Optimal Multi-Way Number Partitioning

Abstract

The NP-hard number-partitioning problem is to separate a multiset S of n positive integers into k subsets such that the largest sum of the integers assigned to any subset is minimized. The classic application is scheduling a set of n jobs with different runtimes on k identical machines such that the makespan, the elapsed time to complete the schedule, is minimized. The two-way number-partitioning decision problem is one of the original 21 problems that Richard Karp proved NP-complete. It is also one of Garey and Johnson's six fundamental NP-complete problems and the only one based on numbers. This article explores algorithms for solving multi-way number-partitioning problems optimally. We explore previous algorithms as well as our own algorithms, which fall into three categories: sequential number partitioning (SNP), a branch-and-bound algorithm; binary-search improved bin completion (BSIBC), a bin-packing algorithm; and cached iterative weakening (CIW), an iterative weakening algorithm. We show experimentally that, for large random numbers, SNP and CIW are state-of-the-art algorithms depending on the values of n and k. Both algorithms outperform the previous state of the art by up to seven orders of magnitude in terms of runtime.

A comparison of instance-level counterfactual explanation algorithms for behavioral and textual data: SEDC, LIME-C and SHAP-C

“…But the algorithm could keep searching, and would be biased towards trying the best features found by LIME (SHAP) before the others, which likely would lead to finding even better explanations faster. Furthermore, we see optimized search algorithms performing quite well for computationally hard problems (Schreiber et al 2018); we conjecture that similar algorithms could be applied in the context of classification from big, sparse data to find optimal explanations fast.…”

Rss++

“…We then solve an optimization problem which assigns buckets of all overloaded cores to either stay in place or to migrate to an underloaded core, according to their fractional load. Techniques for solving multi-way number partitioning, a similar optimization problem (without the transfer constraint and without considering the existing load of underloaded cores), would take tens of seconds [56]. However, an optimal solution is unnecessary.…”

Efficient Distributed Workload (Re-)Embedding

“…However, since we prove our results in the online model of computation, which allows unlimited computational power, the algorithm can solve this NP-complete problem. We note that this problem has also been studied in practice, see, e.g., Schreiber et al [48] and references therein. See Section 5.2.3 for how this problem can be solved approximately at the cost of a constant in the competitive ratio of the algorithm.…”

“…The model and algorithms we study in this paper can also be used to solve an online variant of the k-way partition problem [48]. In the static version of the k-way partition problem one is given a (multi-)set of integers S and the task is to partition S into k subsets S 1 , .…”

“…At the same time, as we will show, the problem features interesting connections to several classic problems. Specifically, the problem can be seen as a distributed version of classic online caching problems [51] or an online version of the k-way partitioning problem [48].…”