The Complexity of Approximating Bounded-Degree Boolean \sharp CSP
Martin Dyer, Leslie Ann Goldberg, Markus Jalsenius, David Richerby
备注：
22 pages. Full classification now applies to degree bounds of at least 6 (formerly at least 25) but no other significant changes from previous version (reduction in page count is just different typesetting)
The degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial-time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP=RP. For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs.
Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a $(\mathbb{Q}\cup\{\infty\})$-valued objective function given as a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on labels from $D=\{0,1\}$ and an optimal assignment is required to use both labels from $D$. Examples include the classical global Min-Cut problem in graphs and the Minimum Distance problem studied in coding theory. We establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs with respect to exact solvability. Our wo…
We prove a complexity classification theorem that classifies all counting constraint satisfaction problems ($\#$CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) $\#$P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) $\#$P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).
Constraint Satisfaction Problems (CSP) constitute a convenient way to capture many combinatorial problems. The general CSP is known to be NP-complete, but its complexity depends on a template, usually a set of relations, upon which they are constructed. Following this template, there exist tractable and intractable instances of CSPs. It has been proved that for each CSP problem over a given set of relations there exists a corresponding CSP problem over graphs of unary functions belonging to the same complexity class. In this short note we show a dichotomy theorem for every finite domain D of C…
We give a complexity dichotomy for the problem of computing the partition function of a weighted Boolean constraint satisfaction problem. Such a problem is parameterized by a set of rational-valued functions, which generalize constraints. Each function assigns a weight to every assignment to a set of Boolean variables. Our dichotomy extends previous work in which the weight functions were restricted to being non-negative. We represent a weight function as a product of the form (-1)^s g, where the polynomial s determines the sign of the weight and the non-negative function g determines its magn…
A counting constraint satisfaction problem (#CSP) asks for the number of ways to satisfy a given list of constraints, drawn from a fixed constraint language Γ. We study how hard it is to evaluate this number approximately. There is an interesting partial classification, due to Dyer, Goldberg, Jalsenius and Richerby, of Boolean constraint languages when the degree of instances is bounded by d>=3 - every variable appears in at most d constraints - under the assumption that "pinning" is allowed as part of the instance. We study the d=2 case under the stronger assumption that "variable weights" ar…
We analyse the complexity of approximate counting constraint satisfactions problems $\mathrm{\#CSP}(\mathcal{F})$, where $\mathcal{F}$ is a set of nonnegative rational-valued functions of Boolean variables. A complete classification is known in the conservative case, where $\mathcal{F}$ is assumed to contain arbitrary unary functions. We strengthen this result by fixing any permissive strictly increasing unary function and any permissive strictly decreasing unary function, and adding only those to $\mathcal{F}$: this is weak conservativity. The resulting classification is employed to character…
The main result of this paper is a generalization of the classical blossom algorithm for finding perfect matchings. Our algorithm can efficiently solve Boolean CSPs where each variable appears in exactly two constraints (we call it edge CSP) and all constraints are even $Δ$-matroid relations (represented by lists of tuples). As a consequence of this, we settle the complexity classification of planar Boolean CSPs started by Dvorak and Kupec. Using a reduction to even $Δ$-matroids, we then extend the tractability result to larger classes of $Δ$-matroids that we call efficiently coverable. It pro…
The degree of a CSP instance is the maximum number of times that a variable may appear in the scope of constraints. We consider the approximate counting problem for Boolean CSPs with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum degree is at least 25 we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial-time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if eve…
