Moderate exponential-time algorithms for scheduling problems

t'Kindt, Vincent; Croce, Federico Della; Liedloff, Mathieu

doi:10.1007/s10288-022-00525-1

Cited by 5 publications

(3 citation statements)

References 55 publications

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“…Lower bounds conditioned on the Exponential Time Hypothesis are given (among others) by Chen et al in [11]. T'kindt et al give a survey on moderately exponential time algorithms for scheduling problems [47]. Cygan et al [13] gave an O((2 − ε) n ) time algorithm (for some constant ε > 0) for the problem of scheduling jobs of arbitrary length with precedence constraints on one machine.…”

Section: Related Workmentioning

confidence: 99%

A Faster Exponential Time Algorithm for Bin Packing With a Constant Number of Bins via Additive Combinatorics

Nederlof

Pawlewicz

Swennenhuis

et al. 2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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In the Bin Packing problem one is given n items with weights w 1 , . . . , w n and m bins with capacities c 1 , . . . , c m . The goal is to find a partition of the items into sets S 1 , . . . , S m such that w(S j ) c j for every bin j, where w(X) denotes i∈X w i .Björklund, Husfeldt and Koivisto (SICOMP 2009) presented an O (2 n ) time algorithm for Bin Packing. In this paper, we show that for every m ∈ N there exists a constant σ m > 0 such that an instance of Bin Packing with m bins can be solved in O(2 (1−σ m )n ) randomized time. Before our work, such improved algorithms were not known even for m equals 4.A key step in our approach is the following new result in Littlewood-Offord theory on the additive combinatorics of subset sums: For every δ > 0 there exists an ε > 0 such that if |{X ⊆ {1, . . . , n} : w(X) = v}| 2 (1−ε)n for some v then |{w(X) : X ⊆ {1, . . . , n}}| 2 δn .

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Section: Related Workmentioning

confidence: 99%

A Faster Exponential Time Algorithm for Bin Packing With a Constant Number of Bins via Additive Combinatorics

Nederlof

Pawlewicz

Swennenhuis

et al. 2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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“…This adaptation requires to introduce a pseudo-polynomial term in the state space of the dynamic programming as well as the aforementioned additive term. We thus obtain an extension of the Dynamic Programming Across the Subsets (DPAS) that many scheduling problems satisfy [17]. Herein, we focus on single-machine scheduling problems and show that our bounded-error hybrid quantum-classical algorithm improves the best-known classical exponential complexities, where in some cases a pseudo-polynomial factor 𝑝 𝑗 appears.…”

Section: Introductionmentioning

confidence: 99%

“…The purpose of this work is to adapt the seminal idea of [2] to NP-hard scheduling problems [17] that satisfy the following property: for a given set of jobs 𝐽 , the optimal solution for 𝐽 is the best concatenation of optimal solutions for 𝑋 and 𝐽 \ 𝑋 among all 𝑋 ⊂ 𝐽 such that |𝑋 | = |𝐽 |/2 (modulo an additive term that arises in the concatenation). This adaptation requires to introduce a pseudo-polynomial term in the state space of the dynamic programming as well as the aforementioned additive term.…”

Section: Introductionmentioning

confidence: 99%

Quantum Speed-ups for Single-machine Scheduling Problems

Grangé

Bourreau

Poss

et al. 2023

Proceedings of the Companion Conference on Genetic and Evolutionary Computation

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Grover search is currently one of the main approaches to obtain quantum speed-ups for combinatorial optimization problems. The combination of Quantum Minimum Finding (obtained from Grover search) with dynamic programming has proved particularly efficient to improve the worst-case complexity of several NP-hard optimization problems. Specifically, for these problems, the classical dynamic programming complexity (ignoring the polynomial factors) in O * (𝑐 𝑛 ) can be reduced by a bounded-error hybrid quantumclassical algorithm to O * (𝑐 𝑛 𝑞𝑢𝑎𝑛𝑡 ) for 𝑐 𝑞𝑢𝑎𝑛𝑡 < 𝑐. In this paper, we extend the resulting hybrid dynamic programming algorithm to three examples of single-machine scheduling problems: minimizing the total weighted completion time with deadlines, minimizing the total weighted completion time with precedence constraints, and minimizing the total weighted tardiness. The extension relies on the inclusion of a pseudo-polynomial term in the state space of the dynamic programming as well as an additive term in the recurrence. CCS CONCEPTS• Theory of computation → Design and analysis of algorithms; • Hardware → Quantum computation; • Mathematics of computing → Combinatorial optimization.

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