Partial polyhedral description and generation of discrete optimization problems with known optima

Abstract

We detail a random cut concept for generating instances of discrete optimization problems based on a partial description of the polytope of solutions. We show how implementations of this approach have the useful properties that an optimal solution and the form of valid equalities required to solve the problem by cutting methods are both known at the completion of generation. The former makes possible large‐scale testing of heuristics, and the latter facilitates cutting algorithm research. The random cut concept of problem generation is first discussed in general and then details are provided on its implementation for symmetric traveling salesman problems.

Analysis of a Random Cut Test Instance Generator for the TSP

“…These polyhedral considerations led us to propose a random cut generation scheme [10,11] based on creating instances drawn from the large subset that could, in principle, be solved over an appropriate linear programming relaxation. To be more specific, standard linear programming optimality conditions establish that a vector x * is optimal in ROP T k above if there exist dual vectors u 0 , u 1 .…”

“…Pilcher and Rardin [11,12] developed a TIG in use for the TSP based on a random cut method. This TIG was later extended by Rais and Rardin [13].…”

From near to eternity: Spin-glass planting, tiling puzzles, and constraint-satisfaction problems

“…Evaluation of such algorithms often requires access to benchmarking problems of various types; ideally their difficulty should also be a controllable (or tunable) property. Ideas of generating test instances with planted solutions, that is, whose optimizing values are known to the problem constructor, have been explored in various fields for decades [1][2][3]. A method of planting solutions to hard random Boolean satisfiability (SAT) problems based on statistical mechanics was first proposed in Ref.…”