This paper investigates a presolving method for handling symmetries in mixed-binary programs, based on inequalities computed from so-called Schreier-Sims tables. We show that an iterative application of this method together with merging variables will produce an instance for which the symmetry group is trivial. We then prove that the problem structure can be preserved for certain monotone binary problems, e.g., packing and covering integer programs, maximum stable set (or clique) problems and set covering. The result is then a changed instance of the same type such that existing black-box solvers can directly be applied. We showcase our approach on stable set problems via computational experiments. For a mixed set of graphs, the presolving method significantly outperforms a black-box branch-and-cut method, while being close to the performance of state-of-the-art symmetry handling methods. For Johnson and coding theory graphs, we significantly outperform the stand-alone solver with and without its integrated symmetry handling. A similar speed-up can be observed when running the maximum clique solver CliSAT on the Johnson graphs.
The branch-and-cut paradigm has led to effective solvers for a wide range of mathematical optimization problems. Despite the extensive empirical success and significant theoretical work on pure branching and pure cutting, little is theoretically known about their interaction. In this paper, we study an extended abstract branch-and-cut model and investigate tradeoffs between branching and cutting within the framework. We prove that there always exists a minimal-time tree following a simple root-cut structure, where all cuts are placed before the first branching node, for certain families of settings. For others, although the root-cut structure may be suboptimal and differ substantially from the minimal-time tree, it nevertheless provides a reliable approximation that does not deviate arbitrarily from the actual minimal-time tree. In addition, we identify conditions under which minimal-time trees necessarily contain local cuts, i.e., cuts added after some branching decisions, under a single-cut-per-path restriction. Lastly, simulations indicate that trees restricted to only root cuts significantly outperform our proved worst-case approximation guarantee.
We introduce a novel variant of stochastic load balancing that enables a quantitative trade-off between the practical benefits of non-adaptive policies and their performance limitations. Our model describes a solution in two stages. In the first stage, given only job-size distributions, we reserve a set of at most k machines for each job (a k-reservation). In the second stage, after observing job-size realizations, we assign each job to one of its reserved machines (a consistent assignment). The goal is to minimize the expected makespan. If k=1 , we get the standard stochastic load balancing problem of finding a non-adaptive assignment with minimum expected makespan, and if k is equal to the number of machines, then we obtain an all-powerful omniscient optimum that can tailor the assignment arbitrarily to the job-size realizations. We give a number of results that quantify this trade-off. Most saliently, we show that in the setting of identical machines, a 2-reservation suffices to achieve a constant-factor approximation to the omniscient optimum, establishing a “power-of-two-choices” result for stochastic load balancing. We also show that this no longer holds true in the more challenging setting of related machines. Nonetheless, we give a number of positive algorithmic results for this setting: a true O(log m/loglog m) -approximation; a bicriteria O(1)-approximation by reserving twice as many machines per job relative to an optimal k-reservation; and a 2-reservation whose cost is within a constant factor of the adaptive optimum.
We propose new abstract and unified perspectives on a range of scheduling and graph coloring problems with general min-sum objectives. Specifically, we consider various problems where the objective function is the weighted sum of completion times over groups of entities (jobs, vertices, or edges), thereby generalizing two important objectives in scheduling: makespan and the sum of weighted completion times. As one of our main results, we present a best-possible 𝒪(log g) -competitive algorithm in the non-clairvoyant online setting, where g denotes the size of the largest group. This is the first non-trivial competitive bound for several problems with group completion time objective, and it is an exponential improvement over previous results for non-clairvoyant coflow scheduling. For offline scheduling, we provide elegant yet powerful meta-frameworks that, in a unifying way, yield new or stronger approximation algorithms for our new abstract problems as well as for previously well-studied special cases.
Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite relaxation and propose a randomized rounding algorithm to generate feasible solutions from the relaxation. Second, we derive purely multiplicative approximation guarantees for our algorithm. When optimizing for m orthogonal vectors in dimension n, we show that our algorithm achieves a performance ratio of at least max{2π m, 1π (log (2m)+1)} . Our analysis is tight in the sense that we exhibit instances where our algorithm’s performance is at most O(1/log m) . We also show how to compute a tighter constant for finite (n, m) by solving a univariate optimization problem, and this analysis is exact for any n when m=1 .
