Matroid basis polytope software

We will present sufficient conditions on a matroid m so its base polytope p m has a sequence of hyperplane splits. The ppr potentialbased cohesive zone model is implemented in a commercial software, i. In fact, you can characterize matroids in terms of the properties of their circuits, which are dual to the. Im dealing with an identically selfdual matroid m on the vertices e2n, that is, if b is a basis of m also eb is a basis of m itself. Furedi, z maximum degree and fractional matchings in. They give lots of info about the graph, like what a neighborhood can look like, etc. Based on your location, we recommend that you select. The matroid polytope of mis the polytope p m in re whose vertices are the indicator vectors of the bases of m. Using the software polymake 4, we computed these subdivisions for the vectors in the rays and the vectors in the maximal cones of the fan. Lectures on matroids and oriented matroids victor reiner abstract. Albeit primarily a tool to study the combinatorics and the geometry of convex polytopes and polyhedra, it is by now also capable of dealing with simplicial complexes, matroids, polyhedral fans, graphs, tropical objects. In mathematics, a matroid polytope, also called a matroid basis polytope or basis matroid polytope to distinguish it from other polytopes derived from a matroid, is a polytope constructed via the bases of a matroid.

The vertices of the independence polytope or base polytope of a matroid are the independent sets, respectively bases, of the matroid. It deals with polytopes, polyhedra and fans as well as simplicial complexes, matroids, graphs, tropical hypersurfaces, and other objects. It can be formulated as a 0 1 linear program using the socalled rank and line constraints. On matroid parity and matching polytopes konstantinos kaparis adam n. Consider the following linear program and its dual. A definition of equivalence of matroids is given and it is shown that two matroids are equivalent if and only if their base graphs are isomorphic. Surely someone will give a more thorough answer, but ill give a short, intuitive explanation. Ehrhart polynomials of matroid polytopes and polymatroids.

Perhaps the two most fundamental wellsolved models in combinatorial optimization are the optimal matching problem and the optimal matroid intersection problem. The following celebrated result gives a simple way to check whether or not a polytope is a matroid polytope. In mathematics, a matroid polytope, also called a matroid basis polytope or basis matroid polytope to distinguish it from other polytopes derived from a matroid. I am looking for matroids in which all minimal basis exchanges look the same, that is, the matroid is transitive on these. Matroid polytope subdivisions and valuations by alexander ray fink doctor of philosophy in mathematics university of california, berkeley professor bernd sturmfels, chair many important invariants for matroids and polymatroids are valuations or are valuative, which is to say they satisfy certain relations imposed by subdivisions of matroid. Since the bases polytope is completly described by the locked structure of the matroid, so a natural matroid oracle follows. On matroid parity and matching polytopes konstantinos kaparisy, adam n. These lecture notes were prepared for the algebraic combinatorics in europe ace summer school in vienna, july 2005. While we have stated theorem 2 in terms of bases of m0, the following natural. Isotropical linear spaces and valuated deltamatroids. Randomized pipage rounding for matroid polytopes and applications. For any linear program maximization problem exactly one of the.

In this paper, we prove that the nontrivial facets of the bases polytope of m are described by the locked subsets. A full understanding of the text supposes the reader is familiar with matroid theory 40,41 and oriented matroid theory 3. Supported platforms include various flavors of linux, freebsd and mac os. In the first half of the paper we prove that, for fixed rank, ehrhart polynomials of matroid polytopes and polymatroids are computable in polynomial time. A matroid base polytope decomposition of pm is a decomposition of the form pm st i1. This simple concept of matroid polytope turns out to be an equivalent way to define coxeter matroids.

The maximum of a linear function over a polytope is always attained not necessarily uniquely at a vertex. The following lemma is a stronger basis exchange statement. In order to present the proof we state one of the most useful theorems about matroids. More specifically, the polytope im interested in is the convex hull of the indicators of basis of a matroid. Fast graph and matroid based algorithms allow for the identification. A matroid base polytope decomposition of p m is a decomposition p m t a. Matroids essentially abstract the concept of independence. Points in the dressian of mhave an interesting relationship to the matroid.

