25 research outputs found
Weighted Linear Matroid Parity
The matroid parity (or matroid matching) problem, introduced as a common generalization of matching and matroid intersection problems, is so general that it requires an exponential number of oracle calls. Nevertheless, Lovasz (1978) showed that this problem admits a min-max formula and a polynomial algorithm for linearly represented matroids. Since then efficient algorithms have been developed for the linear matroid parity problem.
This talk presents a recently developed polynomial-time algorithm for the weighted linear matroid parity problem. The algorithm builds on a polynomial matrix formulation using Pfaffian and adopts a primal-dual approach based on the augmenting path algorithm of Gabow and Stallmann (1986) for the unweighted problem
An algorithm for weighted fractional matroid matching
Let M be a matroid on ground set E. A subset l of E is called a `line' when
its rank equals 1 or 2. Given a set L of lines, a `fractional matching' in
(M,L) is a nonnegative vector x indexed by the lines in L, that satisfies a
system of linear constraints, one for each flat of M. Fractional matchings were
introduced by Vande Vate, who showed that the set of fractional matchings is a
half-integer relaxation of the matroid matching polytope.
It was shown by Chang et al. that a maximum size fractional matching can be
found in polynomial time. In this paper we give a polynomial time algorithm to
find for any given weights on the lines in L, a maximum weight fractional
matching.Comment: 15 page
Matroid matching with Dilworth truncation
Let be a hypergraph and let and be fixed integers. Let be the matroid with ground-set set is independent if and only if each with spans at most hyperedges of . We prove that if is dense enough, then satisfies the double circuit property, thus the min-max formula of Dress and Lovász on the maximum matroid matching holds for . Our result implies the Berge-Tutte formula on the maximum matching of graphs , generalizes Lovász' graphic matroid (cycle matroid) matching formula to hypergraphs and gives a min-max formula for the maximum matroid matching in the 2-dimensional rigidity matroid
Complexity of packing common bases in matroids
One of the most intriguing unsolved questions of matroid optimization is the
characterization of the existence of disjoint common bases of two matroids.
The significance of the problem is well-illustrated by the long list of
conjectures that can be formulated as special cases, such as Woodall's
conjecture on packing disjoint dijoins in a directed graph, or Rota's beautiful
conjecture on rearrangements of bases.
In the present paper we prove that the problem is difficult under the rank
oracle model, i.e., we show that there is no algorithm which decides if the
common ground set of two matroids can be partitioned into common bases by
using a polynomial number of independence queries. Our complexity result holds
even for the very special case when .
Through a series of reductions, we also show that the abstract problem of
packing common bases in two matroids includes the NAE-SAT problem and the
Perfect Even Factor problem in directed graphs. These results in turn imply
that the problem is not only difficult in the independence oracle model but
also includes NP-complete special cases already when , one of the matroids
is a partition matroid, while the other matroid is linear and is given by an
explicit representation.Comment: 14 pages, 9 figure