In recent years, finding new satisfiability algorithms for various circuit classes has been a very active line of research. Despite considerable progress, we are still far away from a definite answer on which circuit classes allow fast satisfiability algorithms. This survey takes a (far from exhaustive) look at some recent satisfiability algorithms for a range of circuit classes and high- lights common themes. A special focus is given to connections between satisfiability algorithms and circuit lower bounds. A second focus is on reductions from satisfiability algorithms to a range of polynomial time problems, such as matrix multiplication and the Vector Domination Problem.
We give a nontrivial algorithm for the satisfiability problem for cn-wire threshold circuits of depth two which is better than exhaustive search by a factor 2^{sn} where s= 1/c^{O(c^2)}. We believe that this is the first nontrivial satisfiability algorithm for cn-wire threshold circuits of depth two. The independently interesting problem of the feasibility of sparse 0-1 integer linear programs is a special case. To our knowledge, our algorithm is the first to achieve constant savings even for the special case of Integer Linear Programming. The key idea is to reduce the satisfiability problem t…
In arXiv:1710.08163 a generalization of Boolean circuits to arbitrary finite algebras had been introduced and applied to sketch P versus NP-complete borderline for circuits satisfiability over algebras from congruence modular varieties. However the problem for nilpotent (which had not been shown to be NP-hard) but not supernilpotent algebras (which had been shown to be polynomial time) remained open. In this paper we provide a broad class of examples, lying in this grey area, and show that, under the Exponential Time Hypothesis and Strong Exponential Size Hypothesis (saying that Boolean circui…
The following problem is considered. A Turing machine $M$, that accepts a string of fixed length $t$ as input, runs for a time not exceeding a fixed value $n$ and is guaranteed to produce a binary output, is given. It's required to find a string $X$ such that $M(X) = 1$ effectively in terms of $t$, $n$, the size of the alphabet of $M$ and the number of states of $M$. The problem is close to the well-known Circuit Satisfiability Problem. The difference from Circuit Satisfiability Problem is that when reduced to Circuit Satisfiability Problem, we get circuits with a rich internal structure (in p…
Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebra…
$ \newcommand{\cclass}[1]{\normalfont\textsf{##1}} $We show average-case lower bounds for explicit Boolean functions against bounded-depth threshold circuits with a superlinear number of wires. We show that for each integer $d > 1$, there is a constant $\varepsilon_d > 0$ such that the Parity function on $n$ bits has correlation at most $n^{-\varepsilon_d}$ with depth-$d$ threshold circuits which have at most $n^{1+\varepsilon_d}$ wires, and the Generalized Andreev function on $n$ bits has correlation at most $\exp(-{n^{\varepsilon_d}})$ with depth-$d$ threshold circuits which have at most $n^…
For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. In 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs, motivated mainly by research on satisfiability algorithms and the satisfiability threshold. They proved dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connecti…
We consider the problem of efficiently enumerating the satisfying assignments to $\AC^0$ circuits. We give a zero-error randomized algorithm which takes an $\AC^0$ circuit as input and constructs a set of restrictions which partition $\{0,1\}^n$ so that under each restriction the value of the circuit is constant. Let $d$ denote the depth of the circuit and $cn$ denote the number of gates. This algorithm runs in time $|C| 2^{n(1-μ_{c.d})}$ where $|C|$ is the size of the circuit for $μ_{c,d} \ge 1/\bigO[\lg c + d \lg d]^{d-1}$ with probability at least $1-2^{-n}$. As a result, we get improved ex…
We consider the weighted antimonotone and the weighted monotone satisfiability problems on normalized circuits of depth at most $t \geq 2$, abbreviated {\sc wsat$^-[t]$} and {\sc wsat$^+[t]$}, respectively. These problems model the weighted satisfiability of antimonotone and monotone propositional formulas (including weighted anitmonoone/monotone {\sc cnf-sat}) in a natural way, and serve as the canonical problems in the definition of the parameterized complexity hierarchy. We characterize the parameterized complexity of {\sc wsat$^-[t]$} and {\sc wsat$^+[t]$} with respect to the genus of the …