We determine the computational complexity of approximately counting the total weight of variable assignments for every complex-weighted Boolean constraint satisfaction problem (or CSP) with any number of additional unary (i.e., arity 1) constraints, particularly, when degrees of input instances are bounded from above by a fixed constant. All degree-1 counting CSPs are obviously solvable in polynomial time. When the instance's degree is more than two, we present a dichotomy theorem that classifies all counting CSPs admitting free unary constraints into exactly two categories. This classificatio…
In this work we consider the approximability of $\textsf{Max-CSP}(f)$ in the context of sketching algorithms and completely characterize the approximability of all Boolean CSPs. Specifically, given $f$, $γ$ and $β$ we show that either (1) the $(γ,β)$-approximation version of $\textsf{Max-CSP}(f)$ has a linear sketching algorithm using $O(\log n)$ space, or (2) for every $ε> 0$ the $(γ-ε,β+ε)$-approximation version of $\textsf{Max-CSP}(f)$ requires $Ω(\sqrt{n})$ space for any sketching algorithm. We also prove lower bounds against streaming algorithms for several CSPs. In particular, we recover…
Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within \approx 0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within \approx 0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (\approx 0.878 for Max-Cut, and \approx 0.940 for Max-2-Sat). The hardness obtained for Max-2-Sat applies to m…
We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance. Such problems are parameterised by a constraint language specifying the relations that may be used in constraints. If every relation in the constraint language is affine then the number of satisfying assignments can be exactly counted in polynomial time. Otherwise, if every relation in the constraint language is in the co-clone IM_2 from Post's lattice, then the problem of counting satisfying assignments is complete with respect to approximation-preserving r…
We show that for constraint satisfaction problems (CSPs), sub-exponential size linear programming relaxations are as powerful as $n^{Ω(1)}$-rounds of the Sherali-Adams linear programming hierarchy. As a corollary, we obtain sub-exponential size lower bounds for linear programming relaxations that beat random guessing for many CSPs such as MAX-CUT and MAX-3SAT. This is a nearly-exponential improvement over previous results, previously, it was only known that linear programs of size $n^{o(\log n)}$ cannot beat random guessing for any CSP (Chan-Lee-Raghavendra-Steurer 2013). Our bounds are obtain…
In the Boolean maximum constraint satisfaction problem - Max CSP$(Γ)$ - one is given a collection of weighted applications of constraints from a finite constraint language $Γ$, over a common set of variables, and the goal is to assign Boolean values to the variables so that the total weight of satisfied constraints is maximized. There exists an elegant dichotomy theorem providing a criterion on $Γ$ for the problem to be polynomial-time solvable and stating that otherwise it becomes NP-hard. We study the NP hard cases through the lens of kernelization and provide a complete characterization of …
Constraint satisfaction problems (or CSPs) have been extensively studied in, for instance, artificial intelligence, database theory, graph theory, and statistical physics. From a practical viewpoint, it is beneficial to approximately solve those CSPs. When one tries to approximate the total number of truth assignments that satisfy all Boolean-valued constraints for (unweighted) Boolean CSPs, there is a known trichotomy theorem by which all such counting problems are neatly classified into exactly three categories under polynomial-time (randomized) approximation-preserving reductions. In contra…
We investigate the computational complexity of the problem of counting the maximal satisfying assignments of a Constraint Satisfaction Problem (CSP) over the Boolean domain {0,1}. A satisfying assignment is maximal if any new assignment which is obtained from it by changing a 0 to a 1 is unsatisfying. For each constraint language Gamma, #MaximalCSP(Gamma) denotes the problem of counting the maximal satisfying assignments, given an input CSP with constraints in Gamma. We give a complexity dichotomy for the problem of exactly counting the maximal satisfying assignments and a complexity trichotom…
In this paper, I consider a fine-grained dichotomy of Boolean counting constraint satisfaction problem (#CSP), under the exponential time hypothesis of counting version (#ETH). Suppose $\mathscr{F}$ is a finite set of algebraic complex-valued functions defined on Boolean domain. When $\mathscr{F}$ is a subset of either two special function sets, I prove that #CSP($\mathscr{F}$) is polynomial-time solvable, otherwise it can not be computed in sub-exponential time unless #ETH fails. I also improve the result by proving the same dichotomy holds for #CSP with bounded degree (every variable appears…
