26 research outputs found
The Max-Flow Min-Cut Theorem for Countable Networks
We prove a strong version of the Max-Flow Min-Cut theorem for countable
networks, namely that in every such network there exist a flow and a cut that
are "orthogonal" to each other, in the sense that the flow saturates the cut
and is zero on the reverse cut. If the network does not contain infinite trails
then this flow can be chosen to be mundane, i.e. to be a sum of flows along
finite paths. We show that in the presence of infinite trails there may be no
orthogonal pair of a cut and a mundane flow. We finally show that for locally
finite networks there is an orthogonal pair of a cut and a flow that satisfies
Kirchhoff's first law also for ends.Comment: 19 pages, to be published in Journal of Combinatorial Theory, Series
Idealness of k-wise intersecting families
A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that, for some integer k ≥ 4, every k-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for k = 4 for the class of binary clutters. Two key ingredients for our proof are Jaeger’s 8-flow theorem for graphs, and Seymour’s characterization of the binary matroids with the sums of circuits property. As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975. We also discuss connections to the chromatic number of a clutter, projective geometries over the two-element field, uniform cycle covers in graphs, and quarter-integral packings of value two in ideal clutters
Relaxations of the Maximum Flow Minimum Cut Property for Ideal Clutters
Given a family of sets, a covering problem consists of finding a minimum cost collection of elements that hits every set. This objective can always be bound by the maximum number of disjoint sets in the family, we refer to this as the covering dual, since when we allow covers to be fractional and relax the notion of disjoint sets, the natural Linear Programming (LP) formulations become duals and the optimal objective values of the two LPs match. A consequence of the Edmonds-Giles theorem about Totally Dual Integral systems is that if we assume the covering dual always has an optimal integer solution for every cost function, then we can always find an optimal integer cover. The converse does not hold in general, but a still standing conjecture from the mid-1970s states that the existence of an optimal integer cover for every cost function implies the existence of a 1/4-integer optimal solution to the dual for every cost function. In this thesis we discuss weaker versions of the conjecture and build tools allowing us to approach them
The Edmonds-Giles Conjecture and its Relaxations
Given a directed graph, a directed cut is a cut with all arcs oriented in the same direction, and a directed join is a set of arcs which intersects every directed cut at least once. Edmonds and Giles conjectured for all weighted directed graphs, the minimum weight of a directed cut is equal to the maximum size of a packing of directed joins. Unfortunately, the conjecture is false; a counterexample was first given by Schrijver. However its ”dual” statement, that the minimum weight of a dijoin is equal to the maximum number of dicuts in a packing, was shown to be true by Luchessi and Younger.
Various relaxations of the conjecture have been considered; Woodall’s conjecture remains open, which asks the same question for unweighted directed graphs, and Edmond- Giles conjecture was shown to be true in the special case of source-sink connected directed graphs. Following these inquries, this thesis explores different relaxations of the Edmond- Giles conjecture
Rees algebras, Monomial Subrings and Linear Optimization Problems
In this thesis we are interested in studying algebraic properties of monomial
algebras, that can be linked to combinatorial structures, such as graphs and
clutters, and to optimization problems. A goal here is to establish bridges
between commutative algebra, combinatorics and optimization. We study the
normality and the Gorenstein property-as well as the canonical module and the
a-invariant-of Rees algebras and subrings arising from linear optimization
problems. In particular, we study algebraic properties of edge ideals and
algebras associated to uniform clutters with the max-flow min-cut property or
the packing property. We also study algebraic properties of symbolic Rees
algebras of edge ideals of graphs, edge ideals of clique clutters of
comparability graphs, and Stanley-Reisner rings.Comment: PhD thesis, Cinvestav-IPN, June 201
Reconfiguration of basis pairs in regular matroids
In recent years, combinatorial reconfiguration problems have attracted great
attention due to their connection to various topics such as optimization,
counting, enumeration, or sampling. One of the most intriguing open questions
concerns the exchange distance of two matroid basis sequences, a problem that
appears in several areas of computer science and mathematics. In 1980, White
proposed a conjecture for the characterization of two basis sequences being
reachable from each other by symmetric exchanges, which received a significant
interest also in algebra due to its connection to toric ideals and Gr\"obner
bases. In this work, we verify White's conjecture for basis sequences of length
two in regular matroids, a problem that was formulated as a separate question
by Farber, Richter, and Shan and Andres, Hochst\"attler, and Merkel. Most of
previous work on White's conjecture has not considered the question from an
algorithmic perspective. We study the problem from an optimization point of
view: our proof implies a polynomial algorithm for determining a sequence of
symmetric exchanges that transforms a basis pair into another, thus providing
the first polynomial upper bound on the exchange distance of basis pairs in
regular matroids. As a byproduct, we verify a conjecture of Gabow from 1976 on
the serial symmetric exchange property of matroids for the regular case.Comment: 28 pages, 6 figure