Expressing combinatorial optimization problems by Linear Programs

Expressing Combinatorial Optimization Problems by Linear Programs

Many combinatorial optimization problems call for the optimization of a linear function over a certain polytope. Typically, these polytopes have an exponential number of facets. W e explore the problem of finding small l inear programming formulations when one may use any new variables and constraints. W e show that expressing the matching and the Traveling Salesman Problem by a symmetric linea...

Efficiently expressing feasibility problems in Linear Systems, as feasibility problems in Asymptotic-Linear-Programs

We present a polynomial-time algorithm that obtains a set of Asymptotic Linear Programs (ALPs) from a given linear system S, such that one of these ALPs admits a feasible solution if and only if S admits a feasible solution. We also show how to use the same algorithm to determine whether or not S admits a non-trivial solution for any desired subset of its variables. S is allowed to consist of l...

Expressing Combinatorial Optimization Problems by Systems of Polynomial Equations and the Nullstellensatz

Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g. a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution. In the first part of this paper, we construct new polynomial encodings for the problems of finding in a graph its lon...

Linear, Semidefinite Programming and Randomization Methods for Combinatorial Optimization Problems

Semidefinite programs and combinatorial optimization

7 Constraint generation and quadratic inequalities 29 7.1 Example: the stable set polytope again . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.2 Strong insolvability of quadratic equations . . . . . . . . . . . . . . . . . . . . . . . . 30 7.3 Inference rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.4 Algorithmic aspects of inference rules . ...