**Hardness of Approximation for Stochastic Problems via Interactive Oracle Proofs**

*Gal Arnon and Alessandro Chiesa and Eylon Yogev*

**Abstract: **Hardness of approximation aims to establish lower bounds on the approximability of optimization problems in NP and beyond. We continue the study of hardness of approximation for problems beyond NP, specifically for \emph{stochastic} constraint satisfaction problems (SCSPs). An SCSP with $k$ alternations is a list of constraints over variables grouped into $2k$ blocks, where each constraint has constant arity.
An assignment to the SCSP is defined by two players who alternate in setting values to a designated block of variables, with one player choosing their assignments uniformly at random and the other player trying to maximize the number of satisfied constraints.

In this paper, we establish hardness of approximation for SCSPs based on interactive proofs. For $k \leq O(\log n)$, we prove that it is $AM[k]$-hard to approximate, to within a constant, the value of SCSPs with $k$ alternations and constant arity. Before, this was known only for $k = O(1)$.

Furthermore, we introduce a natural class of $k$-round interactive proofs, denoted $IR[k]$ (for \emph{interactive reducibility}), and show that several protocols (e.g., the sumcheck protocol) are in $IR[k]$. Using this notion, we extend our inapproximability to all values of $k$: we show that for every $k$, approximating an SCSP instance with $O(k)$ alternations and constant arity is $IR[k]$-hard.

While hardness of approximation for CSPs is achieved by constructing suitable PCPs, our results for SCSPs are achieved by constructing suitable IOPs (interactive oracle proofs). We show that every language in $AM[k \leq O(\log n)]$ or in $IR[k]$ has an $O(k)$-round IOP whose verifier has \emph{constant} query complexity (\emph{regardless} of the number of rounds $k$). In particular, we derive a ``sumcheck protocol'' whose verifier reads $O(1)$ bits from the entire interaction transcript.

**Category / Keywords: **foundations / hardness of approximation; interactive oracle proofs; stochastic satisfaction problems

**Date: **received 14 Feb 2022, last revised 14 Feb 2022

**Contact author: **gal arnon at weizmann ac il, alessandro chiesa at epfl ch, eylon yogev at biu ac il

**Available format(s): **PDF | BibTeX Citation

**Version: **20220220:200720 (All versions of this report)

**Short URL: **ia.cr/2022/168

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