A classic result due to Schaefer (1978) classifies all constraint satisfaction problems (CSPs) over the Boolean domain as being either in $\mathsf{P}$ or $\mathsf{NP}$-hard. This paper considers a promise-problem variant of CSPs called PCSPs. A PCSP over a finite set of pairs of constraints $Γ$ consists of a pair $(Ψ_P, Ψ_Q)$ of CSPs with the same set of variables such that for every $(P, Q) \in Γ$, $P(x_{i_1}, ..., x_{i_k})$ is a clause of $Ψ_P$ if and only if $Q(x_{i_1}, ..., x_{i_k})$ is a clause of $Ψ_Q$. The promise problem $\operatorname{PCSP}(Γ)$ is to distinguish, given $(Ψ_P, Ψ_Q)$, b…
We continue the investigation of polynomial-time sparsification for NP-complete Boolean Constraint Satisfaction Problems (CSPs). The goal in sparsification is to reduce the number of constraints in a problem instance without changing the answer, such that a bound on the number of resulting constraints can be given in terms of the number of variables n. We investigate how the worst-case sparsification size depends on the types of constraints allowed in the problem formulation (the constraint language). Two algorithmic results are presented. The first result essentially shows that for any arity …
This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterised by a finite set F of non-negative functions that may be used to assign weights to the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems correspond to the special case of 0,1-valued functions. We show that the partition function, i.e. the sum of the weights of all configurations, can be computed in polynomial time if either (1) every function in F is of ``…
We study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language $Γ$ and a degree bound $Δ$, we study the complexity of #CSP$_Δ(Γ)$, which is the problem of counting satisfying assignments to CSP instances with constraints from $Γ$ and whose variables can appear at most $Δ$ times. Our main result shows that: (i) if every function in $Γ$ is affine, then #CSP$_Δ(Γ)$ is in FP for all $Δ$, (ii) otherwise, if every function in $Γ$ is in a class called IM$_2$, then for all sufficiently large $Δ$, #…
An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Post's lattice gives a complete classification of all Boolean relational clones, and this has been used to classify the computational difficulty of CSPs. Motivated by a desire to understand the computational complexity of (weighted) counting CSPs, we develop an analogous notion of functional clones and study the landscape of these clones. One of th…
A computational problem exhibits a "gap property" when there is no tractable boundary between two disjoint sets of instances. We establish a Gap Trichotomy Theorem for a family of constraint problem variants, completely classifying the complexity of possible ${\bf NP}$-hard gaps in the case of Boolean domains. As a consequence, we obtain a number of dichotomies for the complexity of specific variants of the constraint satisfaction problem: all are either polynomial-time tractable or $\mathbf{NP}$-complete. Schaefer's original dichotomy for $\textsf{SAT}$ variants is a notable particular case. …
We study the complexity of approximately solving the weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, there is a classification known for the case in which the domain of functions in F is Boolean. In this paper, we give a classification for the more general problem where functions in F have an arbitrary finite domain. We define the notions of weak log-modularity and weak log-supermodularity. We show that if F is weakly log-modular, then #CSP(F)is in FP. Otherwise, it is at least as difficult to approximate as #BIS, the p…
An active topic in the study of random constraint satisfaction problems (CSPs) is the geometry of the space of satisfying or almost satisfying assignments as the function of the density, for which a precise landscape of predictions has been made via statistical physics-based heuristics. In parallel, there has been a recent flurry of work on refuting random constraint satisfaction problems, via nailing refutation thresholds for spectral and semidefinite programming-based algorithms, and also on counting solutions to CSPs. Inspired by this, the starting point for our work is the following questi…
