Faster Space-Efficient Algorithms for Subset Sum,  $k$-Sum, and Related Problems

Bansal, Nikhil; Garg, Swati; Nederlof, Jesper; Vyas, Nikhil

doi:10.1137/17m1158203

Cited by 16 publications

(23 citation statements)

References 32 publications

(66 reference statements)

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“…More recently, these conjectures have become even more prominent as core problems in fine-grained complexity since their interesting consequences have expanded beyond geometry into purely combinatorial problems [102,113,38,72,12,6,83,7,62,72]. Note that k-SUM inherits its hardness from Subset Sum, by a simple reduction: to answer Open Question 1 positively it is enough to solve k-SUM in T 1−ε · n o(k) or n k/2−ε · T o(1) time.Entire books [91,79] are dedicated to the algorithmic approaches that have been used to attack Subset Sum throughout many decades, and, quite astonishingly, major algorithmic advances are still being discovered in our days, e.g., [88,68,26,51,14,108,77,61,44,15,16,85,56,22,94,82,33], not to mention the recent developments on generalized versions (see [24]) and other computational models (see [107,41]). At STOC'17 an algorithm was presented that beats the trivial 2 n bound while using polynomial space, under certain assumptions on access to random bits [22].…”

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confidence: 99%

“…Note that k-SUM inherits its hardness from Subset Sum, by a simple reduction: to answer Open Question 1 positively it is enough to solve k-SUM in T 1−ε · n o(k) or n k/2−ε · T o(1) time.Entire books [91,79] are dedicated to the algorithmic approaches that have been used to attack Subset Sum throughout many decades, and, quite astonishingly, major algorithmic advances are still being discovered in our days, e.g., [88,68,26,51,14,108,77,61,44,15,16,85,56,22,94,82,33], not to mention the recent developments on generalized versions (see [24]) and other computational models (see [107,41]). At STOC'17 an algorithm was presented that beats the trivial 2 n bound while using polynomial space, under certain assumptions on access to random bits [22]. At SODA'17 we have seen the first improvements (beyond log factors [100]) over the O(T n) algorithm, reducing the bound toÕ(T + n) [82,33].…”

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confidence: 99%

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SETH-Based Lower Bounds for Subset Sum and Bicriteria Path

Abboud

Bringmann

Hermelin

et al. 2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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Subset Sum and k-SAT are two of the most extensively studied problems in computer science, and conjectures about their hardness are among the cornerstones of fine-grained complexity.An important open problem in this area is to base the hardness of one of these problems on the other.Our main result is a tight reduction from k-SAT to Subset Sum on dense instances, proving that Bellman's 1962 pseudo-polynomial O * (T )-time algorithm for Subset Sum on n numbers and target T cannot be improved to time T 1−ε · 2 o(n) for any ε > 0, unless the Strong Exponential Time Hypothesis (SETH) fails.As a corollary, we prove a "Direct-OR" theorem for Subset Sum under SETH, offering a new tool for proving conditional lower bounds: It is now possible to assume that deciding whether one out of N given instances of Subset Sum is a YES instance requires time (N T ) 1−o(1) . As an application of this corollary, we prove a tight SETH-based lower bound for the classical Bicriteria s, t-Path problem, which is extensively studied in Operations Research. We separate its complexity from that of Subset Sum: On graphs with m edges and edge lengths bounded by L, we show that the O(Lm) pseudopolynomial time algorithm by Joksch from 1966 cannot be improved toÕ(L + m), in contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017).Subset Sum. Subset Sum is one of the most fundamental problems in computer science. Its most basic form is the following: given n integers x 1 , . . . , x n ∈ N, and a target value T ∈ N, decide whether there is a subset of the numbers that sums to T . The two most classical algorithms for the problem are the pseudo-polynomial O(T n) algorithm using dynamic programming [28], and the O(2 n/2 · poly(n, log T )) algorithm via "meet-in-the-middle" [67]. A central open question in Exact Algorithms [114] is whether faster algorithms exist, e.g., can we combine the two approaches to get a T 1/2 · n O(1) time algorithm? Such a bound was recently found in a Merlin-Arthur setting [94].The status of Subset Sum as a major problem has been established due to many applications, deep connections to other fields, and educational value. The O(T n) algorithm from 1957 is an illuminating example of dynamic programming that is taught in most undergraduate algorithms courses, and the NP-hardness proof (from Karp's original paper [78]) is a prominent example of a reduction to a problem on numbers. Interestingly, one of the earliest cryptosystems by Merkle and Hellman was based on Subset Sum [92], and was later extended to a host of Knapsack-type cryptosystems 4 (see [106,31,97,46,69] and the references therein).The version of Subset Sum where we ask for k numbers that sum to zero (the k-SUM problem) is conjectured to have n ⌈k/2⌉±o(1) time complexity. Most famously, the k = 3 case is the 3-SUM conjecture highlighted in the seminal work of Gajentaan and Overmars [57]. It has been shown that this problem lies at the core and captures the difficulty of dozens of problems in computational geometry. Searching in Google Scholar for "3su...

