Exponential Lower Bounds for the Running Time of DPLL Algorithms on Satisfiable Formulas

SAT 2005 ◽  
2007 ◽  
pp. 51-72
Author(s):  
Michael Alekhnovich ◽  
Edward A. Hirsch ◽  
Dmitry Itsykson
Keyword(s):  
Lower Bounds ◽  
Running Time

scholarly journals Tight Running Time Lower Bounds for Vertex Deletion Problems

2018 ◽  
Vol 10 (2) ◽  
pp. 1-18 ◽  
Author(s):  
Christian Komusiewicz
Keyword(s):  
Lower Bounds ◽  
Running Time ◽  
Vertex Deletion

scholarly journals Clique Is Hard on Average for Regular Resolution

2021 ◽  
Vol 68 (4) ◽  
pp. 1-26
Author(s):  
Albert Atserias ◽  
Ilario Bonacina ◽  
Susanna F. De Rezende ◽  
Massimo Lauria ◽  
Jakob Nordström ◽  
...  
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
State Of The Art ◽  
Edge Density ◽  
Multiplicative Constant ◽  
Running Time ◽  
Maximum Cliques

We prove that for k ≪ 4√ n regular resolution requires length n Ω( k ) to establish that an Erdős–Rényi graph with appropriately chosen edge density does not contain a k -clique. This lower bound is optimal up to the multiplicative constant in the exponent and also implies unconditional n Ω( k ) lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.

scholarly journals On The Independence Number Of Some Strong Products Of Cycle-Powers

2015 ◽  
Vol 40 (2) ◽  
pp. 133-141 ◽  
Author(s):  
Marcin Jurkiewicz ◽  
Marek Kubale ◽  
Krzysztof Ocetkiewicz
Keyword(s):  
Lower Bounds ◽  
Independence Number ◽  
Computational Results ◽  
Shannon Capacity ◽  
Noisy Channels ◽  
Running Time ◽  
The Third ◽  
Independence Numbers

Abstract In the paper we give some theoretical and computational results on the third strong power of cycle-powers, for example, we have found the independence numbers α((C102)√3) = 30 and α((C144)√3) = 14. A number of optimizations have been introduced to improve the running time of our exhaustive algorithm used to establish the independence number of the third strong power of cycle-powers. Moreover, our results establish new exact values and/or lower bounds on the Shannon capacity of noisy channels.

Exponential Lower Bounds for the Running Time of DPLL Algorithms on Satisfiable Formulas

2006 ◽  
Vol 35 (1-3) ◽  
pp. 51-72 ◽  
Author(s):  
Michael Alekhnovich ◽  
Edward A. Hirsch ◽  
Dmitry Itsykson
Keyword(s):  
Lower Bounds ◽  
Running Time

The number of solutions of the Erdős-Straus Equation and sums of k unit fractions

2019 ◽  
Vol 150 (3) ◽  
pp. 1401-1427
Author(s):  
Christian Elsholtz ◽  
Stefan Planitzer
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Rational Number ◽  
Residue Class ◽  
Upper Bounds ◽  
Log P ◽  
Number Of Solutions ◽  
Running Time ◽  
Unit Fractions ◽  
All Solutions

AbstractWe prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed m there are at most ${\cal O}_{\epsilon }(n^{{3}/{5}+\epsilon })$ solutions of ${m}/{n} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$. This improves upon a result of Browning and Elsholtz (2011) and extends a result of Elsholtz and Tao (2013) who proved this when m=4 and n is a prime. Moreover, there exists an algorithm finding all solutions in expected running time ${\cal O}_{\epsilon }(n^{\epsilon }({n^3}/{m^2})^{{1}/{5}})$, for any $\epsilon \gt 0$. We also improve a bound on the maximum number of representations of a rational number as a sum of k unit fractions. Furthermore, we also improve lower bounds. In particular, we prove that for given $m \in {\open N}$ in every reduced residue class e mod f there exist infinitely many primes p such that the number of solutions of the equation ${m}/{p} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$ is $\gg _{f,m} \exp (({5\log 2}/({12\,{\rm lcm} (m,f)}) + o_{f,m}(1)) {\log p}/{\log \log p})$. Previously, the best known lower bound of this type was of order $(\log p)^{0.549}$.

Lower bounds of temporal and spatial complexity of the substring search problem

2014 ◽  
Vol 24 (6) ◽  
Author(s):  
Evgeniy M. Perper
Keyword(s):  
Lower Bounds ◽  
Search Algorithms ◽  
Search Problem ◽  
Spatial Complexity ◽  
Running Time ◽  
Temporal And Spatial

AbstractWe consider the problem of substring search in a set of strings. The problem is the following: given a set of strings and an arbitrary substring, list all strings from the set that contain this substring.We describe search algorithms and obtain lower bounds for the running time and for the memory volume required by the fastest algorithms

scholarly journals Search on a Line by Byzantine Robots

2021 ◽  
pp. 1-19
Author(s):  
Jurek Czyzowicz ◽  
Konstantinos Georgiou ◽  
Evangelos Kranakis ◽  
Danny Krizanc ◽  
Lata Narayanan ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Fault Tolerant ◽  
Sufficient Evidence ◽  
Maximum Speed ◽  
Parallel Search ◽  
Not Given ◽  
Running Time ◽  
Byzantine Faults ◽  
Infinite Line ◽  
Design Algorithms

We consider the problem of fault-tolerant parallel search on an infinite line by [Formula: see text] robots. Starting from the origin, the robots are required to find a target at an unknown location. The robots can move with maximum speed [Formula: see text] and can communicate wirelessly among themselves. However, among the [Formula: see text] robots, there are [Formula: see text] robots that exhibit byzantine faults. A faulty robot can fail to report the target even after reaching it, or it can make malicious claims about having found the target when in fact it has not. Given the presence of such faulty robots, the search for the target can only be concluded when the non-faulty robots have sufficient evidence that the target has been found. We aim to design algorithms that minimize the value of [Formula: see text], the time to find a target at a (unknown) distance [Formula: see text] from the origin by [Formula: see text] robots among which [Formula: see text] are faulty. We give several different algorithms whose running time depends on the ratio [Formula: see text], the density of faulty robots, and also prove lower bounds. Our algorithms are optimal for some densities of faulty robots.

A UNIFYING LOOK AT SEMIGROUP COMPUTATIONS ON MESHES WITH MULTIPLE BROADCASTING

1994 ◽  
Vol 04 (01n02) ◽  
pp. 73-82 ◽  
Author(s):  
D. BHAGAVATHI ◽  
S. OLARIU ◽  
W. SHEN ◽  
L. WILSON
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Entire Range ◽  
Optimal Algorithm ◽  
Running Time ◽  
Associative Operation ◽  
Time Optimal ◽  
First Time

Given a sequence of m items α1, α2,…, αm from a semigroup S with an associative operation ⊕, the semigroup computation problem involves computing α1 ⊕ α2 ⊕…⊕ αm. We consider the semigroup computation problem involving m (2≤m≤n) items on a mesh with multiple broadcasting of size [Formula: see text]. Our contribution is to present the first lower bound and the first time-optimal algorithm which apply to the entire range of m (2≤m≤n). Specifically, we show that any algorithm that solves the semigroup computation problem must take Ω( log m) time if [Formula: see text] and [Formula: see text] time for [Formula: see text]. We then show that our bound is tight by designing an algorithm whose running time matches the lower bound. These results unify and generalize all semigroup lower bounds and algorithms known to the authors.

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