A polynomial algorithm for the extendability problem in bipartite graphs

On the NP-completeness of the perfect matching free subgraph problem

We consider two generalizations of the edge coloring problem in bipartite graphs. The first problem we consider is the weighted bipartite edge coloring problem where we are given an edge-weighted bipartite graph G = (V,E) with weights w:E→[0,1]. The task is to find a proper weighted coloring of the edges with as few colors as possible. An edge coloring of the weighted graph is called a proper weighted coloring if the sum of the weights of the edges incident to a vertex of any color is at most one. We give a polynomial time algorithm for the weighted bipartite edge coloring problem which returns a proper weighted coloring using at most ⌈2.25 n⌉ colors where n is the maximum total weight incident at any vertex. This improves on the previous best bound of Correa and Goemans [5] which returned a coloring using 2.557n + o(n) colors. The second problem we consider is the Balanced Decomposition of Bipartite graphs problem where we are given a bipartite graph G = (V,E) and α 1,...,α k ∈ (0,1) summing to one. The task is to find a partition E 1,..., E k of E such that \(deg_{E_i}(v)\) is close to α i deg E (v) for each 1 ≤ i ≤ k and v ∈ V. We give an alternate proof of the result of Correa and Goemans [5] that there is a decomposition such that \(\lfloor\alpha_i deg_E(v) \rfloor -2 \leq deg_{E_i}(v) \leq \lceil \alpha_i deg_E(v) \rceil +2\) for each v ∈ V and 1 ≤ i ≤ k. Moreover, we show that the additive error can be improved from two to one if only upper bounds or only lower bounds on the degree are present. All our results hold also for bipartite multigraphs, and the decomposition results hold also for general graphs.KeywordsBipartite GraphWeight GraphEdge IncidentGeneral GraphEdge ColoringThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On the robust assignment problem under a fixed number of cost scenarios

A graph theoretical representation for a class of interconnection networks is suggested. The idea is based on a definition of orthogonal binary vectors and leads to a construction rule for a class of orthogonal graphs. An orthogonal graph is first defined as a set of 2/sup m/ nodes, which in turn are linked by 2/sup m-n/ edges for every link model defined in an integer set Q*. The degree and diameter of an orthogonal graph are determined in terms of the parameters n, m, and the number of link modes defined in Q*. Routing in orthogonal graphs is shown to reduce to the node covering problem in bipartite graphs. The proposed theory is applied to describe a number of well-known interconnection networks such as the binary m-cube and spanning-bus meshes. Multidimensional access (MDA) memories are also shown as examples of orthogonal shared memory multiprocessing systems. Finally, orthogonal graphs are applied to the construction of multistage interconnection networks. Connectivity and placement rules are given and shown to yield a number of well-known networks. >

Online request server matching

The shortest bibranching problem is a common generalization of the minimum-weight edge cover problem in bipartite graphs and the minimum-weight arborescence problem in directed graphs. For the shortest bibranching problem, an efficient primal-dual algorithm is given by Keijsper and Pendavingh (J Comb Theory Ser B 73:130–145, 1998), and the tractability of the problem is ascribed to total dual integrality in a linear programming formulation by Schrijver (Ann Discret Math 16:261–280, 1982). Another view on the tractability of this problem is afforded by a valuated matroid intersection formulation by Takazawa (Jpn J Ind Appl Math 29:561–573, 2012). In the present paper, we discuss the relationship between these two formulations for the shortest bibranching problem. We first demonstrate that the valuated matroid intersection formulation can be derived from the linear programming formulation through the Benders decomposition, where integrality is preserved in the decomposition process and the resulting convex programming is endowed with discrete convexity. We then show how a pair of primal and dual optimal solutions of one formulation is constructed from that of the other formulation, thereby providing a connection between polyhedral combinatorics and discrete convex analysis.

This article proposes a graph-theoretic methodology for query approximation in Geographic Information Systems, enabling the relaxation of three kinds of query constraints: topological, semantic and structural. An approximate query is associated with a value corresponding to the degree of similarity with the original query. Such a value is computed for topological constraints on the basis of the topological distance between configurations, for semantic constraints using the information content approach, and for structural constraints revisiting the maximum weighted matching problem in bipartite graphs. Finally, the high correlation of our proposal with human judgment is demonstrated by an experiment.

