Nullity and Loop Complementation for Delta-Matroids

We show that the symmetric difference distance measure for set systems, and more specifically for delta-matroids, corresponds to the notion of nullity for symmetric and skew-symmetric matrices. In particular, as graphs (i.e., symmetric matrices over GF(2)) may be seen as a special class of delta-matroids, this distance measure generalizes the notion of nullity in this case. We characterize delta-matroids in terms of equicardinality of minimal sets with respect to inclusion (in addition we obtain similar characterizations for matroids). In this way, we find that, e.g., the delta-matroids obtained after loop complementation and after pivot on a single element together with the original delta-matroid fulfill the property that two of them have equal "null space" while the third has a larger dimension.Comment: Changes w.r.t. v4: different style, Section 8 is extended, and in addition a few small changes are made in the rest of the paper. 15 pages, no figure

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

Polynomials with the half-plane property and matroid theory

A polynomial f is said to have the half-plane property if there is an open half-plane H, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions regarding multivariate polynomials with the half-plane property and matroid theory. * We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous. * We characterize multivariate multi-affine polynomial with real coefficients that have the half-plane property (with respect to the upper half-plane) in terms of inequalities. This is used to answer two open questions posed by Choe and Wagner regarding strongly Rayleigh matroids. * We prove that the Fano matroid is not the support of a polynomial with the half-plane property. This is the first instance of a matroid which does not appear as the support of a polynomial with the half-plane property and answers a question posed by Choe et al. We also discuss further directions and open problems.Comment: 17 pages. To appear in Adv. Mat

Isotropical Linear Spaces and Valuated Delta-Matroids

The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an n x n skew-symmetric matrix. Its points correspond to n-dimensional isotropic subspaces of a 2n-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type D

Finding a Maximum Restricted tt-Matching via Boolean Edge-CSP

The problem of finding a maximum $2$-matching without short cycles has received significant attention due to its relevance to the Hamilton cycle problem. This problem is generalized to finding a maximum $t$-matching which excludes specified complete $t$-partite subgraphs, where $t$ is a fixed positive integer. The polynomial solvability of this generalized problem remains an open question. In this paper, we present polynomial-time algorithms for the following two cases of this problem: in the first case the forbidden complete $t$-partite subgraphs are edge-disjoint; and in the second case the maximum degree of the input graph is at most $2t-1$. Our result for the first case extends the previous work of Nam (1994) showing the polynomial solvability of the problem of finding a maximum $2$-matching without cycles of length four, where the cycles of length four are vertex-disjoint. The second result expands upon the works of B\'{e}rczi and V\'{e}gh (2010) and Kobayashi and Yin (2012), which focused on graphs with maximum degree at most $t+1$. Our algorithms are obtained from exploiting the discrete structure of restricted $t$-matchings and employing an algorithm for the Boolean edge-CSP.Comment: 20 pages, 2 figure