Ajai Kapoor

The excluded minors for GF (4)-representable matroids

Abstract There are exactly seven excluded minors for the class of GF (4)-representable matroids.

A Mildly Exponential Time Algorithm for Approximating the Number of Solutions to a Multidimensional Knapsack Problem

We describe a time randomized algorithm that estimates the number of feasible solutions of a multidimensional knapsack problem within 1 ± e of the exact number. (Here r is the number of constraints and n is the number of integer variables.) The algorithm uses a Markov chain to generate an almost uniform random solution to the problem.

Even and odd holes in cap-free graphs

In [J Combin Theory Ser B, 26 (1979), 205–216], Jaeger showed that every graph with 2 edge-disjoint spanning trees admits a nowhere-zero 4-flow. In [J Combin Theory Ser B, 56 (1992), 165–182], Jaeger et al. extended this result by showing that, if A is an abelian group with |A| = 4, then every graph with 2 edge-disjoint spanning trees is A-connected. As graphs with 2 edge-disjoint spanning trees are all collapsible, we in this note improve the latter result by showing that, if A is an abelian group with |A| = 4, then every collapsible graph is A-connected. This allows us to prove the following generalization of Jaegers theorem: Let G be a graph with 2 edge-disjoint spanning trees and let M be an edge cut of G with |M| ≥ 4. Then either any partial nowhere-zero 4-flow on M can be extended to a nowhere-zero 4-flow of the whole graph G, or G can be contracted to one of three configurations, including the wheel of 5 vertices, in which cases certain partial nowhere-zero 4-flows on M cannot be extended. Our results also improve a theorem of Catlin in [J Graph Theory, 13 (1989), 465–483].

Edge-Coloring Bipartite Graphs

Given a bipartite graph G with n nodes, m edges, and maximum degree ?, we find an edge-coloring for G using ? colors in time T+O(mlog?), where T is the time needed to find a perfect matching in a k-regular bipartite graph with O(m) edges and k??. Together with best known bounds for T this implies on O(mlog?+m?logm?log2?) edge-coloring algorithm which improves on the O(mlog?+m?logm?log3?) algorithm of Hopcroft and Cole. Our algorithm can also be used to find a (?+2)-edge-coloring for G in time O(mlog?). The previous best approximation algorithm with the same time bound needed ?+log? colors.

Universally signable graphs

In a graph, a chordless cycle of length greater than three is called a hole. Let γ be a {0, 1} vector whose entries are in one-to-one correspondence with the holes of a graphG. We characterize graphs for which, for all choices of the vector γ, we can pick a subsetF of the edge set ofG such that |F ∪H| эγH (mod 2), for all holesH ofG and |F ∪T| э 1 for all trianglesT ofG. We call these graphsuniversally signable. The subsetF of edges is said to be labelledodd. All other edges are said to be labelledeven. Clearly graphs with no holes (triangulated graphs) are universally signable with a labelling of odd on all edges, for all choices of the vector γ. We give a decomposition theorem which leads to a good characterization of graphs that are universally signable. This is a generalization of a theorem due to Hajnal and Surányi [3] for triangulated graphs.

Balanced 0, ±1 Matrices I. Decomposition

A 0, ±1 matrix is balanced if, in every square submatrix with two nonzero entries per row and column, the sum of the entries is a multiple of four. This paper extends the decomposition of balanced 0, 1 matrices obtained by Conforti, Cornuejols, and Rao (1999, J. Combin. Theory Ser. B77, 292?406) to the class of balanced 0, ±1 matrices. As a consequence, we obtain a polynomial time algorithm for recognizing balanced 0, ±1 matrices.

A theorem of Truemper

An important theorem due to Truemper characterizes the graphs whose edges can be labeled so that all chordless cycles have prescribed parities. This theorem has proven to be an essential tool in the study of various objects like balanced matrices, graphs with no even length chordless cycle and graphs with no odd length chordless cycle with at least five edges. In this paper we prove this theorem in a novel and elementary way and derive some of its consequences. In particular, we show an easy way to obtain Tutte’s characterization of regular matrices.

Perfect, ideal and balanced matrices

Abstract In this paper, we survey results and open problems on perfect, ideal and balanced matrices. These matrices arise in connection with the set packing and set covering problems, two integer programming models with a wide range of applications. We concentrate on some of the beautiful polyhedral results that have been obtained in this area in the last 30 years. This survey first appeared in Ricerca Operativa.

A Mickey-Mouse Decomposition Theorem

A graph is signable to be without odd holes if we can assign labels “even” or “odd” to its edges in such a way that the number of edges labelled “odd” in every triangle is odd and the number of edges labelled “odd” in every chordless cycle of length greater than three is even. Note that a graph has no odd holes if it is signed to be without odd holes with every edge having an “odd” label. We derive a co-NP characterization of such graphs.

Balanced 0, ±1 Matrices II. Recognition Algorithm

In this paper we give a polynomial time recognition algorithm for balanced 0, ±1 matrices. This algorithm is based on a decomposition theorem proved in a companion paper.