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).

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 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 Matching Cut in Graphs with Large Minimum Degree

Algorithmica ◽  
2020 ◽  
Author(s):  
Chi-Yeh Chen ◽  
Sun-Yuan Hsieh ◽  
Hoang-Oanh Le ◽  
Van Bang Le ◽  
Sheng-Lung Peng
Keyword(s):  
Polynomial Time ◽  
Minimum Degree ◽  
Positive Root ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Exponential Time ◽  
Running Time ◽  
Subexponential Time ◽  
Exponential Time Hypothesis ◽  
Exponential Time Algorithm

AbstractIn a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be $${\mathsf {NP}}$$ NP -complete. While Matching Cut is trivial for graphs with minimum degree at most one, it is $${\mathsf {NP}}$$ NP -complete on graphs with minimum degree two. In this paper, we show that, for any given constant $$c>1$$ c > 1 , Matching Cut is $${\mathsf {NP}}$$ NP -complete in the class of graphs with minimum degree c and this restriction of Matching Cut has no subexponential-time algorithm in the number of vertices unless the Exponential-Time Hypothesis fails. We also show that, for any given constant $$\epsilon >0$$ ϵ > 0 , Matching Cut remains $${\mathsf {NP}}$$ NP -complete in the class of n-vertex (bipartite) graphs with unbounded minimum degree $$\delta >n^{1-\epsilon }$$ δ > n 1 - ϵ . We give an exact branching algorithm to solve Matching Cut for graphs with minimum degree $$\delta \ge 3$$ δ ≥ 3 in $$O^*(\lambda ^n)$$ O ∗ ( λ n ) time, where $$\lambda$$ λ is the positive root of the polynomial $$x^{\delta +1}-x^{\delta }-1$$ x δ + 1 - x δ - 1 . Despite the hardness results, this is a very fast exact exponential-time algorithm for Matching Cut on graphs with large minimum degree; for instance, the running time is $$O^*(1.0099^n)$$ O ∗ ( 1 . 0099 n ) on graphs with minimum degree $$\delta \ge 469$$ δ ≥ 469 . Complementing our hardness results, we show that, for any two fixed constants $$1< c <4$$ 1 < c < 4 and $$c^{\prime }\ge 0$$ c ′ ≥ 0 , Matching Cut is solvable in polynomial time for graphs with large minimum degree $$\delta \ge \frac{1}{c}n-c^{\prime }$$ δ ≥ 1 c n - c ′ .

scholarly journals Additive-error fine-grained quantum supremacy

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 329
Author(s):  
Tomoyuki Morimae ◽  
Suguru Tamaki
Keyword(s):  
Quantum Computing ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Boolean Circuits ◽  
Exponential Time ◽  
Fine Grained ◽  
Additive Error ◽  
Universal Quantum ◽  
The One ◽  
Computing Models

It is known that several sub-universal quantum computing models, such as the IQP model, the Boson sampling model, the one-clean qubit model, and the random circuit model, cannot be classically simulated in polynomial time under certain conjectures in classical complexity theory. Recently, these results have been improved to ``fine-grained" versions where even exponential-time classical simulations are excluded assuming certain classical fine-grained complexity conjectures. All these fine-grained results are, however, about the hardness of strong simulations or multiplicative-error sampling. It was open whether any fine-grained quantum supremacy result can be shown for a more realistic setup, namely, additive-error sampling. In this paper, we show the additive-error fine-grained quantum supremacy (under certain complexity assumptions). As examples, we consider the IQP model, a mixture of the IQP model and log-depth Boolean circuits, and Clifford+T circuits. Similar results should hold for other sub-universal models.

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
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