Approximating the discrete time-cost tradeoff problem with bounded depth

• 成果类型: Article
• 署名作者: Daboul, Siad; Held, Stephan; Vygen, Jens
• 署名单位: University of Bonn; University of Bonn
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01777-9
• 发表日期: 2023
• 页码: 529-547
• 关键词: vertex cover; complexity; algorithm; hardness

摘要

We revisit the deadline version of the discrete time-cost tradeoff problem for the special case of bounded depth. Such instances occur for example in VLSI design. The depth of an instance is the number of jobs in a longest chain and is denoted by d. We prove new upper and lower bounds on the approximability. First we observe that the problem can be regarded as a special case of finding aminimum-weight vertex cover in a d-partite hypergraph. Next, we study the natural LP relaxation, which can be solved in polynomial time for fixed d and-for time-cost tradeoff instances-up to an arbitrarily small error in general. Improving on priorwork ofLovasz and of Aharoni, Holzman and Krivelevich, we describe a deterministic algorithm with approximation ratio slightly less than d/2 for minimum-weight vertex cover in d-partite hypergraphs for fixed d and given d-partition. This is tight and yields also a d 2 -approximation algorithm for general time-cost tradeoff instances, even if d is not fixed. We also study the inapproximability and show that no better approximation ratio than d+2/4 is possible, assuming the Unique Games Conjecture and P not equal NP. This strengthens a result of Svensson [21], who showed that under the same assumptions no constant-factor approximation algorithm exists for general time-cost tradeoff instances (of unbounded depth). Previously, only APX-hardness was known for bounded depth.