Abstract
AbstractWe 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 a minimum-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 prior work of Lovász and of Aharoni, Holzman and Krivelevich, we describe a deterministic algorithm with approximation ratio slightly less than $$\frac{d}{2}$$
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 $$\frac{d}{2}$$
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 $$\frac{d+2}{4}$$
d
+
2
4
is possible, assuming the Unique Games Conjecture and $$\text {P}\ne \text {NP}$$
P
≠
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.
Funder
Rheinische Friedrich-Wilhelms-Universität Bonn
Publisher
Springer Science and Business Media LLC
Subject
General Mathematics,Software
Reference23 articles.
1. Aharoni, R., Holzman, R., Krivelevich, M.: On a theorem of Lovász on covers in $$r$$-partite hypergraphs. Combinatorica 16(2), 149–174 (1996)
2. Alimonti, P., Kann, V.: Some APX-completeness results for cubic graphs. Theoret. Comput. Sci. 237(1–2), 123–134 (2000)
3. Bar-Yehuda, R., Even, S.: A linear-time approximation algorithm for the weighted vertex cover problem. J. Algorithms 2(2), 198–203 (1981)
4. Brešar, B., Kardoš, F., Katrenič, J., Semanišin, G.: Minimum $$k$$-path vertex cover. Discret. Appl. Math. 159(12), 1189–1195 (2011)
5. Daboul, S., Held, S., Vygen, J., Wittke, S.: An approximation algorithm for threshold voltage optimization. Trans. Des. Autom. Electron. Syst. 23(6), 68 (2018)
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献