Exact Algorithms and Strong Exponential Time Hypothesis

2016 ◽  
pp. 657-661 ◽  
Author(s):  
Joshua R. Wang ◽  
Ryan Williams
Keyword(s):  
Exact Algorithms ◽  
Exponential Time ◽  
Exponential Time Hypothesis

scholarly journals Results on a Super Strong Exponential Time Hypothesis

2020 ◽  
Vol 34 (09) ◽  
pp. 13700-13703
Author(s):  
Nikhil Vyas ◽  
Ryan Williams
Keyword(s):  
Randomized Algorithm ◽  
Time Algorithm ◽  
Critical Threshold ◽  
Polynomial Method ◽  
Variable Ratio ◽  
Sat Solving ◽  
Worst Case ◽  
Exponential Time ◽  
Exponential Time Hypothesis ◽  
Special Case

All known SAT-solving paradigms (backtracking, local search, and the polynomial method) only yield a 2n(1−1/O(k)) time algorithm for solving k-SAT in the worst case, where the big-O constant is independent of k. For this reason, it has been hypothesized that k-SAT cannot be solved in worst-case 2n(1−f(k)/k) time, for any unbounded ƒ : ℕ → ℕ. This hypothesis has been called the “Super-Strong Exponential Time Hypothesis” (Super Strong ETH), modeled after the ETH and the Strong ETH. We prove two results concerning the Super-Strong ETH:1. It has also been hypothesized that k-SAT is hard to solve for randomly chosen instances near the “critical threshold”, where the clause-to-variable ratio is 2k ln 2 −Θ(1). We give a randomized algorithm which refutes the Super-Strong ETH for the case of random k-SAT and planted k-SAT for any clause-to-variable ratio. In particular, given any random k-SAT instance F with n variables and m clauses, our algorithm decides satisfiability for F in 2n(1−Ω( log k)/k) time, with high probability (over the choice of the formula and the randomness of the algorithm). It turns out that a well-known algorithm from the literature on SAT algorithms does the job: the PPZ algorithm of Paturi, Pudlak, and Zane (1998).2. The Unique k-SAT problem is the special case where there is at most one satisfying assignment. It is natural to hypothesize that the worst-case (exponential-time) complexity of Unique k-SAT is substantially less than that of k-SAT. Improving prior reductions, we show the time complexities of Unique k-SAT and k-SAT are very tightly related: if Unique k-SAT is in 2n(1−f(k)/k) time for an unbounded f, then k-SAT is in 2n(1−f(k)(1−ɛ)/k) time for every ɛ > 0. Thus, refuting Super Strong ETH in the unique solution case would refute Super Strong ETH in general.

scholarly journals Lower Bounds Based on the Exponential-Time Hypothesis

2015 ◽  
pp. 467-521 ◽  
Author(s):  
Marek Cygan ◽  
Fedor V. Fomin ◽  
Łukasz Kowalik ◽  
Daniel Lokshtanov ◽  
Dániel Marx ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Exponential Time ◽  
Exponential Time Hypothesis

scholarly journals From Gap-Exponential Time Hypothesis to Fixed Parameter Tractable Inapproximability: Clique, Dominating Set, and More

2020 ◽  
Vol 49 (4) ◽  
pp. 772-810
Author(s):  
Parinya Chalermsook ◽  
Marek Cygan ◽  
Guy Kortsarz ◽  
Bundit Laekhanukit ◽  
Pasin Manurangsi ◽  
...  
Keyword(s):  
Dominating Set ◽  
Fixed Parameter Tractable ◽  
Exponential Time ◽  
Exponential Time Hypothesis ◽  
Fixed Parameter

scholarly journals Refining complexity analyses in planning by exploiting the exponential time hypothesis

2016 ◽  
Vol 78 (2) ◽  
pp. 157-175
Author(s):  
Meysam Aghighi ◽  
Christer Bäckström ◽  
Peter Jonsson ◽  
Simon Ståhlberg
Keyword(s):  
Exponential Time ◽  
Exponential Time Hypothesis

scholarly journals Chamberlin--Courant Rule with Approval Ballots: Approximating the MaxCover Problem with Bounded Frequencies in FPT Time

