Vertex adjacencies in the set covering polyhedron

We describe the adjacency of vertices of the (unbounded version of the) set covering polyhedron, in a similar way to the description given by Chvatal for the stable set polytope. We find a sufficient condition for adjacency, and characterize it with similar conditions in the case where the underlying matrix is row circular. We apply our findings to show a new infinite family of minimally nonideal matrices.Comment: Minor revision, 22 pages, 3 figure

Structure of Cubic Lehman Matrices

A pair $(A,B)$ of square $(0,1)$-matrices is called a \emph{Lehman pair} if $AB^T=J+kI$ for some integer $k\in\{-1,1,2,3,\ldots\}$. In this case $A$ and $B$ are called \emph{Lehman matrices}. This terminology arises because Lehman showed that the rows with the fewest ones in any non-degenerate minimally nonideal (mni) matrix $M$ form a square Lehman submatrix of $M$. Lehman matrices with $k=-1$ are essentially equivalent to \emph{partitionable graphs} (also known as $(\alpha,\omega)$-graphs), so have been heavily studied as part of attempts to directly classify minimal imperfect graphs. In this paper, we view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite graph, focusing in particular on the case where the graph is cubic. From this perspective, we identify two constructions that generate cubic Lehman graphs from smaller Lehman graphs. The most prolific of these constructions involves repeatedly replacing suitable pairs of edges with a particular $6$-vertex subgraph that we call a $3$-rung ladder segment. Two decades ago, L\"{u}tolf \& Margot initiated a computational study of mni matrices and constructed a catalogue containing (among other things) a listing of all cubic Lehman matrices with $k =1$ of order up to $17 \times 17$. We verify their catalogue (which has just one omission), and extend the computational results to $20 \times 20$ matrices. Of the $908$ cubic Lehman matrices (with $k=1$) of order up to $20 \times 20$, only two do not arise from our $3$-rung ladder construction. However these exceptions can be derived from our second construction, and so our two constructions cover all known cubic Lehman matrices with $k=1$

An extension of Lehman's theorem and ideal set functions

Lehman’s theorem on the structure of minimally nonideal clutters is a fundamental result in polyhedral combinatorics. One approach to extending it has been to give a common generalization with the characterization of minimally imperfect clutters (Sebő, 1998; Gasparyan et al., 2003). We give a new generalization of this kind, which combines two types of covering inequalities and works well with the natural definition of minors. We also show how to extend the notion of idealness to unit-increasing set functions, in a way that is compatible with minors and blocking operations

Opposite elements in clutters

Let E be a finite set of elements, and let L be a clutter over ground set E. We say distinct elements e, f are opposite if every member and every minimal cover of L contains at most one of e, f. In this paper, we investigate opposite elements and reveal a rich theory underlying such a seemingly simple restriction. The clutter C obtained from L after identifying some opposite elements is called an identification of L; inversely, L is called a split of C. We will show that splitting preserves three clutter properties, i.e., idealness, the max-flow min-cut property, and the packing property. We will also display several natural examples in which a clutter does not have these properties but a split of them does. We will develop tools for recognizing when splitting is not a useful operation, and as well, we will characterize when identification preserves the three mentioned properties. We will also make connections to spanning arborescences, Steiner trees, comparability graphs, degenerate projective planes, binary clutters, matroids, as well as the results of Menger, Ford and Fulkerson, the Replication Conjecture, and a conjecture on ideal, minimally nonpacking clutters

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

Ideal Clutters

Let E be a finite set of elements, and let C be a family of subsets of E called members. We say that C is a clutter over ground set E if no member is contained in another. The clutter C is ideal if every extreme point of the polyhedron { x>=0 : x(C) >= 1 for every member C } is integral. Ideal clutters are central objects in Combinatorial Optimization, and they have deep connections to several other areas. To integer programmers, they are the underlying structure of set covering integer programs that are easily solvable. To graph theorists, they are manifest in the famous theorems of Edmonds and Johnson on T-joins, of Lucchesi and Younger on dijoins, and of Guenin on the characterization of weakly bipartite graphs; not to mention they are also the set covering analogue of perfect graphs. To matroid theorists, they are abstractions of Seymour’s sums of circuits property as well as his f-flowing property. And finally, to combinatorial optimizers, ideal clutters host many minimax theorems and are extensions of totally unimodular and balanced matrices. This thesis embarks on a mission to develop the theory of general ideal clutters. In the first half of the thesis, we introduce and/or study tools for finding deltas, extended odd holes and their blockers as minors; identically self-blocking clutters; exclusive, coexclusive and opposite pairs; ideal minimally non-packing clutters and the τ = 2 Conjecture; cuboids; cube-idealness; strict polarity; resistance; the sums of circuits property; and minimally non-ideal binary clutters and the f-Flowing Conjecture. While the first half of the thesis includes many broad and high-level contributions that are accessible to a non-expert reader, the second half contains three deep and technical contributions, namely, a character- ization of an infinite family of ideal minimally non-packing clutters, a structure theorem for ±1-resistant sets, and a characterization of the minimally non-ideal binary clutters with a member of cardinality three. In addition to developing the theory of ideal clutters, a main goal of the thesis is to trigger further research on ideal clutters. We hope to have achieved this by introducing a handful of new and exciting conjectures on ideal clutters

A catalog of minimally nonideal matrices

This paper describes a backtracking algorithm for the enumeration of nonisomorphic minimally nonideal (n2n) matrices that are not degenerate projective planes. The application of this algorithm for nh12 yielded 20 such matrices, adding 5 matrices to the 15 previously known. For greater dimensions, n=14 and n=17, 13 new matrices are given. For nonsquare matrices, 38 new minimally nonideal matrices are described

A Catalog of Minimally Nonideal Matrices

This paper describes a backtracking algorithm for the enumeration of nonisomorphic minimally nonideal (n × n) matrices that are not degenerate projective planes. The application of this algorithm for n ≤ 12 yielded 20 such matrices, adding 5 matrices to the 15 previously known. For greater dimensions, n = 14 and n = 17, 13 new matrices are given. For nonsquare matrices, 38 new minimally nonindeal matrices are described