This paper addresses the Minimum Linear Ordering Problem (MLOP): Given a nonnegative set function f on a finite set V, find a linear ordering on V such that the sum of the function values for all the suffixes is minimized. This problem generalizes well-known problems such as the Minimum Linear Arrangement, Min Sum Set Cover, Minimum Latency Set Cover, and Multiple Intents Ranking. Extending a result of Feige, Lovász, and Tetali (2004) on Min Sum Set Cover, we show that the greedy algorithm provides a factor 4 approximate optimal solution when the cost function f is supermodular. We also present a factor 2 rounding algorithm for MLOP with a monotone submodular cost function, using the convexity of the Lovász extension. These are among very few constant factor approximation algorithms for NP-hard minimization problems formulated in terms of submodular/supermodular functions. In contrast, when f is a symmetric submodular function, the problem has an information theoretic lower bound of 2 on the approximability. Feige, Lovász, and Tetali (2004) also devised a factor 2 LP-rounding algorithm for the Min Sum Vertex Cover. In this paper, we present an improved approximation algorithm with ratio 1.79. The algorithm performs multi-stage randomized rounding based on the same LP relaxation, which provides an answer to their open question on the integrality gap. © 2012 Springer-Verlag.
CITATION STYLE
Iwata, S., Tetali, P., & Tripathi, P. (2012). Approximating minimum linear ordering problems. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7408 LNCS, pp. 206–217). https://doi.org/10.1007/978-3-642-32512-0_18
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