Approximating the discrete time-cost tradeoff problem with bounded depth

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 Lov\'asz 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 $\text{P}\neq\text{NP}$. This strengthens a result of Svensson, 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.