Klaus Truemper

Alpha-balanced graphs and matrices and GF(3)-representability of matroids

Abstract A graph with {±1} labels on the edges is α-balanced if the sum of the labels on each induced cycle is congruent (mod 4) to the related entry of a given vector α. A {0, ±1} matrix is α-balanced if an associated graph with {±1} labels on the edges is α-balanced. The condition of α-balancedness for a matrix turns out to be equivalent to the requirement that certain elementary submatrices have absolute determinant as specified by the entries of α. First the graphs that may be labelled to become α-balanced for a given vector α are characterized. Subsequently a concept of almost representation of matroids is introduced, and necessary and sufficient conditions for a matroid to be almost represented by a matrix are given. Then these results on almost representation are combined with the characterization of α-balancedness to establish a new characterization of GF (3)-representable and (as a special case) regular matroids. A new characterization of totally unimodular matrices is a corollary. These results imply known characterizations of GF (3)-representable and regular matroids by R. Reid and W.T. Tutte, respectively. They also unify W.T. Tuttes work on regular matroids and P. Camions results for totally unimodular matrices.

A MINSAT Approach for Learning in Logic Domains

This paper describes a method for learning logic relationships that correctly classify a given data set. The method derives from given logic data certain minimum cost satisfiability problems, solves these problems, and deduces from the solutions the desired logic relationships. Uses of the method include data mining, learning logic in expert systems, and identification of critical characteristics for recognition systems. Computational tests have proved that the method is fast and effective.

On the delta-wye reduction for planar graphs

We provide an elementary proof of an important theorem by G. V. Epifanov, according to which every two-terminal planar graph satisfying certain connectivity restrictions can by some sequence of series/parallel reductions and delta-wye exchanges be reduced to the graph consisting of the two terminals and just one edge.

Partial Matroid Representations

A central theorem of matroid 3-connectivity is established that has a number of new and old connectivity results as corollaries. The proof of this theorem relies on a matrix theory developed here for partial matroid representations.

A decomposition theory for matroids. V. Testing of matrix total unimodularity

Abstract We present an O((m + n)3) algorithm for deciding total unimodularity of any real m × n matrix, i.e., for deciding whether or not every square submatrix of the given matrix has determinant 0 or ±1. The algorithm relies on the well-known reduction to the binary case, but from then on is quite different from any method we know of, in particular, from the prior algorithm by W. H. Cunningham and J. Edmonds (Decomposition of linear systems, in preparation) and the recent algorithm by R. E. Bixby, W. H. Cunningham, and A. Rajan (“A Decomposition Algorithm for Matroids,” Working Paper, Rice University 1986), which are of order O((m + n)5) and O((m + n)4.5 (log(m + n))0.5), respectively. The most difficult part of the algorithm, where regularity of a 3-connected, binary, nongraphic, and noncographic matroid must be decided, is handled by a search procedure that relies on the concept of induced decompositions of Parts III and IV. The efficacy of that search procedure crucially depends on asymmetries between certain circuit results and cocircuit results of graphs. In other settings these asymmetries can be quite annoying, but here they turn out to be most beneficial.

On Whitney's 2-isomorphism theorem for graphs

Let G and H be 2-connected 2-isomorphic graphs with n nodes. Whitneys 2-isomorphism theorem states that G may be transformed to a graph G* isomorphic to H by repeated application of a simple operation, which we will term “switching”. We present a proof of Whitneys theorem that is much shorter than the original one, using a graph decomposition by Tutte. The proof also establishes a surprisingly small upper bound, namely n-2, on the minimal number of switchings required to derive G* from G. The bound is sharp in the sense that for any integer N there exist graphs G and H with n ≥ N nodes for which the minimal number of switchings is n-2.

Polynomial algorithms for a class of linear programs

This paper describes several algorithms for solution of linear programs. The algorithms are polynomial when the problem data satisfy certain conditions.

Decomposition and optimization over cycles in binary matroids

Abstract For k = 2 and 3, we define several k-sums of binary matroids and of polytopes arising from cycles of binary matroids. We then establish relationships between these k-sums, and use these results to give a direct proof that a certain LP-relaxation of the cycle polytope is the polytope itself if and only if M does not have certain minors. The latter theorem was proved earlier by Barahona and Grotschel via Seymours deep theorem characterizing the matroids with the sum of circuits property. We also exploit the relationships between matroid and polytope k-sums to construct polynomial time algorithms for the solution of the maximum weight cycle problem for some classes of binary matroids and for the solution of the separation problem of the LP-relaxation mentioned above.

A decomposition theory for matroids. I. General results

Abstract A new matroid decomposition with several attractive properties leads to a new theorem of alternatives for matroids. A strengthened version of this theorem for binary matroids says roughly that to any binary matroid at least one of the following statements must apply: (1) the matroid is decomposable, (2) several elements can be removed (in any order) without destroying 3-connectivity, (3) the matroid belongs to one of 2 well-specified classes or has 10 elements or less. The latter theorem is easily specialized to graphic matroids. These theorems seem particularly useful for the determination of minimal violation matroids, a subject discussed in part II.

A decomposition of the matroids with the max-flow min-cut property

Abstract A matroid has the max-flow min-cut property if a certain circuit packing problem has an optimal solution that is integral whenever the capacities assigned to the elements of the matroid are integral. P.D. Seymour characterized the matroids with this property in terms of minimal forbidden minors. Here we employ a variant of a previously developed decomposition algorithm to produce two decomposition theorems for this matroid class. The first theorem roughly says that any 3-connected matroid of the class is regular, or equal to the Fano matroid, or is a 3-sum. The second theorem is quite similar, but involves a more detailed analysis of the 3-sum case and includes an additional case.