We prove the #P-hardness of the counting problems associated with various satisfiability, graph and combinatorial problems, when restricted to planar instances. These problems include \begin{romannum} \item[{}] {\sc 3Sat, 1-3Sat, 1-Ex3Sat, Minimum Vertex Cover, Minimum Dominating Set, Minimum Feedback Vertex Set, X3C, Partition Into Triangles, and Clique Cover.} \end{romannum} We also prove the {\sf NP}-completeness of the {\sc Ambiguous Satisfiability} problems \cite{Sa80} and the {\sf D$^P$}-completeness (with respect to random polynomial reducibility) of the unique satisfiability problems \…
In this paper two algorithms solving circuit satisfiability problem over supernilpotent algebras are presented. The first one is deterministic and is faster than fastest previous algorithm presented by Aichinger. The second one is probabilistic with linear time complexity. Application of the former algorithm to finite groups provides time complexity that is usually lower than in previously best (given by Földvári) and application of the latter leads to corollary, that circuit satisfiability problem for group G is either tractable in probabilistic linear time if G is nilpotent or is NP-complete…
Efficient solutions to NP-complete problems would significantly benefit both science and industry. However, such problems are intractable on digital computers based on the von Neumann architecture, thus creating the need for alternative solutions to tackle such problems. Recently, a deterministic, continuous-time dynamical system (CTDS) was proposed (Nat.Phys. {\bf 7}(12), 966 (2011)) to solve a representative NP-complete problem, Boolean Satisfiability (SAT). This solver shows polynomial analog time-complexity on even the hardest benchmark $k$-SAT ($k \geq 3$) formulas, but at an energy cost …
We prove that weighted circuit satisfiability for monotone or antimonotone circuits has no fixed-parameter tractable approximation algorithm with any approximation ratio function $ρ$, unless $FPT\neq W[1]$. In particular, not having such an fpt-approximation algorithm implies that these problems have no polynomial-time approximation algorithms with ratio $ρ(OPT)$ for any nontrivial function $ρ$.
We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in $\text{Quasi-NP} = \text{NTIME}[n^{(\log n)^{O(1)}}]$ and $\text{NEXP}$ do not have small circuits from various circuit classes ${\cal C}$, by showing that ${\cal C}$ admits non-trivial satisfiability and/or $\#$SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of non-trivial $\#$SAT algorithm for a circuit class ${\mathcal C}$. Say a symmetr…
It has been known for almost three decades that many $\mathrm{NP}$-hard optimization problems can be solved in polynomial time when restricted to structures of constant treewidth. In this work we provide the first extension of such results to the quantum setting. We show that given a quantum circuit $C$ with $n$ uninitialized inputs, $\mathit{poly}(n)$ gates, and treewidth $t$, one can compute in time $(\frac{n}δ)^{\exp(O(t))}$ a classical assignment $y\in \{0,1\}^n$ that maximizes the acceptance probability of $C$ up to a $δ$ additive factor. In particular, our algorithm runs in polynomial ti…
We study the class of non-commutative Unambiguous circuits or Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees (FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel [LMP16] and Lagarde, Limaye and Srinivasan [LLS17]) and give the following constructions: (1) An explicit hitting set of quasipolynomial size for UPT circuits, (2) An explicit hitting set of quasipolynomial size for FewPT circuits (circuits with constantly many parse tree shapes), (3) An explicit hitting set of polynomial size for UPT circuits (of known parse tree shape), when a para…
Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit $d^{O(1)}$-variate and degree $d$ polynomial $P_{d}\in VNP$ such that if any depth four circuit $C$ of bounded formal degree $d$ which computes a polynomial of bounded individual degree $O(1)$, that is functionally equivalent to $P_d$, then $C$ must have size $2^{Ω(\sqrt{d}\log{d})}$. The motivation for their work comes from Boolean Circuit Complexity. Based on a characterization for $ACC^0$ circuits by Yao [FOCS, 1985] and Beigel and Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed tha…
