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Linear Space Streaming Lower Bounds for Approximating CSPs

Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Ameya Velingker, Santhoshini Velusamy

TL;DR

This work establishes near-optimal linear-space lower bounds for streaming algorithms approximating Max-CSPs, extending Kapralov–Krachun-type hardness to a broad class of CSPs beyond Max Cut. The authors design a unified communication framework (IRMD/IFRMD) and a robust Fourier-analytic boundedness toolkit to show that improving over trivial approximability by a factor at most q requires Ω(n) space in one-pass streaming. Central to the approach is a tight q^{-(k-1)}-inapproximability for Max k-EQ(q), analyzed via a multi-player folded-encoding reduction and a delicate induction on posterior sets, controlled by a trio of boundedness lemmas. The results imply approximation resistance for wide CSP families and give a sharp q-factor bound on any sublinear-space approximation for general CSPs, with concrete implications for Max q-coloring, Unique Games, and Max-Lin systems. Overall, the paper significantly advances our understanding of the boundaries between sublinear-space streaming capabilities and CSP approximability.

Abstract

We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on $n$ variables taking values in $\{0,\ldots,q-1\}$, we prove that improving over the trivial approximability by a factor of $q$ requires $Ω(n)$ space even on instances with $O(n)$ constraints. We also identify a broad subclass of problems for which any improvement over the trivial approximability requires $Ω(n)$ space. The key technical core is an optimal, $q^{-(k-1)}$-inapproximability for the Max $k$-LIN-mod $q$ problem, which is the Max CSP problem where every constraint is given by a system of $k-1$ linear equations $\bmod q$ over $k$ variables. Our work builds on and extends the breakthrough work of Kapralov and Krachun (Proc. STOC 2019) who showed a linear lower bound on any non-trivial approximation of the MaxCut problem in graphs. MaxCut corresponds roughly to the case of Max $k$-LIN-mod $q$ with ${k=q=2}$. For general CSPs in the streaming setting, prior results only yielded $Ω(\sqrt{n})$ space bounds. In particular no linear space lower bound was known for an approximation factor less than $1/2$ for any CSP. Extending the work of Kapralov and Krachun to Max $k$-LIN-mod $q$ to $k>2$ and $q>2$ (while getting optimal hardness results) is the main technical contribution of this work. Each one of these extensions provides non-trivial technical challenges that we overcome in this work.

Linear Space Streaming Lower Bounds for Approximating CSPs

TL;DR

This work establishes near-optimal linear-space lower bounds for streaming algorithms approximating Max-CSPs, extending Kapralov–Krachun-type hardness to a broad class of CSPs beyond Max Cut. The authors design a unified communication framework (IRMD/IFRMD) and a robust Fourier-analytic boundedness toolkit to show that improving over trivial approximability by a factor at most q requires Ω(n) space in one-pass streaming. Central to the approach is a tight q^{-(k-1)}-inapproximability for Max k-EQ(q), analyzed via a multi-player folded-encoding reduction and a delicate induction on posterior sets, controlled by a trio of boundedness lemmas. The results imply approximation resistance for wide CSP families and give a sharp q-factor bound on any sublinear-space approximation for general CSPs, with concrete implications for Max q-coloring, Unique Games, and Max-Lin systems. Overall, the paper significantly advances our understanding of the boundaries between sublinear-space streaming capabilities and CSP approximability.

Abstract

We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on variables taking values in , we prove that improving over the trivial approximability by a factor of requires space even on instances with constraints. We also identify a broad subclass of problems for which any improvement over the trivial approximability requires space. The key technical core is an optimal, -inapproximability for the Max -LIN-mod problem, which is the Max CSP problem where every constraint is given by a system of linear equations over variables. Our work builds on and extends the breakthrough work of Kapralov and Krachun (Proc. STOC 2019) who showed a linear lower bound on any non-trivial approximation of the MaxCut problem in graphs. MaxCut corresponds roughly to the case of Max -LIN-mod with . For general CSPs in the streaming setting, prior results only yielded space bounds. In particular no linear space lower bound was known for an approximation factor less than for any CSP. Extending the work of Kapralov and Krachun to Max -LIN-mod to and (while getting optimal hardness results) is the main technical contribution of this work. Each one of these extensions provides non-trivial technical challenges that we overcome in this work.

Paper Structure

This paper contains 53 sections, 38 theorems, 143 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Prior work
  5. Techniques and new contributions
  6. Comparison with CGSV21-conference.
  7. Summary of KK19.
  8. Our Analysis.
  9. Incremental contribution over KK19.
  10. Organization of the rest of the paper
  11. Preliminaries
  12. Total variation distance
  13. Concentration inequality
  14. Fourier analysis
  15. Communication problems
  16. ...and 38 more sections

Key Result

theorem 1.1

For every $q,k$ and every wide family $\mathcal{F}$, $\textsf{Max-CSP}(\mathcal{F})$ is approximation-resistant.

Figures (4)

  • Figure 1: A pictorial overview of the proof of \ref{['lem:hybrid']}. Each posterior set $B_t$ (the blue sets) is both $(M_t,\mathbf{c}_t)$-restricted (followed from \ref{['def:set and pdf']}) and $(M_t,C_0,s)$-reduced (followed from \ref{['lem:boundedness base case']}). Each aggregated posterior set $B_{1:t}$ (the orange sets) is $(C_t,s)$-bounded (followed from \ref{['lem:induction step']}).
  • Figure 2: An example of $A_\mathbf{c},\tilde{A}_\mathbf{c},A^{(1)}_\mathbf{c},\tilde{A}^{(1)}_\mathbf{c}$ with $m=1$, $n=7$, $k=4$, $e_1=\{1,3,5,7\}$ and $c_1=\{5\}$.
  • Figure 3: Upper bound the cardinality of $T_\ell$. In this example, $q=3, k=5, m=5$, and $\mathbf{v}$ is specified by integers in red. Note that $E=\{2,3,5\}$ and $O=\{1,4\}$. Next, we consider $h=8,\ell=4$ and a $\mathbf{u}\in T_\ell$ specified by integers in blue. Note that by definition we have $\eta=o=2$. In particular, the tuple $(W_e(\mathbf{u}),W_o(\mathbf{u}))$ is described on the right and $\mathbf{u}+\mathbf{v}$ is specified by integers in green. It is immediate to see that $(W_e(\mathbf{u}),W_o(\mathbf{u}),\mathbf{v})$ uniquely specifies $\mathbf{u}$ because one can subtract $W_e(\mathbf{u})$ and $W_o(\mathbf{u})$ by $\mathbf{v}$ to get the value of $\mathbf{u}$ in those coordinates. In the rest of the coordinates, $\mathbf{u}$ has the same values as $\mathbf{v}$. Moreover, observe that every hyperedge in $W_e(\mathbf{u})$ should contain at least $2$ non-zero points because both $\mathbf{u}$ and $\mathbf{v}$ sum up to $0$ mod $q$ within those hyperedges.
  • Figure 4: A graphical intuition for the parameters appeared in \ref{['def:pcomb']}.

Theorems & Definitions (90)

  • theorem 1.1
  • theorem 1.2
  • definition 2.1: Total variation distance of discrete random variables
  • proposition 2.2: E.g.,KKS
  • lemma 2.3
  • proof
  • lemma 2.4: KK19
  • lemma 2.5: KK19
  • lemma 2.6: Parseval's identity
  • lemma 2.8: Convolution Theorem
  • ...and 80 more