A Performance Study of Some Approximation Algorithms for Computing a Small Dominating Set in a Graph

We implement and test the performances of several approximation algorithms for computing the minimum dominating set of a graph. These algorithms are the standard greedy algorithm, the recent Linear programming (LP) rounding algorithms and a hybrid algorithm that we design by combining the greedy and LP rounding algorithms. Over the range of test data, all algorithms perform better than anticipated in theory, and have small performance ratios, measured as the size of output divided by the LP objective lower bound. However, each have advantages over the others. For instance, LP rounding algorithm normally outperforms the other algorithms on sparse real-world graphs. On a graph with 400,000+ vertices, LP rounding took less than 15 s of CPU time to generate a solution with performance ratio 1.011, while the greedy and hybrid algorithms generated solutions of performance ratio 1.12 in similar time. For synthetic graphs, the hybrid algorithm normally outperforms the others, whereas for hypercubes and k-Queens graphs, greedy outperforms the rest. Another advantage of the hybrid algorithm is to solve very large problems that are suitable for application of LP rounding (sparse graphs) but LP formulations become formidable in practice and LP solvers crash, as we observed on a real-world graph with 7.7 million+ vertices and a planar graph on 1,000,000 vertices.

A Primal-Dual Approach to Analyzing ATO Systems

We study assemble-to-order (ATO) problems from the literature. ATO problems with general structure and integrality constraints are well known to be difficult to solve, and we provide new insight into these issues by establishing worst-case approximation guarantees through primal-dual analyses and linear programming (LP) rounding. First, we relax the one-period ATO problem using a natural newsvendor decomposition and use the dual solution for the relaxation to derive a lower bound on optimal cost, providing a tight approximation guarantee that grows with the maximum product size in the system. Then, we present an LP rounding algorithm that achieves both asymptotic optimality as demand grows large, and a 1.8 approximation factor for any problem instance. In addition to theoretical guarantees, we perform comprehensive numerical simulations and find that our rounding algorithm outperforms existing techniques and is close to optimal. Finally, we demonstrate that our one-period LP rounding results can be used to develop an asymptotically optimal integral policy for dynamic ATO problems with backlogging and identical component lead-times. This paper was accepted by Yinyu Ye, optimization.