These yield to decompositions of p m with two or more pieces for infinitely many matroids m. Two bases are adjacent if they differ by exactly one element. Computers and internet mathematics matrices research matrices mathematics quadratic functions vector spaces vectors mathematics. Matching, matroids, and extensions lehigh university. When 2 s e, m\z denotes the matroid obtained by deleting 2, mz is the matroid obtained by contracting 2. Randomized rounding in matroid polytopes chandra chekuri1 jan vondrak2 rico zenklusen3 1u illinois, urbanachampaign 2ibm almaden research center 3eth zurich april, 2010 chekuri, vondrak, zenklusen matroid rounding april, 2010 1 20. A matroid is called a paving matroid if all of its circuits have size at least equal to its rank. Sometimes this is also referred to as the basis matroid polytope. A base graph of a matroid is the graph whose points are the bases of the matroid. Letchford z, ioannis mourtos x march 2017 abstract the matroid parity mp problem is a natural extension of the matching problem to the matroid setting. Therefore maximizing a linear function over a base polytope is like maximizing it over the collection of bases, for which the greedy algorithm can be used.

For visualization and calculations you can use software such as polymake. Ii both published in jctb free online now since they are older papers. For uniform matroids, the basis polytope is precisely the hypersimplex. We express the matroid polytope p m of a matroid mas a signed minkowski sum of simplices, and obtain a formula for the volume of p. Choose a web site to get translated content where available and see local events and offers.

Let pm be the matroid base polytope of a matroid m. Polymake is software for the algorithmic treatment of convex polyhedra. In mathematics, matroid polytopes are a family of polytopes constructed via the bases of a matroid. June, 2001 published online december 6, 2001 springerverlag 2001 abstract. A matroid m is called a frame matroid if it, or a matroid that contains it, has a basis such that all the points of m are contained in the lines that join pairs of basis elements. For example, there are many rules on how to compute the facets of a polytope. As an application, we give a 1 1e approximation algorithm for the problem of. Polymatroid subdivision polymatroids and their polytopes. A matroid can be specified by its bases, nonbases, circuits, from a matrix, graph, or ideal, or via a collection of predefined matroids. Letchfordy ioannis mourtosz to appear in discrete applied mathematics abstract the. Ipacen is a structure that captures and generalizes the notion of linear independence in vector spaces. The system has all basic matroid oracles and lists of circuits, bases.

The source code of the uel subroutine is provided for a twodimensional linear cohesive element for educational purposes. Pmis said to bedecomposable if it admits a matroid base polytope decomposition with t. Newest matroids questions mathematics stack exchange. For more on how to construct a matroid, see matroid reference oxley, matroid theory, second edition. We investigate properties of ehrhart polynomials for matroid polytopes, independence matroid polytopes, and polymatroids. We assume that we have an efficient independence oracle for the matroid, meaning that we obtain a separation oracle for the polytope via the techniques in testing membership in matroid polyhedra by cunningham. This is a special case of submodular function minimization. Matrix description text the matroid basis polytope of a matroid on n elements lives in rn, and is the convex hull of the indicator vectors of the bases. Alternative definition of and opposite concept to a matroid. The matroid polytope is the convex hull of the indicator vectors of. Can lp for matroid polytopes be solved using the greedy. I need simple combinatorial properties of these, things like. This is a continuation of an early paper chatelain et al. In combinatorics, a branch of mathematics, a matroid template.

Letchfordy ioannis mourtosz to appear in discrete applied mathematics abstract the matroid parity mp problem is a powerful and nphard. Matroids are a combinatorial structure that generalizes, for instance, the concept of families of subspaces. Given a matroid, the matroid polytope is the convex hull of the indicator vectors of the bases of. The known description of the polytope p m by inequalities makes it apparent that it is a generalized permutohedron. This simple concept of matroid polytope turns out to be an. The facets of the bases polytope of a matroid and two. A quasisymmetric function for matroids sciencedirect. A matroid basis polytope is normal in the lattice that it spans. Randomized pipage rounding for matroid polytopes and applications chandra chekuri jan vondr aky september 23, 2009 abstract we present concentration bounds for linear functions of random variables arising from the pipage rounding procedure on matroid polytopes. Let m be a matroid defined on a finite set e and l. Aug 26, 2014 continuing to connect matroid language to linear algebra and graph theory, the maximal independent sets of a matroid are called bases, the size of any basis is the rank of the matroid, and the minimal dependent sets are called circuits.