Voting and Bribing in Single-Exponential Time

Knop, Dušan; Koutecký, Martin; Mnich, Matthias

doi:10.1145/3396855

Cited by 33 publications

(31 citation statements)

References 48 publications

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“…Faster algorithms for these block structured IPs thus immediately improve the running time for other problems. Applications include string algorithms [22], social choice games [23], scheduling [16,21,17], and bin packing problems [25].…”

Section: Related Resultsmentioning

confidence: 99%

Collapsing the Tower -- On the Complexity of Multistage Stochastic IPs

Klein¹,

Reuter²

2021

Preprint

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In this paper we study the computational complexity of solving a class of block structured integer programs (IPs) -so called multistage stochastic IPs. A multistage stochastic IP is an IP of the form max{c T x | Ax = b, l ≤ x ≤ u, x integral} where the constraint matrix A consists of small block matrices ordered on the diagonal line and for each stage there are larger blocks with few columns connecting the blocks in a tree like fashion. Over the last years there was enormous progress in the area of block structured IPs. For many of the known block IP classes -such as n-fold, tree-fold, and two-stage stochastic IPs, nearly matching upper and lower bounds are known concerning their computational complexity. One of the major gaps that remained however was the parameter dependency in the running time for an algorithm solving multistage stochastic IPs. Previous algorithms require a tower of t exponentials, where t is the number of stages, while only a double exponential lower bound was known. In this paper we show that the tower of t exponentials is actually not necessary. We can show an improved running time for the algorithm solving multistage stochastic IPs with a running time of 2• poly(d, n), where d is the sum of columns in the connecting blocks and n is the number of blocks on the lowest stage. Hence, we obtain the first bound by an elementary function for the running time of an algorithm solving multistage stochastic IPs. In contrast to previous works, our algorithm has only a triple exponential dependency on the parameters and only doubly exponential for every constant t. By this we come very close the known double exponential bound (based on the exponential time hypothesis) that holds already for two-stage stochastic IPs, i.e. multistage stochastic IPs with only two stages.The improved running time of the algorithm is based on new bounds for the proximity of multistage stochastic IPs. The idea behind the bound is based on generalization for a structural lemma originally used for two-stage stochastic IPs. While the structural lemma requires iteration to be applied to multistage stochastic IPs, our generalization directly applies to inherent combinatorial properties of multiple stages. Already a special case of our lemma yields an improved bound for the Graver Complexity of multistage stochastic IPs.

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Section: Related Resultsmentioning

confidence: 99%

Collapsing the Tower -- On the Complexity of Multistage Stochastic IPs

Klein¹,

Reuter²

2021

Preprint

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“…We refer to [8,14,19] for more on how the special sparsity structure of block-structured matrices can be used to derive faster algorithms for integer programs. Block-structured integer programs can be applied in various problems such as scheduling and social choice; see, e.g., [21,25,30].…”

Section: Corollary 1 Let (U Imentioning

confidence: 99%

A Colorful Steinitz Lemma with Applications to Block Integer Programs

Oertel¹,

Paat²,

Weismantel³

2022

Preprint

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The Steinitz constant in dimension d is the smallest value c(d) such that for any norm on R d and for any finite zero-sum sequence in the unit ball, the sequence can be permuted such that the norm of each partial sum is bounded by c(d). Grinberg and Sevastyanov prove that c(d) ≤ d and that the bound of d is best possible for arbitrary norms; we refer to their result as the Steinitz Lemma. We present a variation of the Steinitz Lemma that permutes multiple sequences at one time. Our result, which we term a colorful Steinitz Lemma, demonstrates upper bounds that are independent of the number of sequences.Many results in the theory of integer programming are proved by permuting vectors of bounded norm; this includes proximity results, Graver basis algorithms, and dynamic programs. Due to a recent paper of Eisenbrand and Weismantel, there has been a surge of research on how the Steinitz Lemma can be used to improve integer programming results. As an application we prove a proximity result for block-structured integer programs.1 Introduction.Let • : R d → R be an arbitrary norm with corresponding unit ballGrinberg and Sevastyanov prove the following result on zero-sum sequences. We refer to

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“…The result of Hemmecke et al [20] quickly led to multiple improvements in the best-known upper bounds for several parameterized problems, where the technique of configuration ILPs is applicable [27][28][29]. Recently, the technique was also applied to improve the running times of several approximation schemes for scheduling problems [23].…”

Section: Parameterization By the Dual Treedepthmentioning

confidence: 99%

Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints

Knop

Pilipczuk

Wrochna

2020

ACM Trans. Comput. Theory

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• (k + + log b ∞) O(1). We present a streamlined proof of this fact and prove that, again, this running time is probably optimal: even assuming that all entries of A and b are in {−1, 0, 1}, the existence of an algorithm with running time 2 2 o (td D (A)) • (k +) O(1) would contradict the exponential time hypothesis. CCS Concepts: • Theory of computation → Fixed parameter tractability; Integer programming; This work is a part of projects CUTACOMBS, PowAlgDO (M. Wrochna), and TOTAL (M. Pilipczuk), which have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreements 714704, 714532, and 677651, respectively). The authors acknowledge the support of the OP VVV MEYS funded project CZ.

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Voting and Bribing in Single-Exponential Time

Cited by 33 publications

References 48 publications

Collapsing the Tower -- On the Complexity of Multistage Stochastic IPs

Collapsing the Tower -- On the Complexity of Multistage Stochastic IPs

A Colorful Steinitz Lemma with Applications to Block Integer Programs

Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints

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