A constraint satisfaction problem (CSP), $\textsf{Max-CSP}(\mathcal{F})$, is specified by a finite set of constraints $\mathcal{F} \subseteq \{[q]^k \to \{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$ variables is given by $m$ applications of constraints from $\mathcal{F}$ to subsequences of the $n$ variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the $(\gamma,\beta)$-approximation version of the problem for parameters $0 \leq \beta < \gamma \leq 1$, the goal is to distinguish instances where at least $\gamma$ fraction of the constraints can be satisfied from instances where at most $\beta$ fraction of the constraints can be satisfied.

In this work we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family $\mathcal{F}$ and every $\beta < \gamma$, we show that either a linear sketching algorithm solves the problem in polylogarithmic space, or the problem is not solvable by any sketching algorithm in $o(\sqrt{n})$ space.

We also extend previously known lower bounds for general streaming algorithms
to a wide variety of problems, and in particular the case of $q=k=2$ where we get a dichotomy and the case when the satisfying assignments of $f$ support a distribution on $[q]^k$ with uniform marginals.

Prior to this work, other than sporadic examples, the only systematic class of CSPs that were analyzed considered the setting of Boolean variables $q=2$, binary constraints $k=2$, singleton families $|\mathcal{F}|=1$ and only considered the setting where constraints are placed on literals rather than variables.
Our positive results show wide applicability of bias-based algorithms used previously by [Guruswami-Velingker-Velusamy APPROX'17] and [Chou-Golovnev-Velusamy FOCS'20], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [Kapralov-Khanna-Sudan SODA'15], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results.
In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general $q$-ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.

The previous version claimed Theorem 1.1 (the dichotomy theorem) in the dynamic streaming setting. The new version replaces it with a dichotomy theorem for approximability of CSPs with sketching algorithms.

A constraint satisfaction problem (CSP), Max-CSP$({\cal F})$, is specified by a finite set of constraints ${\cal F} \subseteq \{[q]^k \to \{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$ variables is given by $m$ applications of constraints from ${\cal F}$ to subsequences of the $n$ variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the $(\gamma,\beta)$-approximation version of the problem for parameters $0 \leq \beta < \gamma \leq 1$, the goal is to distinguish instances where at least $\gamma$ fraction of the constraints can be satisfied from instances where at most $\beta$ fraction of the constraints can be satisfied.

In this work we consider the approximability of this problem in the context of streaming algorithms and give a dichotomy result in the dynamic setting, where constraints can be inserted or deleted. Specifically, for every family ${\cal F}$ and every $\beta < \gamma$, we show that either the approximation problem is solvable with polylogarithmic space in the dynamic setting, or not solvable with $o(\sqrt{n})$ space. We also establish tight inapproximability results for a broad subclass in the streaming insertion-only setting. Our work builds on, and significantly extends previous work by the authors who consider the special case of Boolean variables ($q=2$), singleton families ($|{\cal F}| = 1$) and where constraints may be placed on variables or their negations. Our framework extends non-trivially the previous work allowing us to appeal to richer norm estimation algorithms to get our algorithmic results. For our negative results we introduce new variants of the communication problems studied in the previous work, build new reductions for these problems, and extend the technical parts of previous works. In particular, previous works used Fourier analysis over the Boolean cube to prove their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general $q$-ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.

A constraint satisfaction problem (CSP), Max-CSP$({\cal F})$, is specified by a finite set of constraints ${\cal F} \subseteq \{[q]^k \to \{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$ variables is given by $m$ applications of constraints from ${\cal F}$ to subsequences of the $n$ variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the $(\gamma,\beta)$-approximation version of the problem for parameters $0 \leq \beta < \gamma \leq 1$, the goal is to distinguish instances where at least $\gamma$ fraction of the constraints can be satisfied from instances where at most $\beta$ fraction of the constraints can be satisfied.

In this work we consider the approximability of this problem in the context of streaming algorithms and give a dichotomy result in the dynamic setting, where constraints can be inserted or deleted. Specifically, for every family ${\cal F}$ and every $\beta < \gamma$, we show that either the approximation problem is solvable with polylogarithmic space in the dynamic setting, or not solvable with $o(\sqrt{n})$ space. We also establish tight inapproximability results for a broad subclass in the streaming insertion-only setting. Our work builds on, and significantly extends previous work by the authors who consider the special case of Boolean variables ($q=2$), singleton families ($|{\cal F}| = 1$) and where constraints may be placed on variables or their negations. Our framework extends non-trivially the previous work allowing us to appeal to richer norm estimation algorithms to get our algorithmic results. For our negative results we introduce new variants of the communication problems studied in the previous work, build new reductions for these problems, and extend the technical parts of previous works. In particular, previous works used Fourier analysis over the Boolean cube to prove their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general $q$-ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.

Streamlined the presentation

A constraint satisfaction problem (CSP), Max-CSP$({\cal F})$, is specified by a finite set of constraints ${\cal F} \subseteq \{[q]^k \to \{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$ variables is given by $m$ applications of constraints from ${\cal F}$ to subsequences of the $n$ variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the $(\gamma,\beta)$-approximation version of the problem for parameters $0 \leq \beta < \gamma \leq 1$, the goal is to distinguish instances where at least $\gamma$ fraction of the constraints can be satisfied from instances where at most $\beta$ fraction of the constraints can be satisfied.

In this work we consider the approximability of this problem in the context of streaming algorithms and give a dichotomy result in the dynamic setting, where constraints can be inserted or deleted. Specifically, for every family ${\cal F}$ and every $\beta < \gamma$, we show that either the approximation problem is solvable with polylogarithmic space in the dynamic setting, or not solvable with $o(\sqrt{n})$ space. We also establish tight inapproximability results for a broad subclass in the streaming insertion-only setting. Our work builds on, and significantly extends previous work by the authors who consider the special case of Boolean variables ($q=2$), singleton families ($|{\cal F}| = 1$) and where constraints may be placed on variables or their negations. Our framework extends non-trivially the previous work allowing us to appeal to richer norm estimation algorithms to get our algorithmic results. For our negative results we introduce new variants of the communication problems studied in the previous work, build new reductions for these problems, and extend the technical parts of previous works. In particular, previous works used Fourier analysis over the Boolean cube to prove their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general $q$-ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.