A Boolean constraint satisfaction problem (CSP), Max-CSP($f$), is a maximization problem specified by a constraint $f:\{-1,1\}^k\to\{0,1\}$. An instance of the problem consists of $m$ constraint applications on $n$ Boolean variables, where each constraint application applies the constraint to $k$ literals chosen from the $n$ variables and their negations. The goal is to compute the maximum number of constraints that can be satisfied by a Boolean assignment to the $n$~variables. In the $(\gamma,\beta)$-approximation version of the problem for parameters $\gamma \geq \beta \in [0,1]$, 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 Max-CSP($f$) in the context of sketching algorithms and completely characterize the approximability of all Boolean CSPs. Specifically, given $f$, $\gamma$, and $\beta$ we show that either (1) the $(\gamma,\beta)$-approximation version of Max-CSP($f$) has a linear sketching algorithm using $O(\log n)$ space, or (2) for every $\epsilon > 0$ the $(\gamma-\epsilon,\beta+\epsilon)$-approximation version of Max-CSP($f$) requires $\Omega(\sqrt{n})$ space for any sketching algorithm. We also prove lower bounds against streaming algorithms for several CSPs. In particular, we recover the streaming dichotomy of [Chou-Golovnev-Velusamy FOCS'20] for $k=2$ and show streaming approximation resistance of all CSPs for which $f^{-1}(1)$ supports a distribution with uniform marginals.
Our positive results show wider applicability of bias-based algorithms used previously by [Guruswami-Velingker-Velusamy APPROX'17] and [Chou-Golovnev-Velusamy FOCS'20] 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.
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 Boolean constraint satisfaction problem (CSP), Max-CSP$(f)$, is a maximization problem specified by a constraint $f:\{-1,1\}^k\to\{0,1\}$. An instance of the problem consists of $m$ constraint applications on $n$ Boolean variables, where each constraint application applies the constraint to $k$ literals chosen from the $n$ variables and their negations. The goal is to compute the maximum number of constraints that can be satisfied by a Boolean assignment to the $n$~variables. In the $(\gamma,\beta)$-approximation version of the problem for parameters $\gamma \geq \beta \in [0,1]$, 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 Max-CSP$(f)$ in the (dynamic) streaming setting, where constraints are inserted (and may also be deleted in the dynamic setting) one at a time. We completely characterize the approximability of all Boolean CSPs in the dynamic streaming setting. Specifically, given $f$, $\gamma$ and $\beta$ we show that either (1) the $(\gamma,\beta)$-approximation version of Max-CSP$(f)$ has a probabilistic dynamic streaming algorithm using $O(\log n)$ space, or (2) for every $\varepsilon > 0$ the $(\gamma-\varepsilon,\beta+\varepsilon)$-approximation version of Max-CSP$(f)$ requires $\Omega(\sqrt{n})$ space for probabilistic dynamic streaming algorithms. We also extend previously known results in the insertion-only setting to a wide variety of cases, and in particular the case of $k=2$ where we get a dichotomy and the case when the satisfying assignments of $f$ support a distribution on $\{-1,1\}^k$ with uniform marginals.
Our positive results show wider applicability of bias-based algorithms used previously by [Guruswami-Velingker-Velusamy APPROX'17] and
[Chou-Golovnev-Velusamy FOCS'20] 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.
A Boolean constraint satisfaction problem (CSP), Max-CSP$(f)$, is a maximization problem specified by a constraint $f:\{-1,1\}^k\to\{0,1\}$. An instance of the problem consists of $m$ constraint applications on $n$ Boolean variables, where each constraint application applies the constraint to $k$ literals chosen from the $n$ variables and their negations. The goal is to compute the maximum number of constraints that can be satisfied by a Boolean assignment to the $n$~variables. In the $(\gamma,\beta)$-approximation version of the problem for parameters $\gamma \geq \beta \in [0,1]$, 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 Max-CSP$(f)$ in the (dynamic) streaming setting, where constraints are inserted (and may also be deleted in the dynamic setting) one at a time. We completely characterize the approximability of all Boolean CSPs in the dynamic streaming setting. Specifically, given $f$, $\gamma$ and $\beta$ we show that either (1) the $(\gamma,\beta)$-approximation version of Max-CSP$(f)$ has a probabilistic dynamic streaming algorithm using $O(\log n)$ space, or (2) for every $\varepsilon > 0$ the $(\gamma-\varepsilon,\beta+\varepsilon)$-approximation version of Max-CSP$(f)$ requires $\Omega(\sqrt{n})$ space for probabilistic dynamic streaming algorithms. We also extend previously known results in the insertion-only setting to a wide variety of cases, and in particular the case of $k=2$ where we get a dichotomy and the case when the satisfying assignments of $f$ support a distribution on $\{-1,1\}^k$ with uniform marginals.
