This paper explores generalizations of the Goemans-Williamson randomization technique. It establishes a simple equivalence of binary linear feasibility problems and max-cut problems and presents an analysis of the semidefinite max-cut relaxation for the case of a single linear equation. Numerical examples for feasible random binary problems indicate that the randomization technique is efficient when the … Read more
Inspired by a question of Lov\’asz, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lov{\’a}sz’s theta body of the graph. … Read more
We study minimizing the sum of weighted completion times in a concurrent open shop. We give a primal-dual 2-approximation algorithm for this problem. We also show that several natural linear programming relaxations for this problem have an integrality gap of 2. Finally, we show that this problem is inapproximable within a factor strictly less than … Read more
Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. In this paper, we give the first constant-factor approximation algorithm for maximizing any … Read more
The number partitioning problem (NPP) is to divide n numbers a_1,…,a_n into two disjoint subsets such that the difference between the two subset sums – the discrepancy, D, is minimized. In the balanced version of NPP (BalNPP), the subsets must have the same cardinality. With $a_j$s chosen uniformly from $[1,R]$, R > 2^n gives the … Read more
This work addresses the computation of free-energy di fferences between protein conformations by using morphing (i.e., transformation) of a source conformation into a target conformation. To enhance the morph- ing procedure, we employ permutations of atoms; we transform atom n in the source conformation into atom \sigma(n) in the target conformation rather than directly transforming atom … Read more
We consider a quadratic programming (QP) problem ($\Pi$) of the form $\min x^T C x$ subject to $Ax \ge b$ where $C\in {\mathbb R}^{n\mbox{\tiny\texttimes} n}_+, rank(C)=1$ and $A\in {\mathbb R}^{m\mbox{\tiny\texttimes} n}, b\in {\mathbb R}^m$. We present an FPTAS for this problem by reformulating the QP ($\Pi$) as a parametrized LP and “rounding” the optimal solution. … Read more
Particle swarm optimization (PSO) has received increasing interest from the optimization community due to its simplicity in implementation and its inexpensive computational overhead. However, PSO has premature convergence, especially in complex multimodal functions. Extremal Optimization (EO) is a recently developed local-search heuristic method and has been successfully applied to a wide variety of hard optimization … Read more
This paper is concerned with a scheduling problem that occurs in service systems, where customers are classified as `ordinary’ and `special’. Ordinary customers can be served on any service facility, while special customers can be served only on the flexible service facilities. Customers arrive dynamically over time and their needs become known upon arrival. We … Read more
This paper presents a general technique to develop approximation algorithms for allocation problems with integral assignment constraints. The core of the method is a randomized dependent rounding scheme, called geometric rounding, which yields termwise rounding ratios (in expectation), while emphasizing the strong correlation between events. We further explore the intrinsic geometric structure and general theoretical … Read more