Algorithmic Combinatorics Based on Slicing Posets

Abstract

We show that some recent results in slicing of a distributed computation can be applied to developing algorithms to solve problems in combinatorics. A combinatorial problem usually requires enumerating, counting or ascertaining existence of structures that satisfy a given property B. We cast the combinatorial problem as a distributed computation such that there is a bijection between combinatorial structures satisfying B and the global states that satisfy a property equivalent to B. We then apply results in slicing a computation with respect to a predicate to obtain a small representation of only those global states that satisfy B. The slicing results are based on a generalization of Birkhoff’s Theorem of representation of finite distributive lattices. This gives us an efficient (polynomial time) algorithm to enumerate, count or detect structures that satisfy B when the total set of structures is large but the set of structures satisfying B is small. We illustrate our techniques by analyzing problems in integer partitions, set families, and set of permutations.

Supported in part by the NSF Grants ECS-9907213, CCR-9988225, Texas Education Board Grant ARP-320, an Engineering Foundation Fellowship, and an IBM grant.