32 research outputs found
Transitive Packing: A Unifying Concept in Combinatorial Optimization
This paper attempts to give a better understanding of the facial structure of previously separately investigated polyhedra. It introduces the notion of transitive packing and the transitive packing polytope. Polytopes that turn out to be special cases of the transitive packing polytope are, among others, the node packing polytope, the acyclic subdigraph polytope, the bipartite subgraph polytope, the planar subgraph polytope, the clique partitioning polytope, the partition polytope, the transitive acyclic subdigraph polytope, the interval order polytope, and the relatively transitive subgraph polytope. We give cutting plane proofs for several rich classes of valid inequalities of the transitive packing polytope,in this way introducing generalized cycle, generalized clique, generalized antihole, generalized antiweb, and odd partition inequalities. These classes subsume several known classes of valid inequalities for several of the special cases and give also many new inequalities for several other special cases. For some of the classes we also prove a lower bound for their Gomory-Chvdtal rank. Finally, we relate the concept of transitive packing to generalized (set) packing and covering as well as to balanced and ideal matrices
The Complexity of Computing Medians of Relations
The Complexity of Computing Medians of Relation
Computing Optimal Morse Matchings
Morse matchings capture the essential structural information of discrete
Morse functions. We show that computing optimal Morse matchings is NP-hard and
give an integer programming formulation for the problem. Then we present
polyhedral results for the corresponding polytope and report on computational
results
On the typical and atypical solutions to the Kuramoto equations
The Kuramoto model is a dynamical system that models the interaction of
coupled oscillators. There has been much work to effectively bound the number
of equilibria to the Kuramoto model for a given network. By formulating the
Kuramoto equations as a system of algebraic equations, we first relate the
complex root count of the Kuramoto equations to the combinatorics of the
underlying network by showing that the complex root count is generically equal
to the normalized volume of the corresponding adjacency polytope of the
network. We then give explicit algebraic conditions under which this bound is
strict and show that there are conditions where the Kuramoto equations have
infinitely many equilibria.Comment: 28 page
The reversing number of a diagraph
AbstractA minimum reversing set of a diagraph is a smallest sized set of arcs which when reversed makes the diagraph acyclic. We investigate a related issue: Given an acyclic diagraph D, what is the size of a smallest tournament T which has the arc set of D as a minimun reversing set? We show that such a T always exists and define the reversing number of an acyclic diagraph to be the number of vertices in T minus the number of vertices in D. We also derive bounds and exact values of the reversing number for certain classes of acyclic diagraphs
Tropicalizing the simplex algorithm
We develop a tropical analog of the simplex algorithm for linear programming.
In particular, we obtain a combinatorial algorithm to perform one tropical
pivoting step, including the computation of reduced costs, in O(n(m+n)) time,
where m is the number of constraints and n is the dimension.Comment: v1: 35 pages, 7 figures, 4 algorithms; v2: improved presentation, 39
pages, 9 figures, 4 algorithm