56 research outputs found
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 Ď„ = 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
Embedded Associated Primes of Powers of Square-free Monomial Ideals
An ideal I in a Noetherian ring R is normally torsion-free if
Ass(R/I^t)=Ass(R/I) for all natural numbers t. We develop a technique to
inductively study normally torsion-free square-free monomial ideals. In
particular, we show that if a square-free monomial ideal I is minimally not
normally torsion-free then the least power t such that I^t has embedded primes
is bigger than beta_1, where beta_1 is the monomial grade of I, which is equal
to the matching number of the hypergraph H(I) associated to I. If in addition I
fails to have the packing property, then embedded primes of I^t do occur when
t=beta_1 +1. As an application, we investigate how these results relate to a
conjecture of Conforti and Cornu\'ejols.Comment: 15 pages, changes have been made to the title, introduction, and
background material, and an example has been added. To appear in JPA
Cuboids, a class of clutters
The Ď„=2 Conjecture, the Replication Conjecture and the f-Flowing Conjecture, and the classification of binary matroids with the sums of circuits property are foundational to Clutter Theory and have far-reaching consequences in Combinatorial Optimization, Matroid Theory and Graph Theory. We prove that these conjectures and result can equivalently be formulated in terms of cuboids, which form a special class of clutters. Cuboids are used as means to (a) manifest the geometry behind primal integrality and dual integrality of set covering linear programs, and (b) reveal a geometric rift between these two properties, in turn explaining why primal integrality does not imply dual integrality for set covering linear programs. Along the way, we see that the geometry supports the Ď„=2 Conjecture. Studying the geometry also leads to over 700 new ideal minimally non-packing clutters over at most 14 elements, a surprising revelation as there was once thought to be only one such clutter
Intersecting restrictions in clutters
A clutter is intersecting if the members do not have a common element yet every two members intersect. It has been conjectured that for clutters without an intersecting minor, total primal integrality and total dual integrality of the corresponding set covering linear system must be equivalent. In this paper, we provide a polynomial characterization of clutters without an intersecting minor. One important class of intersecting clutters comes from projective planes, namely the deltas, while another comes from graphs, namely the blockers of extended odd holes. Using similar techniques, we provide a poly- nomial algorithm for finding a delta or the blocker of an extended odd hole minor in a given clutter. This result is quite surprising as the same problem is NP-hard if the input were the blocker instead of the clutter
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
From coordinate subspaces over finite fields to ideal multipartite uniform clutters
Take a prime power , an integer , and a coordinate subspace
over the Galois field . One can associate with
an -partite -uniform clutter , where every part has size
and there is a bijection between the vectors in and the members of
.
In this paper, we determine when the clutter is ideal, a
property developed in connection to Packing and Covering problems in the areas
of Integer Programming and Combinatorial Optimization. Interestingly, the
characterization differs depending on whether is , a higher power of
, or otherwise. Each characterization uses crucially that idealness is a
minor-closed property: first the list of excluded minors is identified, and
only then is the global structure determined. A key insight is that idealness
of depends solely on the underlying matroid of .
Our theorems also extend from idealness to the stronger max-flow min-cut
property. As a consequence, we prove the Replication and Conjectures
for this class of clutters.Comment: 32 pages, 6 figure
Resistant sets in the unit hypercube
Ideal matrices and clutters are prevalent in Combinatorial Optimization, ranging from balanced matrices, clutters of T-joins, to clutters of rooted arborescences. Most of the known examples of ideal clutters are combinatorial in nature. In this paper, rendered by the recently developed theory of cuboids, we provide a different class of ideal clutters, one that is geometric in nature. The advantage of this new class of ideal clutters is that it allows for infinitely many ideal minimally non-packing clutters. We characterize the densest ideal minimally non-packing clutters of the class. Using the tools developed, we then verify the Replication Conjecture for the class
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