2017 ◽  
Vol 60 ◽  
pp. 687-716 ◽  
Author(s):  
Piotr Skowron ◽  
Piotr Faliszewski
Keyword(s):  
Approximation Algorithm ◽  
Exact Algorithms ◽  
The Other ◽  
Winner Determination ◽  
Exponential Time ◽  
Time Approximation ◽  
Polynomial Time Approximation Algorithm ◽  
Voting Rule ◽  
Np Complete ◽  
Better Than

We consider the problem of winner determination under Chamberlin--Courant's multiwinner voting rule with approval utilities. This problem is equivalent to the well-known NP-complete MaxCover problem and, so, the best polynomial-time approximation algorithm for it has approximation ratio 1 - 1/e. We show exponential-time/FPT approximation algorithms that, on one hand, achieve arbitrarily good approximation ratios and, on the other hand, have running times much better than known exact algorithms. We focus on the cases where the voters have to approve of at most/at least a given number of candidates.

scholarly journals Structural Parameterizations of Clique Coloring

Algorithmica ◽  
2021 ◽  
Author(s):  
Lars Jaffke ◽  
Paloma T. Lima ◽  
Geevarghese Philip
Keyword(s):  
Lower Bound ◽  
Maximal Clique ◽  
Time Algorithm ◽  
Input Graph ◽  
Exponential Time ◽  
Exponential Time Hypothesis ◽  
Tree Decompositions

AbstractA clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed $$q \ge 2$$ q ≥ 2 , we give an $$\mathscr {O}^{\star }(q^{{\mathsf {tw}}})$$ O ⋆ ( q tw ) -time algorithm when the input graph is given together with one of its tree decompositions of width $${\mathsf {tw}} $$ tw . We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is $$\mathsf {XP}$$ XP parameterized by clique-width.

scholarly journals Fine-Grained Reductions from Approximate Counting to Decision

2021 ◽  
Vol 13 (2) ◽  
pp. 1-24
Author(s):  
Holger Dell ◽  
John Lapinskas
Keyword(s):  
General Framework ◽  
Time Algorithm ◽  
Approximate Counting ◽  
Counting Problems ◽  
Negative Weight ◽  
Exponential Time ◽  
Fine Grained ◽  
Exponential Time Hypothesis ◽  
All Pairs Shortest Path ◽  
Orthogonal Pairs

In this article, we introduce a general framework for fine-grained reductions of approximate counting problems to their decision versions. (Thus, we use an oracle that decides whether any witness exists to multiplicatively approximate the number of witnesses with minimal overhead.) This mirrors a foundational result of Sipser (STOC 1983) and Stockmeyer (SICOMP 1985) in the polynomial-time setting, and a similar result of Müller (IWPEC 2006) in the FPT setting. Using our framework, we obtain such reductions for some of the most important problems in fine-grained complexity: the Orthogonal Vectors problem, 3SUM, and the Negative-Weight Triangle problem (which is closely related to All-Pairs Shortest Path). While all these problems have simple algorithms over which it is conjectured that no polynomial improvement is possible, our reductions would remain interesting even if these conjectures were proved; they have only polylogarithmic overhead and can therefore be applied to subpolynomial improvements such as the n 3 / exp(Θ (√ log n ))-time algorithm for the Negative-Weight Triangle problem due to Williams (STOC 2014). Our framework is also general enough to apply to versions of the problems for which more efficient algorithms are known. For example, the Orthogonal Vectors problem over GF( m ) d for constant  m can be solved in time n · poly ( d ) by a result of Williams and Yu (SODA 2014); our result implies that we can approximately count the number of orthogonal pairs with essentially the same running time. We also provide a fine-grained reduction from approximate #SAT to SAT. Suppose the Strong Exponential Time Hypothesis (SETH) is false, so that for some 1 < c < 2 and all k there is an O ( c n )-time algorithm for k -SAT. Then we prove that for all k , there is an O (( c + o (1)) n )-time algorithm for approximate # k -SAT. In particular, our result implies that the Exponential Time Hypothesis (ETH) is equivalent to the seemingly weaker statement that there is no algorithm to approximate #3-SAT to within a factor of 1+ɛ in time 2 o ( n )/ ɛ 2 (taking ɛ > 0 as part of the input).

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