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DOI: 10.4230/LIPIcs.STACS.2024.19

The Complexity of Homomorphism Reconstructibility

Authors: Jan Böker, Louis Härtel, Nina Runde, Tim Seppelt, and Christoph Standke

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Abstract

Representing graphs by their homomorphism counts has led to the beautiful theory of homomorphism indistinguishability in recent years. Moreover, homomorphism counts have promising applications in database theory and machine learning, where one would like to answer queries or classify graphs solely based on the representation of a graph G as a finite vector of homomorphism counts from some fixed finite set of graphs to G. We study the computational complexity of the arguably most fundamental computational problem associated to these representations, the homomorphism reconstructability problem: given a finite sequence of graphs and a corresponding vector of natural numbers, decide whether there exists a graph G that realises the given vector as the homomorphism counts from the given graphs. We show that this problem yields a natural example of an NP^#𝖯-hard problem, which still can be NP-hard when restricted to a fixed number of input graphs of bounded treewidth and a fixed input vector of natural numbers, or alternatively, when restricted to a finite input set of graphs. We further show that, when restricted to a finite input set of graphs and given an upper bound on the order of the graph G as additional input, the problem cannot be NP-hard unless 𝖯 = NP. For this regime, we obtain partial positive results. We also investigate the problem’s parameterised complexity and provide fpt-algorithms for the case that a single graph is given and that multiple graphs of the same order with subgraph instead of homomorphism counts are given.

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Jan Böker, Louis Härtel, Nina Runde, Tim Seppelt, and Christoph Standke. The Complexity of Homomorphism Reconstructibility. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{boker_et_al:LIPIcs.STACS.2024.19,
  author =	{B\"{o}ker, Jan and H\"{a}rtel, Louis and Runde, Nina and Seppelt, Tim and Standke, Christoph},
  title =	{{The Complexity of Homomorphism Reconstructibility}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{19:1--19:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.19},
  URN =		{urn:nbn:de:0030-drops-197298},
  doi =		{10.4230/LIPIcs.STACS.2024.19},
  annote =	{Keywords: graph homomorphism, counting complexity, parameterised complexity}
}

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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2021.32

Graph Similarity and Homomorphism Densities

Authors: Jan Böker

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract

We introduce the tree distance, a new distance measure on graphs. The tree distance can be computed in polynomial time with standard methods from convex optimization. It is based on the notion of fractional isomorphism, a characterization based on a natural system of linear equations whose integer solutions correspond to graph isomorphism. By results of Tinhofer (1986, 1991) and Dvořák (2010), two graphs G and H are fractionally isomorphic if and only if, for every tree T, the number of homomorphisms from T to G equals the corresponding number from T to H, which means that the tree distance of G and H is zero. Our main result is that this correspondence between the equivalence relations "fractional isomorphism" and "equal tree homomorphism densities" can be extended to a correspondence between the associated distance measures. Our result is inspired by a similar result due to Lovász and Szegedy (2006) and Borgs, Chayes, Lovász, Sós, and Vesztergombi (2008) that connects the cut distance of graphs to their homomorphism densities (over all graphs), which is a fundamental theorem in the theory of graph limits. We also introduce the path distance of graphs and take the corresponding result of Dell, Grohe, and Rattan (2018) for exact path homomorphism counts to an approximate level. Our results answer an open question of Grohe (2020) and help to build a theoretical understanding of vector embeddings of graphs. The distance measures we define turn out be closely related to the cut distance. We establish our main results by generalizing our definitions to graphons, which are limit objects of sequences of graphs, as this allows us to apply techniques from functional analysis. We prove the fairly general statement that, for every "reasonably" defined graphon pseudometric, an exact correspondence to homomorphism densities can be turned into an approximate one. We also provide an example of a distance measure that violates this reasonableness condition. This incidentally answers an open question of Grebík and Rocha (2021).

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Jan Böker. Graph Similarity and Homomorphism Densities. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{boker:LIPIcs.ICALP.2021.32,
  author =	{B\"{o}ker, Jan},
  title =	{{Graph Similarity and Homomorphism Densities}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{32:1--32:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.32},
  URN =		{urn:nbn:de:0030-drops-141014},
  doi =		{10.4230/LIPIcs.ICALP.2021.32},
  annote =	{Keywords: graph similarity, homomorphism densities, cut distance}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2019.54

The Complexity of Homomorphism Indistinguishability

Authors: Jan Böker, Yijia Chen, Martin Grohe, and Gaurav Rattan

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract

For every graph class {F}, let HomInd({F}) be the problem of deciding whether two given graphs are homomorphism-indistinguishable over {F}, i.e., for every graph F in {F}, the number hom(F, G) of homomorphisms from F to G equals the corresponding number hom(F, H) for H. For several natural graph classes (such as paths, trees, bounded treewidth graphs), homomorphism-indistinguishability over the class has an efficient structural characterization, resulting in polynomial time solvability [H. Dell et al., 2018]. In particular, it is known that two non-isomorphic graphs are homomorphism-indistinguishable over the class {T}_k of graphs of treewidth k if and only if they are not distinguished by k-dimensional Weisfeiler-Leman algorithm, a central heuristic for isomorphism testing: this characterization implies a polynomial time algorithm for HomInd({T}_k), for every fixed k in N. In this paper, we show that there is a polynomial-time-decidable class {F} of undirected graphs of bounded treewidth such that HomInd({F}) is undecidable. Our second hardness result concerns the class {K} of complete graphs. We show that HomInd({K}) is co-NP-hard, and in fact, we show completeness for the class C_=P (under P-time Turing reductions). On the algorithmic side, we show that HomInd({P}) can be solved in polynomial time for the class {P} of directed paths. We end with a brief study of two variants of the HomInd({F}) problem: (a) the problem of lexographic-comparison of homomorphism numbers of two graphs, and (b) the problem of computing certain distance-measures (defined via homomorphism numbers) between two graphs.

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Jan Böker, Yijia Chen, Martin Grohe, and Gaurav Rattan. The Complexity of Homomorphism Indistinguishability. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{boker_et_al:LIPIcs.MFCS.2019.54,
  author =	{B\"{o}ker, Jan and Chen, Yijia and Grohe, Martin and Rattan, Gaurav},
  title =	{{The Complexity of Homomorphism Indistinguishability}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{54:1--54:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.54},
  URN =		{urn:nbn:de:0030-drops-109980},
  doi =		{10.4230/LIPIcs.MFCS.2019.54},
  annote =	{Keywords: graph homomorphism numbers, counting complexity, treewidth}
}

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Documents authored by Böker, Jan

The Complexity of Homomorphism Reconstructibility

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Graph Similarity and Homomorphism Densities

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The Complexity of Homomorphism Indistinguishability

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