In this paper, we deal with both the complexity and the approximability of the labeled perfect matching problem in bipartite graphs. Given a simple graph G = (V,E) with n vertices with a color (or label) function L : E→ {c1,...,cq}, the labeled maximum matching problem consists in finding a maximum matching on G that uses a minimum or a maximum number of colors.

Short Solution of Kotzig's Problem for Bipartite Graphs

In 1982, Papadimitriou and Yannakakis introduced the Exact Matching (EM) problem where given an edge colored graph, with colors red and blue, and an integer k, the goal is to decide whether or not the graph contains a perfect matching with exactly k red edges. Although they conjectured it to be NP-complete, soon after it was shown to be solvable in randomized polynomial time in the seminal work of Mulmuley et al., placing it in the complexity class RP. Since then, all attempts at finding a deterministic algorithm for EM have failed, thus leaving it as one of the few natural combinatorial problems in RP but not known to be contained in P, and making it an interesting instance for testing the hypothesis RP = P. Progress has been lacking even on very restrictive classes of graphs despite the problem being quite well known as evidenced by the number of works citing it. In this paper we aim to gain more insight into EM by studying a new optimization problem we call Top-k Perfect Matching (TkPM) which we show to be polynomially equivalent to EM. By virtue of being an optimization problem, it is more natural to approximate TkPM so we provide approximation algorithms for it. Some of the approximation algorithms rely on a relaxation of EM on bipartite graphs where the output is required to be a perfect matching with a number of red edges differing from k by at most k/2, which is of independent interest and generalizes to the Exact Weight Perfect Matching (EWPM) problem. We also consider parameterized algorithms and show that TkPM can be solved in FPT time parameterized by k and the independence number of the graph. This result again relies on new tools developed for EM which are also of independent interest.

Our topic is an extension of the following classical result of Hall to hypergraphs: A bipartite graph G contains a perfect matching if and only if for each independent set X of vertices, at least |X| vertices of G are adjacent to some vertex of X. Berge generalized the concept of bipartite graphs to hypergraphs by defining a hypergraph G to be balanced if each odd cycle in G has an edge containing at least three vertices of the cycle. Based on this concept, Conforti, Cornuejols, Kapoor, and Vuskovic extended Hall's result by proving that a balanced hypergraph G contains a perfect matching if and only if for any disjoint sets A and B of vertices with |A| > |B|, there is an edge in G containing more vertices in A than in B (for graphs, the latter condition is equivalent to the latter one in Hall's result). Their proof is non-combinatorial and highly based on the theory of linear programming. In the present paper, we give an elementary combinatorial proof.

For any positive integer k, the total k-domatic partition problem is to partition the vertices of a graph G into k pairwise disjoint total dominating sets. In this paper, we study the problem for planar graphs, chordal bipartite graphs, convex bipartite graphs, and bipartite permutation graphs. We show that the total 3-domatic partition problem on planar graphs is NP-complete. Moreover, we give an alternative algorithm to solve the total k-domatic partition problem for chordal bipartite graphs with weak elimination orderings, and adapt it to solve the problem in linear time for bipartite permutation graphs and convex bipartite graphs even if Gamma-free forms of the adjacency matrices of the considered graphs are not given.

A near perfect matching is a matching covering all but one vertex in a graph. Let G be a connected graph with 2n + 3 vertices at least. If any n independent edges in G are contained in a near perfect matching, then G is said to be defect n-extendable. Let G be a defect n-extendable bipartite graph with connectivity not less than two and M be a near perfect matching of G. In this paper, we first characterize G using M-alternating paths, which can be developed into an effective algorithm to determine whether a bipartite graph with connectivity not less than two is defect n-extendable. This problem finds application in the circuit design of allocating jobs to processors when some jobs require specified machines to process. We also study the number of disjoint M-alternating paths between two subsets and between a vertex and a subset of G respectively.

Construction for bicritical graphs and k-extendable bipartite graphs

On the complexity of parallel parsing of general context-free languages