Approximating Minimum Linear Ordering Problems
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.
KeywordsGreedy Algorithm Submodular Function 50th Annual IEEE Symposium Constant Factor Approximation Algorithm Minimum Linear
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