A characterization of a class of non-binary matroids

A well-known result of Tutte is that U2,4, the 4-point line, is the only non-binary matroid M such that, for every element e, both Mβe and M e, the deletion and contraction of e from M, are binary. This paper characterizes those non-binary matroids M such that, for every element e, Mβe or M e is binary. © 1990

Many 2-level polytopes from matroids

The family of 2-level matroids, that is, matroids whose base polytope is 2-level, has been recently studied and characterized by means of combinatorial properties. 2-level matroids generalize series-parallel graphs, which have been already successfully analyzed from the enumerative perspective. We bring to light some structural properties of 2-level matroids and exploit them for enumerative purposes. Moreover, the counting results are used to show that the number of combinatorially non-equivalent (n-1)-dimensional 2-level polytopes is bounded from below by $c \cdot n^{-5/2} \cdot \rho^{-n}$, where $c\approx 0.03791727$ and $\rho^{-1} \approx 4.88052854$.Comment: revised version, 19 pages, 7 figure

On the structure of spikes

Spikes are an important class of 3-connected matroids. For an integer $r\geq 3$, there is a unique binary r-spike denoted by $Z_{r}$. When a circuit-hyperplane of $Z_{r}$ is relaxed, we obtain another spike and repeating this procedure will produce other non-binary spikes. The $es$-splitting operation on a binary spike of rank $r$, may not yield a spike. In this paper, we give a necessary and sufficient condition for the $es$-splitting operation to construct $Z_{r+1}$ directly from $Z_{r}$. Indeed, all binary spikes and many of non-binary spikes of each rank can be derived from the spike $Z_{3}$ by a sequence of The $es$-splitting operations and circuit-hyperplane relaxations

Polymatroid greedoids

AbstractThis paper discusses polymatroid greedoids, a superclass of them, called local poset greedoids, and their relations to other subclasses of greedoids. Polymatroid greedoids combine in a certain sense the different relaxation concepts of matroids as polymatroids and as greedoids. Some characterization results are given especially for local poset greedoids via excluded minors. General construction principles for intersection of matroids and polymatroid greedoids with shelling structures are given. Furthermore, relations among many subclasses of greedoids which are known so far, are demonstrated

Hypergraphic LP Relaxations for Steiner Trees

We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Koenemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. Our results are the following. Structural results: We extend the technique of uncrossing, usually applied to families of sets, to families of partitions. As a consequence we show that any basic feasible solution to the partition LP formulation has sparse support. Although the number of variables could be exponential, the number of positive variables is at most the number of terminals. Relations with other relaxations: We show the equivalence of the partition LP relaxation with other known hypergraphic relaxations. We also show that these hypergraphic relaxations are equivalent to the well studied bidirected cut relaxation, if the instance is quasibipartite. Integrality gap upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap of these hypergraph relaxations in general graphs. In the special case of uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~ 1.216. By our equivalence theorem, the latter result implies an improved upper bound for the bidirected cut relaxation as well.Comment: Revised full version; a shorter version will appear at IPCO 2010

Approximation Algorithms for Traveling Salesman Problems

The traveling salesman problem is the probably most famous problem in combinatorial optimization. Given a graph G and nonnegative edge costs, we want to find a closed walk in G that visits every vertex at least once and has minimum cost. We consider both the symmetric traveling salesman problem (TSP) where G is an undirected graph and the asymmetric traveling salesman problem (ATSP) where G is a directed graph. We also investigate the unit-weight special cases and the more general path versions, where we do not require the walk to be closed, but to start and end in prescribed vertices s and t. In this thesis we give improved approximation algorithms and better upper bounds on the integrality ratio of the classical linear programming relaxations for several of these traveling salesman problems. For this we use techniques arising from various parts of combinatorial optimization such as linear programming, network flows, ear-decompositions, matroids, and T-joins. Our results include a (22 + &epsilon)-approximation algorithm for ATSP (for any &epsilon > 0), the first constant upper bound on the integrality ratio for s-t-path ATSP, a new upper bound on the integrality ratio for s-t-path TSP, and a black-box reduction from s-t-path TSP to TSP

Algebraic matroids and Frobenius flocks

We show that each algebraic representation of a matroid $M$ in positive characteristic determines a matroid valuation of $M$, which we have named the {\em Lindstr\"om valuation}. If this valuation is trivial, then a linear representation of $M$ in characteristic $p$ can be derived from the algebraic representation. Thus, so-called rigid matroids, which only admit trivial valuations, are algebraic in positive characteristic $p$ if and only if they are linear in characteristic $p$. To construct the Lindstr\"om valuation, we introduce new matroid representations called flocks, and show that each algebraic representation of a matroid induces flock representations.Comment: 21 pages, 1 figur

A Comparison of Steiner Tree Relaxations

There are many (mixed) integer programming formulations of the Steiner problem in networks. The corresponding linear programming relaxations are of great interest particularly, but not exclusively, for computing lower bounds; but not much has been known ab out the relative quality of these relaxations. We compare all classical and some new relaxations from a theoretical point of view with respect to their optimal values. Among other things, we prove that the optimal value of a flowclass relaxation (e.g. the multicommodity flow or the dicut relaxation) cannot be worse than the optimal value of a tree-class relaxation (e.g. degree-constrained spanning tree relaxation) and that the ratio of the corresponding optimal values can be arbitrarily large. Furthermore, we present a new flow based relaxation, which is to the authors' knowledge the strongest linear relaxation of polynomial size for the Steiner problem in networks