Tight lower bounds for the Workflow Satisfiability Problem based on the Strong Exponential Time Hypothesis

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.

Download Full-text

The Exponential Time Hypothesis and the Parameterized Clique Problem

Parameterized and Exact Computation - Lecture Notes in Computer Science ◽

10.1007/978-3-642-33293-7_4 ◽

2012 ◽

pp. 13-24 ◽

Cited By ~ 3

Author(s):

Yijia Chen ◽

Kord Eickmeyer ◽

Jörg Flum

Keyword(s):

Exponential Time ◽

Exponential Time Hypothesis ◽

Clique Problem

Download Full-text

Exponential Cosmological Solutions with Three Different Hubble-Like Parameters in (1 + 3 + k1 + k2)-Dimensional EGB Model with a Λ-Term

Symmetry ◽

10.3390/sym12020250 ◽

2020 ◽

Vol 12 (2) ◽

pp. 250

Author(s):

K. K. Ernazarov ◽

V. D. Ivashchuk

Keyword(s):

Lower Bounds ◽

Gravitational Constant ◽

Cosmological Term ◽

Upper And Lower Bounds ◽

Symmetry Groups ◽

Exponential Time ◽

Scale Factors ◽

Exponential Expansion ◽

Cosmological Solutions ◽

Stable Solutions

A D-dimensional Einstein–Gauss–Bonnet model with a cosmological term Λ , governed by two non-zero constants: α 1 and α 2 , is considered. By restricting the metrics to diagonal ones, we study a class of solutions with the exponential time dependence of three scale factors, governed by three non-coinciding Hubble-like parameters: H > 0 , h 1 , and h 2 , obeying 3 H + k 1 h 1 + k 2 h 2 ≠ 0 and corresponding to factor spaces of dimensions: 3, k 1 > 1 , and k 2 > 1 , respectively, with D = 4 + k 1 + k 2 . The internal flat factor spaces of dimensions k 1 and k 2 have non-trivial symmetry groups, which depend on the number of compactified dimensions. Two cases: (i) 3 < k 1 < k 2 and (ii) 1 < k 1 = k 2 = k , k ≠ 3 , are analyzed. It is shown that in both cases, the solutions exist if α = α 2 / α 1 > 0 and α Λ > 0 obey certain restrictions, e.g., upper and lower bounds. In Case (ii), explicit relations for exact solutions are found. In both cases, the subclasses of stable and non-stable solutions are singled out. Case (i) contains a subclass of solutions describing an exponential expansion of 3 d subspace with Hubble parameter H > 0 and zero variation of the effective gravitational constant G.

Download Full-text