Michele Conforti

Balanced matrices

A 0,+/-1 matrix is balanced if, in every submatrix with two nonzero entries per row and column, the sum of the entries is a multiple of 4. This definition was introduced by Truemper and generalizes the notion of balanced 0,1 matrix introduced by Berge. In this tutorial, we survey what is currently known about these matrices: polyhedral results, combinatorial and structural theorems, and recognition algorithms.

Decomposition of Balanced Matrices

A 0, 1 matrix is balanced if it does not contain a square submatrix of odd order with two ones per row and per column. We show that a balanced 0, 1 matrix is either totally unimodular or its bipartite representation has a cutset consisting of two adjacent nodes and some of their neighbors. This result yields a polytime recognition algorithm for balancedness. To prove the result, we first prove a decomposition theorem for balanced 0, 1 matrices that are not strongly balanced.

A branch-and-cut algorithm for the equicut problem

We describe an algorithm for solving the equicut problem on complete graphs. The core of the algorithm is a cutting-plane procedure that exploits a subset of the linear inequalities defining the convex hull of the incidence vectors of the edge sets that define an equicut. The cuts are generated by several separation procedures that will be described in the paper. Whenever the cutting-plane procedure does not terminate with an optimal solution, the algorithm uses a branch-and-cut strategy. We also describe the implementation of the algorithm and the interface with the LP solver. Finally, we report on computational results on dense instances with sizes up to 100 nodes.

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].

Polyhedral approaches to mixed integer linear programming

This survey presents tools from polyhedral theory that are used in integer programming. It applies them to the study of valid inequalities for mixed integer linear sets, such as Gomory’s mixed integer cuts.

Structural properties and decomposition of linear balanced matrices

AbstractClaude Berge defines a (0, 1) matrix A to be linear ifA does not contain a 2 × 2 submatrix of all ones.A(0, 1) matrixA is balanced ifA does not contain a square submatrix of odd order with two ones per row and column.The contraction of a rowi of a matrix consists of the removal of rowi and all the columns that have a 1 in the entry corresponding to rowi. In this paper we show that if a linear balanced matrixA does not belong to a subclass of totally unimodular matrices, thenA orAT contains a rowi such that the submatrix obtained by contracting rowi has a block-diagonal structure.

Square-free perfect graphs

We prove that square-free perfect graphs are bipartite graphs or line graphs of bipartite graphs or have a 2-join or a star cutset.

A class of logic problems solvable by linear programming

In propositional logic, several problems, such as satisfiability, MAX SAT and logical inference, can be formulated as integer programs. In this paper, we consider sets of clauses for which the corresponding integer programs can be solved as linear programs. We prove that balanced sets of clauses have this property.

Compact formulations as a union of polyhedra

We explore one method for finding the convex hull of certain mixed integer sets. The approach is to break up the original set into a small number of subsets, find a compact polyhedral description of the convex hull of each subset, and then take the convex hull of the union of these polyhedra. The resulting extended formulation is then compact, its projection is the convex hull of the original set, and optimization over the mixed integer set is reduced to solving a linear program over the extended formulation.The approach is demonstrated on three different sets: a continuous mixing set with an upper bound and a mixing set with two divisible capacities both arising in lot-sizing, and a single node flow model with divisible capacities that arises as a subproblem in network design.

A polyhedral approach to an integer multicommodity flow problem

Abstract In this paper we propose a branch-and-cut algorithm for the exact solution of an integer multicommodity flow problem. This NP -hard problem finds important applications in transportation, VLSI design, and telecommunications. We consider alternative formulations of the problem and describe several classes of valid inequalities. We describe lifting procedures to extend a given valid inequality for the problem with k commodities, to that having a larger number of commodities. We introduce a new large class of valid constraints, the multi-handle comb inequalities. The polyhedral structure of the integer multicommodity polytope is studied in the case of unit edge capacities. We prove that this polytope is full dimensional and show that some multi-handle comb inequalities are facet defining. Also, the lifting procedures are facet preserving under certain conditions. A branch-and-cut algorithm for the exact solution of the problem is then outlined, and separation algorithms for the main classes of inequalities studied in the paper are proposed. Computational results on several classes of test problems are finally reported, with a comparison between two different formulations.