Problems on Finite Automata and the Exponential Time Hypothesis

2016 ◽  
pp. 89-100 ◽  
Author(s):  
Henning Fernau ◽  
Andreas Krebs
Keyword(s):  
Finite Automata ◽  
Exponential Time ◽  
Exponential Time Hypothesis

scholarly journals Problems on Finite Automata and the Exponential Time Hypothesis

Algorithms ◽  
2017 ◽  
Vol 10 (1) ◽  
pp. 24 ◽  
Author(s):  
Henning Fernau ◽  
Andreas Krebs
Keyword(s):  
Finite Automata ◽  
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 Type Inference with Inequalities

1990 ◽  
Vol 19 (336) ◽  
Author(s):  
Michael I. Schwartzbach
Keyword(s):  
General Result ◽  
Existence Of Solutions ◽  
Finite Automata ◽  
Minimal Solution ◽  
Type Inference ◽  
Constraint Solving ◽  
Multiple Inheritance ◽  
Exponential Time ◽  
Systems Of Inequalities ◽  
Infinite Collection

Type inference can be phrased as constraint-solving over types. We consider an implicitly typed language equipped with recursive types, multiple inheritance, 1st order parametric polymorphism, and assignments. Type correctness is expressed as satisfiability of a possibly infinite collection of (monotonic) inequalities on the types of variables and expressions. A general result about systems of inequalities over semilattices yields a solvable form. We distinguish between deciding <em>typability</em> (the existence of solutions) and <em>type inference</em> (the computation of a minimal solution). In our case, both can be solved by means of nondeterministic finite automata; unusually, the two problems have different complexities: polynomial vs. exponential time.

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.

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