The Workflow Satisfiability Problem (WSP) asks whether there exists an assignment of authorized users to the steps in a workflow specification, subject to certain constraints on the assignment. The problem is NP-hard even when restricted to just not equals constraints. Since the number of steps $k$ is relatively small in practice, Wang and Li (2010) introduced a parametrisation of WSP by $k$. Wang and Li (2010) showed that, in general, the WSP is W[1]-hard, i.e., it is unlikely that there exists a fixed-parameter tractable (FPT) algorithm for solving the WSP. Crampton et al. (2013) and Cohen e…
In this paper, we present a novel algorithm to solve the Boolean Satisfiability (SAT) problem, using noise-based logic (NBL). Contrary to what the name may suggest, NBL is not a random/fuzzy logic system. In fact, it is a completely deterministic logic system. A key property of NBL is that it allows us to apply a superposition of many input vectors to a SAT instance at the same time, circumventing a key restriction and assumption in the traditional approach to solving SAT. By exploiting the superposition property of NBL, our NBL-based SAT algorithm can determine whether an instance is SAT or n…
Aix Marseille Univ（艾克斯-马赛大学）；Université de Toulon（土伦大学）；CNRS（法国国家科学研究中心）；Université Paris Cité（巴黎西岱大学）；Leibniz Universität Hannover（汉诺威莱布尼茨大学）
We refine the complexity landscape for enumeration problems by introducing very low classes defined by using Boolean circuits as enumerators. We locate well-known enumeration problems, e.g., from graph theory, Gray code enumeration, and propositional satisfiability in our classes. In this way we obtain a framework to distinguish between the complexity of different problems known to be in $\mathbf{DelayP}$, for which a formal way of comparison was not possible to this day.
Different techniques have been used to prove several transference theorems of the form "nontrivial algorithms for a circuit class C yield circuit lower bounds against C". In this survey we revisit many of these results. We discuss how circuit lower bounds can be obtained from derandomization, compression, learning, and satisfiability algorithms. We also cover the connection between circuit lower bounds and useful properties, a notion that turns out to be fundamental in the context of these transference theorems. Along the way, we obtain a few new results, simplify several proofs, and show conn…
Transient chaos is an ubiquitous phenomenon characterizing the dynamics of phase space trajectories evolving towards a steady state attractor in physical systems as diverse as fluids, chemical reactions and condensed matter systems. Here we show that transient chaos also appears in the dynamics of certain efficient algorithms searching for solutions of constraint satisfaction problems that include scheduling, circuit design, routing, database problems or even Sudoku. In particular, we present a study of the emergence of hardness in Boolean satisfiability ($k$-SAT), a canonical class of constra…
University of Essex（埃塞克斯大学）；University of Nottingham（诺丁汉大学）；Royal Holloway, University of London（伦敦大学皇家霍洛威学院）；Cardiff University（卡迪夫大学）
The fixed parameter tractable (FPT) approach is a powerful tool in tackling computationally hard problems. In this paper, we link FPT results to classic artificial intelligence (AI) techniques to show how they complement each other. Specifically, we consider the workflow satisfiability problem (WSP) which asks whether there exists an assignment of authorised users to the steps in a workflow specification, subject to certain constraints on the assignment. It was shown by Cohen et al. (JAIR 2014) that WSP restricted to the class of user-independent constraints (UI), covering many practical cases…
A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic black-box identity testing algorithm for univariate polynomials of the form $\sum_{j=0}^t c_j X^{α_j} (a + b X)^{β_j}$. From our algorithm we derive an exponential lower bound for representations of polynomials such as $\prod_{i=1}^{2^n} (X^i-1)$ under this form. It has been conjectured that these polynomials are hard to compute by general arithmetic circuits. Our result shows that the "hardness from derandomization" approach to lower bounds is feasible for…
