136 research outputs found
A Convex-Analysis Perspective on Disjunctive Cuts
An updated version of this paper has appeared in Math. Program., Ser. A 106, pp 567-586 (2006), DOI 10.1007/s10107-005-0670-8We treat the general problem of cutting planes with tools from convex analysis. We emphasize the case of disjunctive polyhedra and the generation of facets. We conclude with some considerations on the design of disjunctive cut generators
Idealness and 2-resistant sets
A subset of the unit hypercube {0,1}n is cube-ideal if its convex hull is described by hypercube and generalized set covering inequalities. In this note, we study sets S⊆{0,1}n such that, for any subset X⊆{0,1}n of cardinality at most 2, S∪X is cube-ideal
Optimality certificates for convex minimization and Helly numbers
We consider the problem of minimizing a convex function over a subset of R^n
that is not necessarily convex (minimization of a convex function over the
integer points in a polytope is a special case). We define a family of duals
for this problem and show that, under some natural conditions, strong duality
holds for a dual problem in this family that is more restrictive than
previously considered duals.Comment: 5 page
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
Arc connectivity and submodular flows in digraphs
Let be a digraph. For an integer , a -arc-connected
flip is an arc subset of such that after reversing the arcs in it the
digraph becomes (strongly) -arc-connected.
The first main result of this paper introduces a sufficient condition for the
existence of a -arc-connected flip that is also a submodular flow for a
crossing submodular function. More specifically, given some integer , suppose for all , where and denote the number of arcs
in leaving and entering , respectively. Let be a crossing
family over ground set , and let be a crossing
submodular function such that for
all . Then has a -arc-connected flip such that
for all . The result has several
applications to Graph Orientations and Combinatorial Optimization. In
particular, it strengthens Nash-Williams' so-called weak orientation theorem,
and proves a weaker variant of Woodall's conjecture on digraphs whose
underlying undirected graph is -edge-connected.
The second main result of this paper is even more general. It introduces a
sufficient condition for the existence of capacitated integral solutions to the
intersection of two submodular flow systems. This sufficient condition implies
the classic result of Edmonds and Giles on the box-total dual integrality of a
submodular flow system. It also has the consequence that in a weakly connected
digraph, the intersection of two submodular flow systems is totally dual
integral.Comment: 29 pages, 4 figure
On packing dijoins in digraphs and weighted digraphs
In this paper, we make some progress in addressing Woodall's Conjecture, and
the refuted Edmonds-Giles Conjecture on packing dijoins in unweighted and
weighted digraphs. Let be a digraph, and let . Suppose every dicut has weight at least , for some integer . Let , where each is
the integer in equal to
mod . In this paper, we prove the following results, amongst others: (1)
If , then can be partitioned into a dijoin and a
-dijoin. (2) If , then there is an
equitable -weighted packing of dijoins of size . (3) If
, then there is a -weighted packing of dijoins of size
. (4) If , , and , then can be
partitioned into three dijoins.
Each result is best possible: (1) and (4) do not hold for general , (2)
does not hold for even if , and (3) does not hold
for . The results are rendered possible by a \emph{Decompose,
Lift, and Reduce procedure}, which turns into a set of
\emph{sink-regular weighted -bipartite digraphs}, each of which
is a weighted digraph where every vertex is a sink of weighted degree or
a source of weighted degree , and every dicut has weight at least
. Our results give rise to a number of approaches for resolving Woodall's
Conjecture, fixing the refuted Edmonds-Giles Conjecture, and the
Conjecture for the clutter of minimal dijoins. They also show an intriguing
connection to Barnette's Conjecture.Comment: 71 page
Unique Minimal Liftings for Simplicial Polytopes
For a minimal inequality derived from a maximal lattice-free simplicial
polytope in , we investigate the region where minimal liftings are
uniquely defined, and we characterize when this region covers . We then
use this characterization to show that a minimal inequality derived from a
maximal lattice-free simplex in with exactly one lattice point in the
relative interior of each facet has a unique minimal lifting if and only if all
the vertices of the simplex are lattice points.Comment: 15 page
Cuboids, a class of clutters
The τ=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 τ=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
Disjunctive cuts for cross-sections of the second-order cone
Abstract In this paper we study general two-term disjunctions on affine cross-sections of the secondorder cone. Under some mild assumptions, we derive a closed-form expression for a convex inequality that is valid for such a disjunctive set, and we show that this inequality is sufficient to characterize the closed convex hull of all two-term disjunctions on ellipsoids and paraboloids and a wide class of two-term disjunctions-including split disjunctions-on hyperboloids. Our approach relies on the work of Kılınç-Karzan and Yıldız which considers general two-term disjunctions on the second-order cone
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