David Hartvigsen

Packing subgraphs in a graph

We show that the problem of packing edges and triangles in a graph in order to cover the maximum number of nodes can be solved in polynomial time. More generally we present results for the problem of packing edges and a family of hypomatchable subgraphs.

An extension of matching theory

Abstract Given a graph G and a family H of hypomatchable subgraphs of G, we introduce the notion of a hypomatching of G relative to H as a collection of node disjoint edges and subgraphs, where the subgraphs all belong to H. Examples include matchings (H = O), fractional matchings (H contains all the hypomatchable subgraphs of G), and edge-and-triangle packings (H is the set of 3-cliques of G). We show that many of the classical theorems about maximum cardinality matchings can be extended to hypomatchings which cover the maximum number of nodes in a graph.

The All-Pairs Min Cut Problem and the Minimum Cycle Basis Problem on Planar Graphs

The all-pairs min cut (APMC) problem on a normegative weighted graph is the problem of finding, for every pair of nodes, the min cut separating the pair. It is shown that on planar graphs, the APMC problem is equivalent to another problem, the minimum cycle basis (MCB) problem, on the dual graph. This is shown by characterizing the structure of MCBs on planar graphs in several ways. This leads to a new algorithm for solving both of these problems on planar graphs. The complexity of this algorithm equals that of the best algorithm for either problem, but is simpler.

Finding maximum square-free 2-matchings in bipartite graphs

A 2-matching in a simple graph is a subset of edges such that every node of the graph is incident with at most two edges of the subset. A maximum 2-matching is a 2-matching of maximum size. The problem of finding a maximum 2-matching is a relaxation of the problem of finding a Hamilton tour in a graph. In this paper we study, in bipartite graphs, a problem of intermediate difficulty: The problem of finding a maximum 2-matching that contains no 4-cycles. Our main result is a polynomial time algorithm for this problem. We also present a min-max theorem.

Recognizing Voronoi Diagrams with Linear Programming

Let P = { P 1 , …, P n } be a finite set of distinct points in ℝ d and let R i be the set of points whose distance from P i is less than or equal to the distance from every other point in P . The collection of regions R 1 , …, R n is called the Voronoi diagram generated by P . Our main result is a polynomial time algorithm for recognizing whether or not a given tessellation of ℝ d into polyhedra is a Voronoi diagram. This is accomplished by describing a linear program that has a solution if and only if the given tessellation is a Voronoi diagram. As a consequence, for each R i of a Voronoi diagram, the set of points in R i contained in some generating set P is either a singleton or the interior of a polyhedron. We also give a polynomial time algorithm for describing this set for each R i . Finally, this leads to a second algorithm for recognizing Voronoi diagrams; this algorithm also relies on linear programming but, for fixed dimension d , it is strongly polynomial and has linear time complexity. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Is every cycle basis fundamental

A collection of (simple) cycles in a graph is called fundamental if they form a basis for the cycle space and if they can be ordered such that Cj(C1 U … U Cj-1) ≠ O for all j. We characterize by excluded minors those graphs for which every cycle basis is fundamental. We also give a constructive characterization that leads to a (polynomial) algorithm for recognizing these graphs. In addition, this algorithm can be used to determine if a graph has a cycle basis that covers every edge two or more times. An equivalent dual characterization for the cutset space is also given.

Representing the strengths and directions of pairwise comparisons

We consider the problem of finding weights that well represent a set of pairwise multiplicative comparisons of a set of objects (as in the AHP and other methods). Our main contribution is a method for deriving such weights that takes into consideration not only the strengths of the pairwise comparisons, but also their directions. For example, if the comparison directions satisfy transitivity, then the weights produced by our method also satisfy transitivity (this is not always true for other methods). We also present a set of reasonable axioms for which our method is the (essentially) unique solution. Our method and axioms are closely related to those of Cook and Kress [Eur. J. Oper. Res. 37 (1988) 335]. Our method, like theirs, reduces to solving a linear program (hence it is different from the approach used in the AHP). For the special case that the comparison directions satisfy transitivity, our method is quite simple and reduces to performing a forward pass as in the critical path method.

When do short cycles generate the cycle space

Let G = (V, E) be a graph with arbitrary (perturbed) edge weights and let C(e) denote the shortest cycle containing the edge e. It is easy to show that the cycles in {C(e) | e ∈ E} are not only independent (over GF(2)) but are also contained in the cycle basis of minimum weight. We characterize, in several ways, those graphs for which {C(e) | e ∈ E} is a cycle basis (hence, the cycle basis of minimum weight) for every perturbed edge weighting. For example, these are the planar graphs such that no dual graph has two non-adjacent nodes connected by three internally node-disjoint paths. Another characterization shows that these graphs can be obtained from cycles, bonds, and K4′s by a special type of 2-sum operation; this leads to a linear time recognition algorithm for this class.

Minimum Path Bases

Abstract In this paper we introduce a new vector space associated with the paths and cycles in a graph (this space properly includes the well-known cycle space). We present a polynomial algorithm for finding a minimum weight basis for this space; we then present an application of this algorithm. The application is an algorithm that finds in a graph a minimum weight subgraph whose cycle space has specified dimension k (when k = 1, this is the problem of finding a minimum weight cycle in a graph). We then show how this algorithm easily provides a new algorithm for the planar k-split problem ; for a planar graph, this is the problem of finding a minimum weight set of edges whose deletion results in a graph with k components (when k = 2, this is the problem of finding a minimum cut in a planar graph). The algorithm for the application is polynomial for fixed k . In particular, for the planar three-split problem, our algorithm can be implemented with worst case time complexity O (| V | 3 ), whereas the best previous algorithm has complexity O (| V | 6 log| V |).

The complexity of lifted inequalities for the knapsack problem

Abstract It is well known that one can obtain facets and valid inequalities for the knapsack polytope by lifting simple inequalities associated with minimal covers. We study the complexity of lifting. We show that recognizing integral lifted facets or valid inequalities can be done in O(n2) time, even if the minimal cover from which they are lifted is not given. We show that the complexities of recognizing nonintegral lifted facets and valid inequalities are similar, respectively, to those of recognizing general (not necessarily lifted) facets and valid inequalities. Finally, we show that recognizing valid inequalities is in co- NPC while recognizing facets is in Dn. The question of whether recognizing facets is complete for Dn is open.