Abstract
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 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 \(\frac{d}{2}\) for minimum-weight vertex cover in d-partite hypergraphs for fixed d and given d-partition. This is tight and yields also a \(\frac{d}{2}\)-approximation algorithm for general time-cost tradeoff instances.
We also study the inapproximability and show that no better approximation ratio than \(\frac{d+2}{4}\) is possible, assuming the Unique Games Conjecture and \(\text {P}\ne \text {NP}\). This strengthens a result of Svensson [17], 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.
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Notes
- 1.
A hypergraph \((V, \mathcal {B})\) is d-partite if there exists a partition \(V = V_1\dot{\cup }V_2\dots \dot{\cup }V_d\) such that \(|P\cap V_i| \le 1\) for all \(P \in \mathcal {B}\) and \(i \in \{1,\dots , d\}\). We call \(\{V_1,\dots , V_d\}\) a d-partition. We do not require the hypergraph to be d-uniform.
- 2.
Note that this is really a special case: for example the 3-partite hypergraph with vertex set \(\{1,2,3,4,5,6\}\) and hyperedges \(\{1,4,6\}, \{2,3,6\}\), and \(\{2,4,5\}\) does not result from a time-cost tradeoff instance of depth 3 with our construction.
- 3.
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Acknowledgement
We thank Nikhil Bansal for fruitful discussions at an early stage of this project.
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Daboul, S., Held, S., Vygen, J. (2021). Approximating the Discrete Time-Cost Tradeoff Problem with Bounded Depth. In: Singh, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2021. Lecture Notes in Computer Science(), vol 12707. Springer, Cham. https://doi.org/10.1007/978-3-030-73879-2_3
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