Non-Model-Based Search Guidance for Set Partitioning Problems

We present a dynamic branching scheme for set partitioning problems. The idea is to trace features of the underlying MIP model and to base search decisions on the features of the current subproblem to be solved. We show how such a system can be trained efficiently by introducing minimal learning bias that traditional model-based machine learning approaches rely on. Experiments on a highly heterogeneous collection of set partitioning instances show significant gains over dynamic search guidance in Cplex as well as instancespecifically tuned pure search heuristics. Search is an integral part of solution approaches for NP-hard combinatorial optimization and decision problems. Once the ability to reason deterministically is exhausted, state-of-theart solvers try out different alternatives which may lead to an improved (in case of optimization) or feasible (in case of satisfaction) solution. This consideration of alternatives may take place highly opportunistically as in local search approaches, or systematically as in backtracking-based methods. Efficiency could be much improved if we could effectively favor alternatives that lead to optimal or feasible solutions and a search space partition that allows short proofs of optimality or infeasibility. After all, the existence of an “oracle” is what distinguishes a non-deterministic from a deterministic Turing machine. This of course means that assuming P != NP , perfect choices are impossible to guarantee. The important insight is to realize that this is a worst-case statement. In practice, we may still hope to be able to make very good choices on average. The view outlined above has motivated research on exploiting statistical methods to guide the search. The idea of using survey propagation in SAT (A. Braunstein 2005) has led to a remarkable performance improvement of systematic solvers for random SAT instances. In stochastic offline programming (Malitsky and Sellmann 2009), biased randomized search decisions are based on an offline training of the solver. This approach later led to the idea of instancespecific algorithm configuration (ISAC (S. Kadioglu 2010)). Here, offline training is used to associate certain features of Copyright c © 2012, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. the problem instance with specific parameter settings for the solver, whereby the latter may include the choice of branching heuristic to be used. In (Samulowitz and Memisevic 2007) branching heuristics for quantified Boolean formulae (QBF) were selected based on the features of the current subproblem which led to more robust performance and solutions to formerly unsolved instances. In this paper, we combine the idea of instance-specific algorithm configuration with the idea of a dynamic branching scheme that bases branching decisions on the features of the current subproblem to be solved. Like the ISAC configuration system, our approach is not based on graphical or any other machine learning models. Instead, we cluster training instances according to their features and determine an assignment of branching heuristics to clusters that results in the best performance when the branching heuristic is dynamically chosen based on the current subproblem’s nearest cluster. We test our approach on the MIP-solver Cplex that we use to tackle set partitioning problems. Our experiments show that this approach can effectively boost search performance even when trained on a rather small set of instances.