Our positive results show wider applicability of bias-based algorithms used previously by [Guruswami-Velingker-Velusamy APPROX'17] and [Chou-Golovnev-Velusamy FOCS'20] 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.
This version of the paper replaces the previous version (now withdrawn). The previous version claimed Theorem 1.1 in the insertion-only streaming setting, the new version proves Theorem 1.1 in the dynamic streaming setting (and also extends previously known results in the insertion-only setting to a wide variety of cases). The status of Theorem 1.1 in the previous version is currently open.
A Boolean constraint satisfaction problem (CSP), Max-CSP$(f)$, is a maximization problem specified by a constraint $f:\{-1,1\}^k\to\{0,1\}$. An instance of the problem consists of $m$ constraint applications on $n$ Boolean variables, where each constraint application applies the constraint to $k$ literals chosen from the $n$ variables and their negations. The goal is to compute the maximum number of constraints that can be satisfied by a Boolean assignment to the $n$ variables. In the $(\gamma,\beta)$-approximation version of the problem for parameters $\gamma \geq \beta \in [0,1]$, 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 completely characterize the approximability of all Boolean CSPs in the streaming model. Specifically, given $f$, $\gamma$ and $\beta$ we show that either (1) the $(\gamma,\beta)$-approximation version of Max-CSP$(f)$ has a probabilistic streaming algorithm using $O(\log n)$ space, or (2) for every $\epsilon > 0$ the $(\gamma-\epsilon,\beta+\epsilon)$-approximation version of Max-CSP$(f)$ requires $\Omega(\sqrt{n})$ space for probabilistic streaming algorithms. Previously such a separation was known only for $k=2$. We stress that for $k=2$, there are only finitely many distinct problems to consider.
Our positive results show wider applicability of bias-based algorithms used previously by [Guruswami-Velingker-Velusamy APPROX'17], [Chou-Golovnev-Velusamy FOCS'20] by giving a systematic way to explore 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.
Improved the presentation.
A Boolean constraint satisfaction problem (CSP), Max-CSP$(f)$, is a maximization problem specified by a constraint $f:\{-1,1\}^k\to\{0,1\}$. An instance of the problem consists of $m$ constraint applications on $n$ Boolean variables, where each constraint application applies the constraint to $k$ literals chosen from the $n$ variables and their negations. The goal is to compute the maximum number of constraints that can be satisfied by a Boolean assignment to the $n$ variables. In the $(\gamma,\beta)$-approximation version of the problem for parameters $\gamma \geq \beta \in [0,1]$, 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 completely characterize the approximability of all Boolean CSPs in the streaming model. Specifically, given $f$, $\gamma$ and $\beta$ we show that either (1) the $(\gamma,\beta)$-approximation version of Max-CSP$(f)$ has a probabilistic streaming algorithm using $O(\log n)$ space, or (2) for every $\epsilon > 0$ the $(\gamma-\epsilon,\beta+\epsilon)$-approximation version of Max-CSP$(f)$ requires $\Omega(\sqrt{n})$ space for probabilistic streaming algorithms. Previously such a separation was known only for $k=2$. We stress that for $k=2$, there are only finitely many distinct problems to consider.
Our positive results show wider applicability of bias-based algorithms used previously by [Guruswami-Velingker-Velusamy APPROX'17], [Chou-Golovnev-Velusamy FOCS'20] by giving a systematic way to explore 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.
We regret that due to a fatal error in this paper, we are retracting the results of this paper. We are grateful to Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh Saxena, Zhao Song, and Huacheng Yu, for pointing out the error (in the proof of Claim 5.6). While some of the results (including the algorithmic result (Theorem 4.1) and the lower bound on the communication complexity of the RMD problem (Theorem 5.3)) continue to hold, the dichotomy claim (Theorem 1.1) is now open. We will post an updated version of this paper shortly.
