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 $\Delta$-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 $\Delta$-matroids, we then extend the tractability result to larger classes of $\Delta$-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 $\Delta$-matroids by lists of tuples, while the last two use a representation by matrices. Since an $n\times n$ matrix can represent exponentially many tuples, our tractability result is not strictly stronger than the known algorithm for linear and binary $\Delta$-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 “soft” 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 “soft” 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 Edmonds’ 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)

Spanning trees of 3-uniform hypergraphs

Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and related to the class of Pfaffian graphs. We prove a complexity result for recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian 3-graphs -- one of these is given by a forbidden subgraph characterization analogous to Little's for bipartite Pfaffian graphs, and the other consists of a class of partial Steiner triple systems for which the property of being 3-Pfaffian can be reduced to the property of an associated graph being Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are not 3-Pfaffian, none of which can be reduced to any other by deletion or contraction of triples. We also find some necessary or sufficient conditions for the existence of a spanning tree of a 3-graph (much more succinct than can be obtained by the currently fastest polynomial-time algorithm of Gabow and Stallmann for finding a spanning tree) and a superexponential lower bound on the number of spanning trees of a Steiner triple system.Comment: 34 pages, 9 figure

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