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confidence: 99%

mentioning

confidence: 99%

SETH-Based Lower Bounds for Subset Sum and Bicriteria Path

Abboud

Bringmann

Hermelin

et al. 2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The BCM algorithm for E D does not straightforwardly extend to L D , and it is still open whether L D can be solved in n o(1) -space and n 2−Ω(1) time, even allowing random oracles. Recently, Bansal, Garg, Nederlof, and Vyas [BGNV18] showed that a variant of the BCM algorithm can be applied to solve L D with an improved running time, provided that the input arrays have small second frequency moment (i.e., there are few collision pairs within each arrays). Formally, define…”

Section: Dmentioning

confidence: 99%

“…Due to complicated dependencies on the paths in this random digraph, it looks difficult to reduce the number of random bits using pseudorandomness. It was asked as an open question [BCM13,BGNV18] whether the BCM algorithm can be modified to work with only "one-way access" to random bits, where we may toss up to O(t) coins in time t, but cannot randomly access arbitrary coins tossed in the past. In particular, [BCM13] stated it "seems plausible" that the random oracle in the BCM algorithm could be replaced by some family of poly log(n)-wise independent hash functions in the analysis.…”

Section: Introductionmentioning

confidence: 99%

Truly Low-Space Element Distinctness and Subset Sum via Pseudorandom Hash Functions

Chen¹,

Jin²,

Williams³

et al. 2021

Preprint

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We consider low-space algorithms for the classic E D problem: given an array of n input integers with O(log n) bit-length, decide whether or not all elements are pairwise distinct. Beame, Clifford, and Machmouchi [FOCS 2013] gave an Õ(n 1.5 )-time randomized algorithm for E -D using only O(log n) bits of working space. However, their algorithm assumes a random oracle (in particular, read-only random access to polynomially many random bits), and it was asked as an open question whether this assumption can be removed.In this paper, we positively answer this question by giving an Õ(n 1.5 )-time randomized algorithm using O(log 3 n log log n) bits of space, with one-way access to random bits. As a corollary, we also obtain a poly(n)-space O * (2 0.86n )-time randomized algorithm for the Subset Sum problem, removing the random oracles required in the algorithm of Bansal, Garg, Nederlof, and Vyas [STOC 2017].The main technique underlying our results is a pseudorandom hash family based on iterative restrictions, which can fool the cycle-finding procedure in the algorithms of Beame et al. and Bansal et al.

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“…Several classic algorithms for SubsetSum are typically taught in undergraduate courses, including the meet-in-the-middle algorithm running in time O(2 n/2 ) [25] and Bellman's dynamic programming algorithm running in pseudopolynomial time O(n • t) [12]. Surprisingly, after decades of research, major algorithmic advances were still discovered in the last 10 years, e.g., [39,33,21,7,24,8,9,32,11,38,14,31,27,1,10]. Among these developments, the most relevant for this paper are improvements over Bellman's O(n•t) algorithm: Koiliaris and Xu [31] designed a deterministic algorithm running in time 1 O(min{ √ n•t, t 4/3 }), and Bringmann [14] devised a randomized algorithm running in time O(t) (which was improved in terms of log factors in [27]).…”

Section: Subset Summentioning

confidence: 99%

Top-𝑘-convolution and the quest for near-linear output-sensitive subset sum

Bringmann

Nakos

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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In the classical SubsetSum problem we are given a set X and a target t, and the task is to decide whether there exists a subset of X which sums to t. A recent line of research has resulted in (t •poly(log t))-time algorithms, which are (near-)optimal under popular complexity-theoretic assumptions. On the other hand, the standard dynamic programming algorithm runs in time O(n • |S(X , t)|), where S(X , t) is the set of all subset sums of X that are smaller than t. All previous pseudopolynomial algorithms actually solve a stronger task, since they actually compute the whole set S(X , t). As the aforementioned two running times are incomparable, in this paper we ask whether one can achieve the best of both worlds: running time |S(X , t)| • poly(log t). In particular, we ask whether S(X , t) can be computed in near-linear time in the output-size. Using a diverse toolkit containing techniques such as color coding, sparse recovery, and sumset estimates, we make considerable progress towards this question and design an algorithm running in time |S(X , t)| 4/3 • poly(log t). Central to our approach is the study of Top-k-Convolution, a natural problem of independent interest: given degree-d sparse polynomials with non-negative coefficients, compute the lowest k non-zero monomials of their product. We design an algorithm running in time k 4/3 poly(log d), by a combination of sparse convolution and sumset estimates considered in Additive Combinatorics. Moreover, we provide evidence that going beyond some of the barriers we have faced requires either an algorithmic breakthrough or possibly new techniques from Additive Combinatorics on how to pass from information on restricted sumsets to information on unrestricted sumsets. CCS CONCEPTS • Theory of computation → Dynamic programming.

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Faster Space-Efficient Algorithms for Subset Sum, $k$-Sum, and Related Problems

Cited by 16 publications

References 32 publications

SETH-Based Lower Bounds for Subset Sum and Bicriteria Path

SETH-Based Lower Bounds for Subset Sum and Bicriteria Path

Truly Low-Space Element Distinctness and Subset Sum via Pseudorandom Hash Functions

Top-𝑘-convolution and the quest for near-linear output-sensitive subset sum

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