117 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
Even Delta-Matroids and the Complexity of Planar Boolean CSPs
The main result of this paper is a generalization of the classical blossom
algorithm for finding perfect matchings. Our algorithm can efficiently solve
Boolean CSPs where each variable appears in exactly two constraints (we call it
edge CSP) and all constraints are even -matroid relations (represented
by lists of tuples). As a consequence of this, we settle the complexity
classification of planar Boolean CSPs started by Dvorak and Kupec.
Using a reduction to even -matroids, we then extend the tractability
result to larger classes of -matroids that we call efficiently
coverable. It properly includes classes that were known to be tractable before,
namely co-independent, compact, local, linear and binary, with the following
caveat: we represent -matroids by lists of tuples, while the last two
use a representation by matrices. Since an matrix can represent
exponentially many tuples, our tractability result is not strictly stronger
than the known algorithm for linear and binary -matroids.Comment: 33 pages, 9 figure
IST Austria Thesis
An instance of the Constraint Satisfaction Problem (CSP) is given by a finite set of
variables, a finite domain of labels, and a set of constraints, each constraint acting on
a subset of the variables. The goal is to find an assignment of labels to its variables
that satisfies all constraints (or decide whether one exists). If we allow more general
âsoftâ constraints, which come with (possibly infinite) costs of particular assignments,
we obtain instances from a richer class called Valued Constraint Satisfaction Problem
(VCSP). There the goal is to find an assignment with minimum total cost.
In this thesis, we focus (assuming that P
6
=
NP) on classifying computational com-
plexity of CSPs and VCSPs under certain restricting conditions. Two results are the core
content of the work. In one of them, we consider VCSPs parametrized by a constraint
language, that is the set of âsoftâ constraints allowed to form the instances, and finish
the complexity classification modulo (missing pieces of) complexity classification for
analogously parametrized CSP. The other result is a generalization of Edmondsâ perfect
matching algorithm. This generalization contributes to complexity classfications in two
ways. First, it gives a new (largest known) polynomial-time solvable class of Boolean
CSPs in which every variable may appear in at most two constraints and second, it
settles full classification of Boolean CSPs with planar drawing (again parametrized by a
constraint language)
A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees
General factors are a generalization of matchings. Given a graph with a
set of feasible degrees, called a degree constraint, for each vertex
of , the general factor problem is to find a (spanning) subgraph of
such that for every of . When all
degree constraints are symmetric -matroids, the problem is solvable in
polynomial time. The weighted general factor problem is to find a general
factor of the maximum total weight in an edge-weighted graph. Strongly
polynomial-time algorithms are only known for weighted general factor problems
that are reducible to the weighted matching problem by gadget constructions.
In this paper, we present the first strongly polynomial-time algorithm for a
type of weighted general factor problems with real-valued edge weights that is
provably not reducible to the weighted matching problem by gadget
constructions
Combinatorial Optimization
Combinatorial Optimization is a very active field that benefits from bringing together ideas from different areas, e.g., graph theory and combinatorics, matroids and submodularity, connectivity and network flows, approximation algorithms and mathematical programming, discrete and computational geometry, discrete and continuous problems, algebraic and geometric methods, and applications. We continued the long tradition of triannual Oberwolfach workshops, bringing together the best researchers from the above areas, discovering new connections, and establishing new and deepening existing international collaborations
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