Bertrand Guenin

The Packing Property

Abstract.A clutter (V, E) packs if the smallest number of vertices needed to intersect all the edges (i.e. a minimum transversal) is equal to the maximum number of pairwise disjoint edges (i.e. a maximum matching). This terminology is due to Seymour 1977. A clutter is minimally nonpacking if it does not pack but all its minors pack. An m×n 0,1 matrix is minimally nonpacking if it is the edge-vertex incidence matrix of a minimally nonpacking clutter. Minimally nonpacking matrices can be viewed as the counterpart for the set covering problem of minimally imperfect matrices for the set packing problem. This paper proves several properties of minimally nonpacking clutters and matrices.

Packing odd circuits in Eulerian graphs

Let b be the clutter of odd circuits of a signed graph (G, Σ). For nonnegative integral edge-weights w, we are interested in the linear program min(wtx: x(C) ≥ 1, for C ∈ b and x ≥ 0), which we denote by (P). The problem of solving the related integer program clearly contains the maximum cut problem, which is NP-hard. Guenin proved that (P) has an optimal solution that is integral so long as (G, Σ) does not contain a minor isomorphic to odd-K5. We generalize this by showing that if (G, Σ) does not contain a minor isomorphic to odd-K5 then (P) has an integral optimal solution and its dual has a half-integral optimal solution.

Perfect and Ideal 0, ±1 Matrices

A 0, ±1 matrix A is said to be perfect (resp. ideal) if the corresponding generalized packing ( resp. covering) polytope is integral. Given a 0, ±1 matrix A, we construct a 0, 1 matrix that is perfect if and only if A is perfect. A similar result is obtained for the generalized covering problem. We also extend some known results on perfect 0, 1 matrices to the 0, ±1 case.

A Characterization of Weakly Bipartite Graphs

A labeled graph is said to be weakly bipartite if the clutter of its odd cycles is ideal. Seymour conjectured that a labeled graph is weakly bipartite if and only if it does not contain a minor called an odd K 5. An outline of the proof of this conjecture is given in this paper.

Two Constructive Methods for Designing Compact Feedforward Networks of Threshold Units

We propose two algorithms for constructing and training compact feedforward networks of linear threshold units. The SHIFT procedure constructs networks with a single hidden layer while the PTI constructs multilayered networks. The resulting networks are guaranteed to perform any given task with binary or real-valued inputs. The various experimental results reported for tasks with binary and real-valued inputs indicate that our methods compare favorably with alternative procedures deriving from similar strategies, both in terms of size of the resulting networks and of their generalization properties.

Packing directed circuits exactly

We give an “excluded minor” and a “structural” characterization of digraphs D that have the property that for every subdigraph H of D, the maximum number of disjoint circuits in H is equal to the minimum cardinality of a set T ⊆ V(H) such that H\T is acyclic.

Ideal Binary Clutters, Connectivity, and a Conjecture of Seymour

A binary clutter is the family of odd circuits of a binary matroid, that is, the family of circuits that intersect with odd cardinality a fixed given subset of elements. Let A denote the 0,1 matrix whose rows are the characteristic vectors of the odd circuits. A binary clutter is ideal if the polyhedron

Lehman matrices

\{ x \geq {\bf 0}: \; Ax \geq {\bf 1} \}

Integral Polyhedra Related to Even-Cycle and Even-Cut Matroids

is integral. Examples of ideal binary clutters are st-paths, st-cuts, T-joins or T-cuts in graphs, and odd circuits in weakly bipartite graphs. In 1977, Seymour [J. Combin. Theory Ser. B, 22 (1977), pp. 289--295] conjectured that a binary clutter is ideal if and only if it does not contain

A short proof of Seymour's characterization of the matroids with the max-flow min-cut property

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