A statistical perspective on non-deterministic polynomial-time hard ordering problems: Making use of design for order-of-addition experiments

Abstract

The need for solving NP-hard problems emerges from many fields, including configuration, process monitoring, scheduling, planning, among others. One type of NP-hard problem is the ordering problem, where the goal is to find the optimal sequence. This is, in fact, identical to the purpose of Order-of-Addition (OofA) problem. Based on this connection, this paper proposes a statistical methodology to infer solutions of NP-hard ordering problems via design of OofA experiments. An efficient pairwise-order (PWO) modeling approach is able to identify statistically significant PWO effects, and consequentially find the optimal order(s). In general, the proposed method could solve any NP-hard ordering problems, as long as its underlying structure can be adequately approximated by the PWO model. Empirical studies (Job scheduling problems and run orders problems) show that the proposed methodology often yields the optimal (or nearly optimal) orders. Theoretical support is given under specific setups. This paper is an interdisciplinary collaboration between statistics and optimization, and it pioneers an approach that connects two seemingly unrelated research topics together: the NP-hard ordering problems and the OofA experiments. We hope that the idea of this paper will provide a fresh perspective on NP-hard ordering problems and also motivate fruitful applications of the techniques related to OofA experiments.