Hardness of Approximation for Stochastic Problems via Interactive Oracle Proofs

Gal Arnon, Alessandro Chiesa, Eylon Yogev

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations


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 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 ≤ 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 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 ≤ O(log n)] or in IR[k] has an O(k)-round IOP whose verifier has constant query complexity (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.

Original languageEnglish
Title of host publication37th Computational Complexity Conference, CCC 2022
EditorsShachar Lovett
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772419
StatePublished - 1 Jul 2022
Event37th Computational Complexity Conference, CCC 2022 - Philadelphia, United States
Duration: 20 Jul 202223 Jul 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference37th Computational Complexity Conference, CCC 2022
Country/TerritoryUnited States

Bibliographical note

Publisher Copyright:
© Gal Arnon, Alessandro Chiesa, and Eylon Yogev


  • hardness of approximation
  • interactive oracle proofs
  • stochastic satisfaction problems


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