A Subquadratic Approximation Scheme for Partition

Mucha, Marcin; Węgrzycki, Karol; Włodarczyk, Michał

doi:10.1137/1.9781611975482.5

Cited by 21 publications

(28 citation statements)

References 54 publications

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“…The journal version of their paper [25] only proves a weaker result, assuming the stricter condition t mx 1/2 X Σ X /n. Nevertheless, their conference version was recently cited and used in [37]. It would therefore be desirable to have an accessible full proof of Theorem 1.1.…”

Section: Reference Running Time Commentsmentioning

confidence: 99%

On Near-Linear-Time Algorithms for Dense Subset Sum

Bringmann¹,

Wellnitz²

2021

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

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In the Subset Sum problem we are given a set of n positive integers X and a target t and are asked whether some subset of X sums to t. Natural parameters for this problem that have been studied in the literature are n and t as well as the maximum input number mxX and the sum of all input numbers ΣX . In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in n. In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time n O(1) .Our main question is: When can dense Subset Sum be solved in near-linear time O(n)? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters n, t, mxX , ΣX for which dense Subset Sum is in time O(n). For notational convenience we assume without loss of generality that t ≥ mxX (as larger numbers can be ignored) and t ≤ ΣX /2 (using symmetry). Then our dichotomy reads as follows:By reviving and improving an additive-combinatorics-based approach by Galil and Margalit [SICOMP'91], we show that Subset Sum is in near-linear time O(n) if t mxX ΣX /n 2 . We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with t mxX ΣX /n 2 , then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis fail. We also generalize our algorithm from sets to multi-sets, albeit with non-matching upper and lower bounds.

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Section: Reference Running Time Commentsmentioning

confidence: 99%

On Near-Linear-Time Algorithms for Dense Subset Sum

Bringmann¹,

Wellnitz²

2021

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

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“…They obtained an O( n ε 2 log n log 2 W )-time algorithm, which they used to design an approximation algorithm for Tree Sparsity. Subsequently, their running time was improved to O( n ε log(n/ε) log W ) [34]. In this section, we start with a simple (1+ε)-approximation algorithm for Min-Plus Convolution that directly follows from our Sum-To-Max-Covering and runs in time O(n 3/2 /ε), see Theorem 8.1.…”

Section: Strongly Polynomial Approximation For Min-plus Convolutionmentioning

confidence: 99%

“…Theorem 8.7 (cf. Theorem 7.1 [34], reformulation of [7]). If (1 + ε)-Approximate Min-Plus Convolution can be solved in time T (n, ε), then (1 + ε)-Approximate Tree Sparsity can be solved in time O n + T (n, ε/ log 2 n) log n .…”

Section: Applications For Tree Sparsitymentioning

confidence: 99%

“…Note, that this reduction runs in strongly polynomial time, and the log W -factors in the running time of [7,34] come exclusively from the use of scaling for approximating Min-Plus Convolution. In consequence, our strongly polynomial (1 + ε)-approximation for Min-Plus Convolution yields a strongly polynomial (1 + ε)-approximation for Tree Sparsity.…”

Section: Applications For Tree Sparsitymentioning

confidence: 99%

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Approximating APSP without scaling: equivalence of approximate min-plus and exact min-max

Bringmann

Künnemann

Węgrzycki

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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Zwick's (1+ε)-approximation algorithm for the All Pairs Shortest Path (APSP) problem runs in time O( n ω ε log W ), where ω ≤ 2.373 is the exponent of matrix multiplication and W denotes the largest weight. This can be used to approximate several graph characteristics including the diameter, radius, median, minimum-weight triangle, and minimum-weight cycle in the same time bound.Since Zwick's algorithm uses the scaling technique, it has a factor log W in the running time. In this paper, we study whether APSP and related problems admit approximation schemes avoiding the scaling technique. That is, the number of arithmetic operations should be independent of W ; this is called strongly polynomial. Our main results are as follows. * Claim 5.6. We have d T r,b [2k], T r,b [2k + 1] > 1 ε for any level r, index k, and b ∈ {1, 2}. Proof. Because of how T r,1 , T r,2 are constructed, the chunks T r,b [2k] and T r,b [2k + 1] correspond to chunks T r [2k ′ ] and T r [2k ′ + 3] for some k ′ . The statement now follows from Claim 5.4. The following analogue of Claim 5.5 is immediate. Claim 5.7. For any x, y ∈ Z, if d(x, y) > 1 ε and x < y, then there exist a level r, index k, and b ∈ {1, 2} such that x ∈ T r,b [2k − 1] and y ∈ T r,b [2k].Proof. Consecutive chunks T r [2k − 1] and T r [2k] are either both added to T r,1 or both added to T r,2 . The statement thus follows from Claim 5.5.

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“…Moving past pattern matching, we want to point that in a closely related problem of computing (min, +)-convolution there exists O( n ε log n ε log U ) time algorithm computing (1 + ε) approximation, cf. Mucha et al [MWW19].…”

Section: Introductionmentioning

confidence: 99%

Approximating Approximate Pattern Matching

Studený,

Uznański

2018

Preprint

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Given a text T of length n and a pattern P of length m, the approximate pattern matching problem asks for computation of a particular distance function between P and every m-substring of T . We consider a (1 ± ε) multiplicative approximation variant of this problem, for ℓ p distance function. In this paper, we describe two (1 + ε)-approximate algorithms with a runtime of O( n ε ) for all (constant) non-negative values of p. For constant p ≥ 1 we show a deterministic (1 + ε)-approximation algorithm. Previously, such run time was known only for the case of ℓ 1 distance, by Gawrychowski and Uznański [ICALP 2018] and only with a randomized algorithm. For constant 0 ≤ p ≤ 1 we show a randomized algorithm for the ℓ p , thereby providing a smooth tradeoff between algorithms of Porat [FOCS 2015, SOSA 2018] for Hamming distance (case of p = 0) and of Gawrychowski and Uznański for ℓ 1 distance.

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A Subquadratic Approximation Scheme for Partition

Cited by 21 publications

References 54 publications

On Near-Linear-Time Algorithms for Dense Subset Sum

On Near-Linear-Time Algorithms for Dense Subset Sum

Approximating APSP without scaling: equivalence of approximate min-plus and exact min-max

Approximating Approximate Pattern Matching

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