lower bound
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2022 ◽  
Vol 69 (1) ◽  
pp. 1-18
Author(s):  
Anupam Gupta ◽  
David G. Harris ◽  
Euiwoong Lee ◽  
Jason Li

In the k -cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into k connected components. Algorithms of Karger and Stein can solve this in roughly O ( n 2k ) time. However, lower bounds from conjectures about the k -clique problem imply that Ω ( n (1- o (1)) k ) time is likely needed. Recent results of Gupta, Lee, and Li have given new algorithms for general k -cut in n 1.98k + O(1) time, as well as specialized algorithms with better performance for certain classes of graphs (e.g., for small integer edge weights). In this work, we resolve the problem for general graphs. We show that the Contraction Algorithm of Karger outputs any fixed k -cut of weight α λ k with probability Ω k ( n - α k ), where λ k denotes the minimum k -cut weight. This also gives an extremal bound of O k ( n k ) on the number of minimum k -cuts and an algorithm to compute λ k with roughly n k polylog( n ) runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight k -clique. The first main ingredient in our result is an extremal bound on the number of cuts of weight less than 2 λ k / k , using the Sunflower lemma. The second ingredient is a fine-grained analysis of how the graph shrinks—and how the average degree evolves—in the Karger process.


2022 ◽  
Vol 309 ◽  
pp. 172-179
Author(s):  
Michael Harp ◽  
Elijah Jackson ◽  
David Jensen ◽  
Noah Speeter
Keyword(s):  

2022 ◽  
Vol 14 (1) ◽  
pp. 1-54
Author(s):  
Victor Lagerkvist ◽  
Magnus Wahlström

We study the fine-grained complexity of NP-complete satisfiability (SAT) problems and constraint satisfaction problems (CSPs) in the context of the strong exponential-time hypothesis (SETH) , showing non-trivial lower and upper bounds on the running time. Here, by a non-trivial lower bound for a problem SAT (Γ) (respectively CSP (Γ)) with constraint language Γ, we mean a value c 0 > 1 such that the problem cannot be solved in time O ( c n ) for any c < c 0 unless SETH is false, while a non-trivial upper bound is simply an algorithm for the problem running in time O ( c n ) for some c < 2. Such lower bounds have proven extremely elusive, and except for cases where c 0 =2 effectively no such previous bound was known. We achieve this by employing an algebraic framework, studying constraint languages Γ in terms of their algebraic properties. We uncover a powerful algebraic framework where a mild restriction on the allowed constraints offers a concise algebraic characterization. On the relational side we restrict ourselves to Boolean languages closed under variable negation and partial assignment, called sign-symmetric languages. On the algebraic side this results in a description via partial operations arising from system of identities, with a close connection to operations resulting in tractable CSPs, such as near unanimity operations and edge operations . Using this connection we construct improved algorithms for several interesting classes of sign-symmetric languages, and prove explicit lower bounds under SETH. Thus, we find the first example of an NP-complete SAT problem with a non-trivial algorithm which also admits a non-trivial lower bound under SETH. This suggests a dichotomy conjecture with a close connection to the CSP dichotomy theorem: an NP-complete SAT problem admits an improved algorithm if and only if it admits a non-trivial partial invariant of the above form.


2022 ◽  
Vol 186 ◽  
pp. 105554
Author(s):  
Christian Kaspers ◽  
Yue Zhou
Keyword(s):  

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 261
Author(s):  
Shaoxiong Hou

This paper introduces the new annulus body to establish the optimal lower bound for the anisotropic logarithmic potential as the complement to the theory of its upper bound estimate which has already been investigated. The connections with convex geometry analysis and some metric properties are also established. For the application, a polynomial dual log-mixed volume difference law is deduced from the optimal estimate.


2022 ◽  
Vol 17 (1) ◽  
Author(s):  
Luiz Augusto G. Silva ◽  
Luis Antonio B. Kowada ◽  
Noraí Romeu Rocco ◽  
Maria Emília M. T. Walter

Abstract Background sorting by transpositions (SBT) is a classical problem in genome rearrangements. In 2012, SBT was proven to be $$\mathcal {NP}$$ NP -hard and the best approximation algorithm with a 1.375 ratio was proposed in 2006 by Elias and Hartman (EH algorithm). Their algorithm employs simplification, a technique used to transform an input permutation $$\pi$$ π into a simple permutation$${\hat{\pi }}$$ π ^ , presumably easier to handle with. The permutation $${\hat{\pi }}$$ π ^ is obtained by inserting new symbols into $$\pi$$ π in a way that the lower bound of the transposition distance of $$\pi$$ π is kept on $${\hat{\pi }}$$ π ^ . The simplification is guaranteed to keep the lower bound, not the transposition distance. A sequence of operations sorting $${\hat{\pi }}$$ π ^ can be mimicked to sort $$\pi$$ π . Results and conclusions First, using an algebraic approach, we propose a new upper bound for the transposition distance, which holds for all $$S_n$$ S n . Next, motivated by a problem identified in the EH algorithm, which causes it, in scenarios involving how the input permutation is simplified, to require one extra transposition above the 1.375-approximation ratio, we propose a new approximation algorithm to solve SBT ensuring the 1.375-approximation ratio for all $$S_n$$ S n . We implemented our algorithm and EH’s. Regarding the implementation of the EH algorithm, two other issues were identified and needed to be fixed. We tested both algorithms against all permutations of size n, $$2\le n \le 12$$ 2 ≤ n ≤ 12 . The results show that the EH algorithm exceeds the approximation ratio of 1.375 for permutations with a size greater than 7. The percentage of computed distances that are equal to transposition distance, computed by the implemented algorithms are also compared with others available in the literature. Finally, we investigate the performance of both implementations on longer permutations of maximum length 500. From the experiments, we conclude that maximum and the average distances computed by our algorithm are a little better than the ones computed by the EH algorithm and the running times of both algorithms are similar, despite the time complexity of our algorithm being higher.


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