather than being a consequence of the hardness of the optimization problem itself