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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2026.43

Markov Chains with Rewinding

Authors: Amir Azarmehr, Soheil Behnezhad, Alma Ghafari, and Madhu Sudan

Published in: LIPIcs, Volume 392, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)

Abstract

Motivated by techniques developed in recent progress on lower bounds for sublinear time algorithms (Behnezhad, Roghani and Rubinstein, STOC 2023, FOCS 2023, and STOC 2024) we introduce and study a new class of randomized algorithmic processes that we call "Markov Chains with Rewinding". In this setting an agent/algorithm interacts with a (partially observable) Markovian random evolution by periodically/strategically rewinding the Markov chain to previous states. Depending on the application this may lead the evolution to desired states faster, or allow the agent to efficiently learn or test properties of the underlying Markov chain that may be infeasible or inefficient with passive observation. We study the task of identifying the initial state in a given partially observable Markov chain. Analysis of this question in specific Markov chains is the central ingredient in the above cited works and we aim to systematize the analysis in our work. Our first result is that any pair of states distinguishable with any rewinding strategy can also be distinguished with a non-adaptive rewinding strategy (i.e., one whose rewinding choices are predetermined before observing any outcomes of the chain). Therefore, while rewinding strategies can be shown to be strictly more powerful than passive strategies (i.e., those that do not rewind back to previous states), adaptivity does not give additional power to a rewinding strategy in the absence of efficiency considerations. The difference becomes apparent however when we introduce a natural efficiency measure, namely the query complexity (i.e., the number of observations they need to identify distinguishable states). Our second main contribution is to quantify this efficiency gap. We present a non-adaptive rewinding strategy whose query complexity is within a polynomial of that of the optimal (adaptive) strategy, and show that such a polynomial loss is necessary in general.

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Amir Azarmehr, Soheil Behnezhad, Alma Ghafari, and Madhu Sudan. Markov Chains with Rewinding. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 43:1-43:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{azarmehr_et_al:LIPIcs.APPROX/RANDOM.2026.43,
  author =	{Azarmehr, Amir and Behnezhad, Soheil and Ghafari, Alma and Sudan, Madhu},
  title =	{{Markov Chains with Rewinding}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
  pages =	{43:1--43:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-449-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{392},
  editor =	{Singh, Mohit and Gur, Tom},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.43},
  URN =		{urn:nbn:de:0030-drops-277603},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2026.43},
  annote =	{Keywords: Partially observable Markov chains, rewinding algorithm, sublinear algorithms}
}

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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2023.14

Robust Communication Complexity of Matching: EDCS Achieves 5/6 Approximation

Authors: Amir Azarmehr and Soheil Behnezhad

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract

We study the robust communication complexity of maximum matching. Edges of an arbitrary n-vertex graph G are randomly partitioned between Alice and Bob independently and uniformly. Alice has to send a single message to Bob such that Bob can find an (approximate) maximum matching of the whole graph G. We specifically study the best approximation ratio achievable via protocols where Alice communicates only Õ(n) bits to Bob. There has been a growing interest on the robust communication model due to its connections to the random-order streaming model. An algorithm of Assadi and Behnezhad [ICALP'21] implies a (2/3+ε₀ ∼ .667)-approximation for a small constant 0 < ε₀ < 10^{-18}, which remains the best-known approximation for general graphs. For bipartite graphs, Assadi and Behnezhad [Random'21] improved the approximation to .716 albeit with a computationally inefficient (i.e., exponential time) protocol. In this paper, we study a natural and efficient protocol implied by a random-order streaming algorithm of Bernstein [ICALP'20] which is based on edge-degree constrained subgraphs (EDCS) [Bernstein and Stein; ICALP'15]. The result of Bernstein immediately implies that this protocol achieves an (almost) (2/3 ∼ .666)-approximation in the robust communication model. We present a new analysis, proving that it achieves a much better (almost) (5/6 ∼ .833)-approximation. This significantly improves previous approximations both for general and bipartite graphs. We also prove that our analysis of Bernstein’s protocol is tight.

Cite as

Amir Azarmehr and Soheil Behnezhad. Robust Communication Complexity of Matching: EDCS Achieves 5/6 Approximation. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{azarmehr_et_al:LIPIcs.ICALP.2023.14,
  author =	{Azarmehr, Amir and Behnezhad, Soheil},
  title =	{{Robust Communication Complexity of Matching: EDCS Achieves 5/6 Approximation}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{14:1--14:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.14},
  URN =		{urn:nbn:de:0030-drops-180666},
  doi =		{10.4230/LIPIcs.ICALP.2023.14},
  annote =	{Keywords: Maximum Matching, Robust Communication Complexity, Edge Degree Constrained Subgraph}
}

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Documents authored by Azarmehr, Amir

Markov Chains with Rewinding

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Robust Communication Complexity of Matching: EDCS Achieves 5/6 Approximation

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