70 research outputs found
A Discrete Convex Min-Max Formula for Box-TDI Polyhedra
A min-max formula is proved for the minimum of an integer-valued separable
discrete convex function where the minimum is taken over the set of integral
elements of a box total dual integral (box-TDI) polyhedron. One variant of the
theorem uses the notion of conjugate function (a fundamental concept in
non-linear optimization) but we also provide another version that avoids
conjugates, and its spirit is conceptually closer to the standard form of
classic min-max theorems in combinatorial optimization. The presented framework
provides a unified background for separable convex minimization over the set of
integral elements of the intersection of two integral base-polyhedra,
submodular flows, L-convex sets, and polyhedra defined by totally unimodular
(TU) matrices. As an unexpected application, we show how a wide class of
inverse combinatorial optimization problems can be covered by this new
framework.Comment: 32 page
Polyhredral techniques in combinatorial optimization I: theory
Combinatorial optimization problems appear in many disciplines ranging
from management and logistics to mathematics, physics, and chemistry. These
problems are usually relatively easy to formulate mathematically, but most
of them are computationally hard due to the restriction that a subset of the
variables have to take integral values. During the last two decades there has
been a remarkable progress in techniques based on the polyhedral description
of combinatorial problems, leading to a large increase in the size of several
problem types that can be solved. The basic idea behind polyhedral techniques
is to derive a good linear formulation of the set of solutions by identifying
linear inequalities that can be proved to be necessary in the description of the
convex hull of feasible solutions. Ideally we can then solve the problem as
a linear programming problem, which can be done eciently. The purpose of
this manuscript is to give an overview of the developments in polyhedral theory,
starting with the pioneering work by Dantzig, Fulkerson and Johnson on the
traveling salesman problem, and by Gomory on integer programming. We also
present some modern applications, and computational experience
Submodular linear programs on forests
A general linear programming model for an order-theoretic analysis of both Edmonds' greedy algorithm for matroids and the NW-corner rule for transportation problems with Monge costs is introduced. This approach includes the model of Queyranne, Spieksma and Tardella (1993) as a special case. We solve the problem by optimal greedy algorithms for rooted forests as underlying structures. Other solvable cases are also discussed
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
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