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### A Near-Linear Pseudopolynomial Time Algorithm for Subset Sum [article]

Karl Bringmann
2017 arXiv   pre-print
A textbook pseudopolynomial time algorithm by Bellman from 1957 solves Subset Sum in time O(nt). This has been improved to O(n Z) by Pisinger [J.  ...  Given a set Z of n positive integers and a target value t, the Subset Sum problem asks whether any subset of Z sums to t.  ...  Acknowledgements The author wants to thank Marvin Künnemann and Jesper Nederlof for providing useful comments on a draft of this paper.  ...

### A Simple Near-Linear Pseudopolynomial Time Randomized Algorithm for Subset Sum [article]

Ce Jin, Hongxun Wu
2018 arXiv   pre-print
In this paper, we present a simple and elegant randomized algorithm for Subset Sum in Õ(n + t) time.  ...  Given a multiset S of n positive integers and a target integer t, the Subset Sum problem asks to determine whether there exists a subset of S that sums up to t.  ...  Our algorithm can solve the modulo p version # p Knapsack in near-linear pseudopolynomial time for prime p > t.  ...

### A Simple Near-Linear Pseudopolynomial Time Randomized Algorithm for Subset Sum

Ce Jin, Hongxun Wu, Michael Wagner
2018 ACM-SIAM Symposium on Discrete Algorithms
In this paper, we present a simple and elegant randomized algorithm for Subset Sum in Õ(n + t) time.  ...  Bringmann [SODA'17] later gave a randomized Õ(n + t) time algorithm using two-stage color-coding. The Õ(n + t) running time is believed to be near-optimal.  ...  [1] recently showed that Subset Sum has no O(t 1− n O (1) ) algorithm for any > 0, unless the Strong Exponential Time Hypothesis (SETH) is false, so the Õ(n + t) time bound is likely to be near-optimal  ...

### A Faster Pseudopolynomial Time Algorithm for Subset Sum [article]

Konstantinos Koiliaris, Chao Xu
2016 arXiv   pre-print
We also present a modified algorithm for cyclic groups, which computes all the realizable subset sums within the group in O({√(n)m,m^5/4}) time, where m is the order of the group.  ...  Given a multiset S of n positive integers and a target integer t, the subset sum problem is to decide if there is a subset of S that sums up to t.  ...  We would like to thank Igor Shparlinski and Arne Winterhof for pointing out a problem in a proof.  ...

### A Faster Pseudopolynomial Time Algorithm for Subset Sum

Konstantinos Koiliaris, Chao Xu
2017 Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms
We also present a modified algorithm for finite cyclic groups, which computes all the realizable subset sums within the group in O(min{ √ nm, m 5/4 }) time, where m is the order of the group.  ...  Given a multiset S of n positive integers and a target integer t, the subset sum problem is to decide if there is a subset of S that sums up to t.  ...  We would like to thank Igor Shparlinski and Arne Winterhof for pointing out a problem in a proof.  ...

### The Subset Sum Problem: Reducing Time Complexity of NP-Completeness with Quantum Search

Bo Moon
2013 Undergraduate Journal of Mathematical Modeling: One + Two
The Subset Sum Problem is a member of the NP-complete class, so no known polynomial time algorithm exists for it.  ...  Quantum computation offers new insights for not only the Subset Sum Problem but also the entire NP-complete class; most notably, Grover's quantum algorithm for an unstructured database search can be tailored  ...  For example, the dynamic programming algorithm for the Subset Sum Problem presented earlier runs in pseudopolynomial time by utilizing optimal substructure and overlapping subproblems.  ...

### Near-optimal Approximate Discrete and Continuous Submodular Function Minimization [article]

Brian Axelrod, Yang P. Liu, Aaron Sidford
2019 arXiv   pre-print
In this paper we provide improved running times and oracle complexities for approximately minimizing a submodular function.  ...  Further, we leverage a generalization of this result to obtain efficient algorithms for minimizing a broad class of nonconvex functions.  ...  Acknowledgements We thank Deeparnab Chakrabarty, Yin Tat Lee, Sahil Singla, Kevin Tian, and Sam Chiu-wai Wong for helpful conversations.  ...