We prove a complexity dichotomy theorem for symmetric complex-weighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #P-hard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generalizes a theorem of Cai, Lu, and Xia for the case of real weights. We also obtain a dichotomy theorem for a symmetric arity 4 signature with complex weights in the planar Holant framework, which we use in the proof of our #CSP dichotomy. In particular, we reduce the problem of evaluating the Tutte polyn…
We study the 2-ary constraint satisfaction problems (2-CSPs), which can be stated as follows: given a constraint graph $G=(V,E)$, an alphabet set $Σ$ and, for each $\{u, v\}\in E$, a constraint $C_{uv} \subseteq Σ\timesΣ$, the goal is to find an assignment $σ: V \to Σ$ that satisfies as many constraints as possible, where a constraint $C_{uv}$ is satisfied if $(σ(u),σ(v))\in C_{uv}$. While the approximability of 2-CSPs is quite well understood when $|Σ|$ is constant, many problems are still open when $|Σ|$ becomes super constant. One such problem is whether it is hard to approximate 2-CSPs to …
The distribution of overlaps of solutions of a random CSP is an indicator of the overall geometry of its solution space. For random $k$-SAT, nonrigorous methods from Statistical Physics support the validity of the ``one step replica symmetry breaking'' approach. Some of these predictions were rigorously confirmed in \cite{cond-mat/0504070/prl} \cite{cond-mat/0506053}. There it is proved that the overlap distribution of random $k$-SAT, $k\geq 9$, has discontinuous support. Furthermore, Achlioptas and Ricci-Tersenghi proved that, for random $k$-SAT, $k\geq 8$. and constraint densities close enou…
In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. It is known that every Boolean (that is, two-valued) Max CSP problem with a finite set of allowed cons…
A $(k \times l)$-birthday repetition $\mathcal{G}^{k \times l}$ of a two-prover game $\mathcal{G}$ is a game in which the two provers are sent random sets of questions from $\mathcal{G}$ of sizes $k$ and $l$ respectively. These two sets are sampled independently uniformly among all sets of questions of those particular sizes. We prove the following birthday repetition theorem: when $\mathcal{G}$ satisfies some mild conditions, $val(\mathcal{G}^{k \times l})$ decreases exponentially in $Ω(kl/n)$ where $n$ is the total number of questions. Our result positively resolves an open question posted b…
In this paper, we prove an almost-optimal hardness for Max $k$-CSP$_R$ based on Khot's Unique Games Conjecture (UGC). In Max $k$-CSP$_R$, we are given a set of predicates each of which depends on exactly $k$ variables. Each variable can take any value from $1, 2, \dots, R$. The goal is to find an assignment to variables that maximizes the number of satisfied predicates. Assuming the Unique Games Conjecture, we show that it is NP-hard to approximate Max $k$-CSP$_R$ to within factor $2^{O(k \log k)}(\log R)^{k/2}/R^{k - 1}$ for any $k, R$. To the best of our knowledge, this result improves on al…
Promise Constraint Satisfaction Problems (PCSP) were proposed recently by Brakensiek and Guruswami arXiv:1704.01937 as a framework to study approximations for Constraint Satisfaction Problems (CSP). Informally a PCSP asks to distinguish between whether a given instance of a CSP has a solution or not even a specified relaxation can be satisfied. All currently known tractable PCSPs can be reduced in a natural way to tractable CSPs. Barto arXiv:1909.04878 presented an example of a PCSP over Boolean structures for which this reduction requires solving a CSP over an infinite structure. We give a fi…
Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahlén [9] and Husfeldt and Taslaman [12], in combination with Curticapean [7], extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line $y=1$, which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all ac…
We give an efficient algorithm to strongly refute \emph{semi-random} instances of all Boolean constraint satisfaction problems. The number of constraints required by our algorithm matches (up to polylogarithmic factors) the best-known bounds for efficient refutation of fully random instances. Our main technical contribution is an algorithm to strongly refute semi-random instances of the Boolean $k$-XOR problem on $n$ variables that have $\widetilde{O}(n^{k/2})$ constraints. (In a semi-random $k$-XOR instance, the equations can be arbitrary and only the right-hand sides are random.) One of our …
We study the complexity of valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a \emph{constraint language}, a fixed set of cost functions over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimise the sum. We consider the case of so-called \emph{conservative} languages; that is, languages containing all unary cost functions, thus allowing arbitrary restrictions on the domains of the variables. This problem has been studied by Bulatov [LICS'03] for $\{0,\infty\}$-valued languages (i.…
