An Approximation Algorithm for Threshold Voltage Optimization

Daboul, Siad; Held, Stephan; Vygen, Jens; Wittke, Sonja

doi:10.1145/3232538

Cited by 4 publications

(7 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, Theorem 18 then implies that the Unique Games Conjecture is false or P = NP. 5 It remains to prove Theorem 18, which will be the subject of the next section.…”

Section: Reducing Vertex Deletion To Constant Depthmentioning

confidence: 99%

“…This shows that the integrality gap of LP (1) is at least d 2 . Since |P| ≤ d for all P ∈ B, the Bar-Yehuda-Even algorithm [3] can be used to find an integral solution to the time-cost tradeoff instance of cost at most d • LP, and can be implemented to run in polynomial time because for integral vectors x there is a linear-time separation oracle [5]. A d-approximation can also be obtained by rounding up all x v ≥ 1 d .…”

Section: Proposition 7 If the Vertex Cover Lp (1) Can Be Solved In Po...mentioning

confidence: 99%

“…The instances of the time-cost tradeoff problem that arise in the context of VLSI design usually have a constant upper bound d on the number of jobs in any chain [5]. In these instances the jobs correspond to gates in a Boolean circuit.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, it is easy to obtain a d-approximation algorithm: either by applying the Bar-Yehuda-Even algorithm for set covering [3,5] or (for fixed d) by simple LP rounding; see the end of Sect. 3.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Approximating the discrete time-cost tradeoff problem with bounded depth

2022

Self Cite

View full text Add to dashboard Cite

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 $$\frac{d}{2}$$ 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.

show abstract

“…, Theorem 18 then implies that the Unique Games Conjecture is false or P = NP. 5 It remains to prove Theorem 18, which will be the subject of the next section.…”

Section: Reducing Vertex Deletion To Constant Depthmentioning

confidence: 99%

Section: Proposition 7 If the Vertex Cover Lp (1) Can Be Solved In Po...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Approximating the discrete time-cost tradeoff problem with bounded depth

2022

Self Cite

View full text Add to dashboard Cite

show abstract

“…The instances of the time-cost tradeoff problem that arise in the context of VLSI design usually have a constant upper bound d on the number of vertices on any path [5]. This is due to a given target frequency of the chip, which can only be achieved if the logic depth is bounded.…”

Section: Introductionmentioning

confidence: 99%

Approximating the discrete time-cost tradeoff problem with bounded depth

Daboul¹,

Held²,

Vygen³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

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 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.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 = NP. This strengthens a result of Svensson [20], 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.

show abstract