🤖 AI Summary
This work investigates the approximability threshold of constraint satisfaction problems (CSPs) in the streaming model. For $n$-variable CSPs over domain ${0,dots,q-1}$ with $O(n)$ constraints, we prove that any algorithm achieving approximation ratio strictly better than the trivial $1/q$ requires $Omega(n)$ space—establishing the first linear-space lower bound for approximation ratios below $1/2$. Methodologically, we extend the Kapralov–Krachun linear lower-bound technique to general CSPs (surpassing prior $Omega(sqrt{n})$ bounds) via modular-$q$ linear equation encoding, communication complexity analysis, and pseudorandom hard-instance reduction. This yields optimal $q^{-(k-1)}$ inapproximability for Max $k$-LIN mod $q$ with $k>2$, $q>2$. Our results uniformly characterize the streaming hardness of broad CSP subclasses: all nontrivial approximation requires essentially linear space, significantly advancing the theoretical understanding of streaming algorithm limitations.
📝 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,…,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 mod 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(√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.
