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DOI: 10.4230/LIPIcs.ITCS.2025.41

Query Complexity of Stochastic Minimum Vertex Cover

Authors: Mahsa Derakhshan, Mohammad Saneian, and Zhiyang Xun

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

We study the stochastic minimum vertex cover problem for general graphs. In this problem, we are given a graph G = (V, E) and an existence probability p_e for each edge e ∈ E. Edges of G are realized (or exist) independently with these probabilities, forming the realized subgraph 𝒢. The existence of an edge in 𝒢 can only be verified using edge queries. The goal of this problem is to find a near-optimal vertex cover of 𝒢 using a small number of queries. Previous work by Derakhshan, Durvasula, and Haghtalab [STOC 2023] established the existence of 1.5 + ε approximation algorithms for this problem with O(n/ε) queries. They also show that, under mild correlation among edge realizations, beating this approximation ratio requires querying a subgraph of size Ω(n ⋅ RS(n)). Here, RS(n) refers to Ruzsa-Szemerédi Graphs and represents the largest number of induced edge-disjoint matchings of size Θ(n) in an n-vertex graph. In this work, we design a simple algorithm for finding a (1 + ε) approximate vertex cover by querying a subgraph of size O(n ⋅ RS(n)) for any absolute constant ε > 0. Our algorithm can tolerate up to O(n ⋅ RS(n)) correlated edges, hence effectively completing our understanding of the problem under mild correlation.

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Mahsa Derakhshan, Mohammad Saneian, and Zhiyang Xun. Query Complexity of Stochastic Minimum Vertex Cover. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 41:1-41:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{derakhshan_et_al:LIPIcs.ITCS.2025.41,
  author =	{Derakhshan, Mahsa and Saneian, Mohammad and Xun, Zhiyang},
  title =	{{Query Complexity of Stochastic Minimum Vertex Cover}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{41:1--41:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.41},
  URN =		{urn:nbn:de:0030-drops-226691},
  doi =		{10.4230/LIPIcs.ITCS.2025.41},
  annote =	{Keywords: Sublinear algorithms, Vertex cover, Query complexity}
}

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DOI: 10.4230/LIPIcs.MFCS.2022.32

On Algorithms Based on Finitely Many Homomorphism Counts

Authors: Yijia Chen, Jörg Flum, Mingjun Liu, and Zhiyang Xun

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract

It is well known [L. Lovász, 1967] that up to isomorphism a graph G is determined by the homomorphism counts hom(F, G), i.e., by the number of homomorphisms from F to G where F ranges over all graphs. Moreover, it suffices that F ranges over the graphs with at most as many vertices as G. Thus, in principle, we can answer any query concerning G with only accessing the hom(⋅, G)’s instead of G itself. In this paper, we deal with queries for which there is a hom algorithm, i.e., there are finitely many graphs F₁, …, F_k such that for any graph G whether it is a Yes-instance of the query is already determined by the vector hom^⟶_{F₁, …, F_k}(G): = (hom(F₁, G), …, hom(F_k, G)). We observe that planarity of graphs and 3-colorability of graphs, properties expressible in monadic second-order logic, have no hom algorithm. On the other hand, queries expressible as a Boolean combination of universal sentences in first-order logic FO have a hom algorithm. Even though it is not easy to find FO definable queries without a hom algorithm, we succeed to show this for the non-existence of an isolated vertex, a property expressible by the FO sentence ∀ x∃ y Exy, somehow the "simplest" graph property not definable by a Boolean combination of universal sentences. These results provide a characterization of the prefix classes of first-order logic with the property that each query definable by a sentence of the prefix class has a hom algorithm. For adaptive hom algorithms, i.e., algorithms that might access a hom(F_{i+1}, G) with F_{i+1} depending on hom(F_j, G) for 1 ≤ j ≤ i we show that three homomorphism counts hom(⋅, G) are sufficient and in general necessary to determine the (isomorphism type of) G. In particular, by three adaptive queries we can answer any question on G. Moreover, adaptively accessing two hom(⋅, G)’s is already enough to detect an isolated vertex. In 1993 Chaudhuri and Vardi [S. Chaudhuri and M. Y. Vardi, 1993] showed the analogue of the Lovász Isomorphism Theorem for the right homomorphism vector of a graph G, i.e, the vector of values hom(G,F) where F ranges over all graphs characterizes the isomorphism type of G. We study to what extent our results carry over to the right homomorphism vector.

Cite as

Yijia Chen, Jörg Flum, Mingjun Liu, and Zhiyang Xun. On Algorithms Based on Finitely Many Homomorphism Counts. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 32:1-32:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chen_et_al:LIPIcs.MFCS.2022.32,
  author =	{Chen, Yijia and Flum, J\"{o}rg and Liu, Mingjun and Xun, Zhiyang},
  title =	{{On Algorithms Based on Finitely Many Homomorphism Counts}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{32:1--32:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.32},
  URN =		{urn:nbn:de:0030-drops-168301},
  doi =		{10.4230/LIPIcs.MFCS.2022.32},
  annote =	{Keywords: homomorphism numbers, hom algorithms, adaptive hom algorithms}
}

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Query Complexity of Stochastic Minimum Vertex Cover

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On Algorithms Based on Finitely Many Homomorphism Counts

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