Heuristic Certificates via Approximations

This paper suggests a new framework in which the quality of a (not necessarily optimal) heuristic solution is certified by an approximation algorithm. Namely, the result of a heuristic solution is accompanied by a scale obtained from an approximation algorithm. The creation of a scale is efficient whereas solutions obtained from an approximation algorithm usually involve long calculation when compared to a heuristic approach. On the other hand, a result obtained by heuristics without a scale might not be useful. We investigate criteria for choosing an approximation scheme for producing a scale. To obtain a scale in practice, we examine approximations not only by their asymptotic behavior, but also examine relations as a function of the input size of a given problem. We examine, as case studies only, heuristic and approximation algorithms for the SINGLE KNAPSACK, MAX 3-SAT, and MAXIMUM BOUNDED 3-DIMENSIONAL MATCHING (MB3DM) NP-hard problems. We obtain certificates for the heuristic runs by using fitting approximations. Within the scope of distributed computing one may execute two distributed Algorithms, one that is based on approximation and another on heuristics. The approximation result is used to certify the heuristic result and stop (the possibly exponential time) heuristic search. Thus, the reliability of the obtained solution can be estimated and certificated.