arXiv subjects

Xavier Allamigeon

Publications and source records attributed to Xavier Allamigeon.

40 records · Page 3Linked to original sources

arXiv2013

Computing the vertices of tropical polyhedra using directed hypergraphs

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.

math.CO↗

arXiv2010

Tropical polar cones, hypergraph transversals, and mean payoff games

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.

math.CO↗

arXiv2010

The tropical double description method

We develop a tropical analogue of the classical double description method allowing one to compute an internal representation (in terms of vertices) of a polyhedron defined externally (by inequalities). The heart of the tropical algorithm is a characterization of the extreme points of a polyhedron in terms of a system of constraints which define it. We show that checking the extremality of a point reduces to checking whether there is only one minimal strongly connected component in an hypergraph. The latter problem can be solved in almost linear time, which allows us to eliminate quickly redundant generators. We report extensive tests (including benchmarks from an application to static analysis) showing that the method outperforms experimentally the previous ones by orders of magnitude. The present tools also lead to worst case bounds which improve the ones provided by previous methods.

cs.CG↗

arXiv2009

The number of extreme points of tropical polyhedra

The celebrated upper bound theorem of McMullen determines the maximal number of extreme points of a polyhedron in terms of its dimension and the number of constraints which define it, showing that the maximum is attained by the polar of the cyclic polytope. We show that the same bound is valid in the tropical setting, up to a trivial modification. Then, we study the natural candidates to be the maximizing polyhedra, which are the polars of a family of cyclic polytopes equipped with a sign pattern. We construct bijections between the extreme points of these polars and lattice paths depending on the sign pattern, from which we deduce explicit bounds for the number of extreme points, showing in particular that the upper bound is asymptotically tight as the dimension tends to infinity, keeping the number of constraints fixed. When transposed to the classical case, the previous constructions yield some lattice path generalizations of Gale's evenness criterion.

math.CO↗