### Fast and Simple Modular Subset Sum [article]

Kyriakos Axiotis, Arturs Backurs, Karl Bringmann, Ce Jin, Vasileios Nakos, Christos Tzamos, Hongxun Wu
2020 arXiv   pre-print
In this work, we present two simple algorithms for the Modular Subset Sum problem running in near-linear time in m, both efficiently implementing Bellman's iteration over ℤ_m.  ...  A series of recent works has provided near-optimal algorithms for this problem under the Strong Exponential Time Hypothesis.  ...  A simple near-linear pseudopolynomial time randomized algorithm for subset sum. In SOSA@SODA, volume 69 of OASICS, pages 17:1-17:6, 2019. 24 Konstantinos Koiliaris and Chao Xu.  ...

### On Near-Linear-Time Algorithms for Dense Subset Sum [article]

Karl Bringmann, Philip Wellnitz
2020 arXiv   pre-print
Σ_X/n^2. - We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with t ≪mx_X Σ_X/n^2, then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis  ...  Our main question is: When can dense Subset Sum be solved in near-linear time Õ(n)?  ...  However, Galil and Margalit's data structure can even reconstruct solutions, namely after preprocessing X and given t they can compute a subset Y ⊆ X summing to t, if it exists, in time O(|Y |).  ...

### Subquadratic Submodular Function Minimization [article]

Deeparnab Chakrabarty and Yin Tat Lee and Aaron Sidford and Sam Chiu-wai Wong
2016 arXiv   pre-print
For integer-valued submodular functions, we give an SFM algorithm which runs in O(nM^3 n·EO) time giving the first nearly linear time algorithm in any known regime.  ...  For real-valued submodular functions with range in [-1,1], we give an algorithm which in Õ(n^5/3·EO/ε^2) time returns an ε-additive approximate solution.  ...  A special thanks to Bobby Kleinberg for asking the question about approximate SFM.  ...

### An improved projection operation for cylindrical algebraic decomposition of three-dimensional space

Scott McCallum
1988 Journal of symbolic computation
Observations suggest that the reduction in the projection set size leads to a substantial decrease in the computing time of the cad algorithm.  ...  A key component of the cylindrical algebraic decomposition (cad) algorithm of Collins (1975) is the projection operation: the projecthm of a set A of r-variate polynomials is defined to be a certain set  ...  Thanks also to Pierre Milman for helping me to better understand Zariski's theorem. I am grateful to the referees for their helpful comments.  ...

### A Pseudopolynomial Algorithm for Alexandrov's Theorem [chapter]

Daniel Kane, Gregory N. Price, Erik D. Demaine
2009 Lecture Notes in Computer Science
We describe an algorithm based on this differential equation to compute the polyhedron to arbitrary precision given the metric, and prove a pseudopolynomial bound on its running time.  ...  Recent work by Bobenko and Izmestiev describes a differential equation whose solution is the polyhedron corresponding to a given metric.  ...  We thank Jeff Erickson and Joseph Mitchell for helpful discussions about shortest paths on non-shortest-path triangulations.  ...

### Bicriteria Network Design Problems [article]

Madhav V. Marathe, R. Ravi, Ravi Sundaram,S. S. Ravi, Daniel J. Rosenkrantz, Harry B. Hunt III
1998 arXiv   pre-print
Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming.  ...  We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.  ...  Bodlaender for pointing out to us the equivalence between treewidth bounded graphs and decomposable graphs. We thank A. Ramesh for bringing [KP+93] to our attention. We also thank Dr. V.  ...

### Page 7564 of Mathematical Reviews Vol. , Issue 2001J [page]

2001 Mathematical Reviews
Finally, polynomial time algorithms for three special cases are derived.  ...  Properties of an optimal schedule for the prob- lems under consideration are presented, and pseudopolynomial algorithms based on dynamic programming are given.  ...

### Құрылымданбаған деректерді ақпараттық іздеу әдістері МЕТОДЫ ИНФОРМАЦИОННОГО ПОИСКА НЕСТРУКТУРИРОВАННЫХ ДАННЫХ

А.С. Ауезова, К.Н. Муратова, Б. Синчев
2022 INTERNATIONAL JOURNAL OF INFORMATION AND COMMUNICATION TECHNOLOGIES
A near-linear pseudopolynomial time algorithm for subset sum. To appear in SODA '17, 2017. //arXiv:1610.04712v2[cs.Ds] 8 Jan 2017.-18p. 5. E. Horowitz, S. Sanni.  ...  A Faster pseudopolynomial time algorithm for subset sum. To appear in SODA '17, 2017. //arXiv:1610.04712v2[cs.Ds] 8 Jan 2017.-18p. 4. Karl Bringmann.  ...  To achieve this goal, the new polynomial algorithms are based on the problem of the sum of subsets, which belongs to the NPcomplete class.  ...
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