What Works Best When? A Systematic Evaluation of Heuristics for Max-Cut and QUBO

Abstract

Though empirical testing is broadly used to evaluate heuristics, there are shortcomings with how it is often applied in practice. In a systematic review of Max-Cut and Quadratic Unconstrained Binary Optimization (QUBO) heuristics papers, we found only 4% publish source code, only 14% compare heuristics with identical termination criteria, and most experiments are performed with an artificial, homogeneous set of problem instances. To address these limitations, we implement and release as open-source a code-base of 10 Max-Cut and 27 QUBO heuristics. We perform heuristic evaluation using cloud computing on a library of 3,296 instances. This large-scale evaluation provides insight into the types of problem instances for which each heuristic performs well or poorly. Because no single heuristic outperforms all others across all problem instances, we use machine learning to predict which heuristic will work best on a previously unseen problem instance, a key question facing practitioners.

Engineering Kernelization for Maximum Cut

“…By examining the performance of each individual reduction rule, we can see that this is solely due to Reduction Rule 1 w=c . These findings could improve the work by de Sousa et al [13], which also affects the work by Dunning et al [14]. In conclusion, our novel reduction rules give us a simple but powerful tool for speeding up existing Table 2: Impact of kernelization on the computation of a maximum cut by LocalSolver (LS) and Biq Mac (BM).…”

“…All algorithms were implemented in C++ and compiled using gcc version 7.3.0 with optimization flag -O3. We use the following state-of-the-art Weighted Max Cut solvers for comparisons: the exact solvers LocalSolver [8] (heuristically finds a large cut, and can then verify if it is maximum), Biq Mac [31] as well as the heuristic solver MqLib [14]. MqLib is unable to determine on its own when it reaches a maximum cut and always exhausts the given time limit.…”

Warm-starting quantum optimization

“…We expect warm-start to be applicable to other problems within Combinatorial Optimization and Integer Programming, for which a good solution can be found through randomized rounding [52], possibly following an encoding into a QUBO [70,71,97], a mixedinteger linear optimization problem [60], or a polynomial unconstrained binary optimization problem [90]. Indeed, both the recipe to obtain SDP relaxations [65] and the analytical tools of Appendix D are applicable to linearly constrained problems equally well.…”

Mining Optimal Policies: A Pattern Recognition Approach to Model Analysis

“…This higher-resolution approach allows us to identify general structural properties rather than predicting a policy for a new instance. Lastly, we note the recent work of Khalil et al (2017a) and Dunning et al (2018), who study the application of ML for designing metaheuristic algorithms for predicting when heuristics work well, and the work of Khalil et al (2017b), who use deep reinforcement learning and solutions to multiple problem instances to develop general heuristics that can solve arbitrary problem instances. These successful applications, however, do not address the issue of interpretabilty and model analysis, which are the main focuses of this work.…”