On Decomposing Knapsack Constraints for Length-Lex Bounds Consistency

Abstract

The length-lex representation for set variables orders all subsets of a given universe of values according to cardinality and lexicography. To achieve length-lex bounds consistency for Knapsack constraints it has been proposed to decompose the constraint into two sum constraints. We provide theoretical and practical evidence which shows that decomposition increases the problem of computing a fixpoint which is intrinsic to the length-lex representation: 1. The fixpoint problem for this domain representation is NP-hard in general. 2. For a tractable sub-family of Knapsack decomposition takes more time than exponential brute-force enumeration. 3. Experimental results on decomposed Knapsack constraints show that exponential-time fixpoint computation is the rule and not some pathological exception.

This work was supported by the National Science Foundation through the Career: Cornflower Project (award number 0644113).