Linear Space Streaming Lower Bounds for Approximating CSPs
Flatiron Institute（平顿研究所）；Georgetown University（乔治城大学）；Harvard University（哈佛大学）；Google Research（谷歌研究院）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
研究在流模型下近似约束满足问题的可行性，证明改进近似因子需线性空间，针对Max k-LIN mod q问题提出最优近似下界。
We consider the problem of approximately solving constraint satisfaction problems with arity $k > 2$ ($k$-CSPs) on instances satisfying certain expansion properties, when viewed as hypergraphs. Random instances of $k$-CSPs, which are also highly expanding, are well-known to be hard to approximate using known algorithmic techniques (and are widely believed to be hard to approximate in polynomial time). However, we show that this is not necessarily the case for instances where the hypergraph is a high-dimensional expander. We consider the spectral definition of high-dimensional expansion used by…
The Role of Regularity in (Hyper-)Clique Detection and Implications for Optimizing Boolean CSPs
INSAIT, Sofia University “St. Kliment Ohridski”（索菲亚大学“圣克利门特·奥赫里德斯基”研究所）；Weizmann Institute of Science（魏茨曼科学研究所）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）
Holant problem is a general framework to study the computational complexity of counting problems. We prove a complexity dichotomy theorem for Holant problems over Boolean domain with non-negative weights. It is the first complete Holant dichotomy where constraint functions are not necessarily symmetric. Holant problems are indeed read-twice $\#$CSPs. Intuitively, some $\#$CSPs that are $\#$P-hard become tractable when restricting to read-twice instances. To capture them, we introduce the Block-rank-one condition. It turns out that the condition leads to a clear separation. If a function set $\…
We explore the intricate interdependent relationship among counting problems, considered from three frameworks for such problems: Holant Problems, counting CSP and weighted H-colorings. We consider these problems for general complex valued functions that take boolean inputs. We show that results from one framework can be used to derive results in another, and this happens in both directions. Holographic reductions discover an underlying unity, which is only revealed when these counting problems are investigated in the complex domain $\mathbb{C}$. We prove three complexity dichotomy theorems, l…
We define and explore a notion of unique prime factorization for constraint functions, and use this as a new tool to prove a complexity classification for counting weighted Eulerian orientation problems with arrow reversal symmetry (ARS). We prove that all such problems are either polynomial-time computable or #P-hard. We show that the class of weighted Eulerian orientation problems subsumes all weighted counting constraint satisfaction problems (#CSP) on Boolean variables. More significantly, we establish a novel connection between #CSP and counting weighted Eulerian orientation problems that…
The theorem states that: Every Boolean function can be $ε-approximated$ by a Disjunctive Normal Form (DNF) of size $O_ε(2^{n}/\log{n})$. This paper will demonstrate this theorem in detail by showing how this theorem is generated and proving its correctness. We will also dive into some specific Boolean functions and explore how these Boolean functions can be approximated by a DNF whose size is within the universal bound $O_ε(2^{n}/\log{n})$. The Boolean functions we interested in are: Parity Function: the parity function can be $ε-approximated$ by a DNF of width $(1 - 2ε)n$ and size $2^{(1 - 2ε…
We design a nonadaptive algorithm that, given oracle access to a function $f: \{0,1\}^n \to \{0,1\}$ which is $α$-far from monotone, makes poly$(n, 1/α)$ queries and returns an estimate that, with high probability, is an $\widetilde{O}(\sqrt{n})$-approximation to the distance of $f$ to monotonicity. The analysis of our algorithm relies on an improvement to the directed isoperimetric inequality of Khot, Minzer, and Safra (SIAM J. Comput., 2018). Furthermore, we rule out a poly$(n, 1/α)$-query nonadaptive algorithm that approximates the distance to monotonicity significantly better by showing th…
We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In~particular, we explore the approximability of monarchy-like functions where the value of the function is determined by a weighted combination of the vote of the first variable (the president) and the sum of the votes of all remaining variables. The pure version of this function is when the president can only be overruled by when all remaining variables agree. For every $k \geq 5$, we show that CSPs where the under…
