We consider the problem of optimizing a multivariate quadratic function where each decision variable is constrained to be a complex m’th root of unity. Such problems have applications in signal processing, MIMO detection, and the computation of ground states in statistical physics, among others. In this paper, we construct an integer binary reformulation of the problem based on zonotopes. Compared to the standard reformulation, the number of binary variables is reduced by half for even m. We also provide, with the help of tools from number theory, lower bounds on the number of variables in a real binary reformulation when m is an odd prime or a power of 2. We show on numerical experiments that zonotope reformulation can result in one order of magnitude speedup in solution time.
We study the problem of counting disruption scenarios in a supply chain network (SCN), modeled as combinatorial link failures, that violate a performance threshold. Using duality, the violation set is represented as a union of threshold-defined subsets, each characterized by a linear inequality, enabling exact dynamic programming for counting. Under rational coefficients, we derive a pseudo-polynomial dynamic program conditional on the generated inequalities and, for nonnegative coefficients, an exact capped variant. Computational experiments on four-echelon SCNs demonstrate the approach’s effectiveness.
We study exponential clocks for weighted Set Cover, previously analyzed as a (lnB+1)-approximation by Buchbinder et al. (2018) and Ma (2023). We show that an optimized analysis using positive correlation gives an H(B)-approximation. Combining independent LP rounding with exponential clocks as alteration yields a simple two-stage algorithm that, when the maximum element frequency satisfies f=Θ(lnB), improves over the current best bound of Saket and Sviridenko (2012).
We address the problem of online preemptive single-machine scheduling to minimize the total weighted completion time. We improve the lower bound on the competitive ratio of deterministic algorithms from 1.0730 to 1.1038. Based on this approach and the worst-case instance for the WSRPT (Weighted Shortest Remaining Processing Time) algorithm, we describe a new adversarial strategy that enables the determination of even better lower bounds.
