5 Search Results for "Varvitsiotis, Antonios"

Document
DOI: 10.4230/LIPIcs.MFCS.2025.37
New Hardness Results for Low-Rank Matrix Completion

Authors: Dror Chawin and Ishay Haviv

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
The low-rank matrix completion problem asks whether a given real matrix with missing values can be completed so that the resulting matrix has low rank or is close to a low-rank matrix. The completed matrix is often required to satisfy additional structural constraints, such as positive semi-definiteness or a bounded infinity norm. The problem arises in various research fields, including machine learning, statistics, and theoretical computer science, and has broad real-world applications. This paper presents new NP-hardness results for low-rank matrix completion problems. We show that for every sufficiently large integer d and any real number ε ∈ [2^{-O(d)},1/7], given a partial matrix A with exposed values of magnitude at most 1 that admits a positive semi-definite completion of rank d, it is NP-hard to find a positive semi-definite matrix that agrees with each given value of A up to an additive error of at most ε, even when the rank is allowed to exceed d by a multiplicative factor of O (1/(ε²⋅log(1/ε))). This strengthens a result of Hardt, Meka, Raghavendra, and Weitz (COLT, 2014), which applies to multiplicative factors smaller than 2 and to ε that decays polynomially in d. We establish similar NP-hardness results for the case where the completed matrix is constrained to have a bounded infinity norm (rather than be positive semi-definite), for which all previous hardness results rely on complexity assumptions related to the Unique Games Conjecture. Our proofs involve a novel notion of nearly orthonormal representations of graphs, the concept of line digraphs, and bounds on the rank of perturbed identity matrices.
Cite as

Dror Chawin and Ishay Haviv. New Hardness Results for Low-Rank Matrix Completion. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chawin_et_al:LIPIcs.MFCS.2025.37,
  author =	{Chawin, Dror and Haviv, Ishay},
  title =	{{New Hardness Results for Low-Rank Matrix Completion}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{37:1--37:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.37},
  URN =		{urn:nbn:de:0030-drops-241448},
  doi =		{10.4230/LIPIcs.MFCS.2025.37},
  annote =	{Keywords: hardness of approximation, low-rank matrix completion, graph coloring}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.61
Quantum Relaxations of CSP and Structure Isomorphism

Authors: Amin Karamlou

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
We investigate quantum relaxations of two key decision problems in computer science: the constraint satisfaction problem (CSP) and the structure isomorphism problem. CSP asks whether a homomorphism exists between two relational structures, while structure isomorphism seeks an isomorphism between them. In recent years, it has become increasingly apparent that many special cases of CSP can be reformulated in terms of the existence of perfect classical strategies in non-local games, a key topic of study in quantum information theory. These games have allowed us to study quantum advantage in relation to many important decision problems, such as the k-colouring problem, and the problem of solving binary constraint systems. Abramsky et al. (2017) have shown that all of these games can be seen as special instances of a non-local CSP game. Moreover, they show that perfect quantum strategies in this CSP game can be viewed as Kleisli morphisms of a graded monad on the category of relational structures, which they dub the quantum monad. In this way, the quantum monad provides a categorical characterisation of quantum advantage for the non-local CSP game. In this work we solidify and expand the results of Abramsky et al., answering several of their open questions. Firstly, we compare the definition of quantum graph homomorphisms arising from this work with an earlier definition of the concept due to Mančinska and Roberson and show that there are graphs which exhibit quantum advantage under one definition but not the other. Our second contribution is to extend the results of Abramsky et al. which only hold in the tensor product framework of quantum mechanics to the commuting operator framework. Next, we study a non-local structure isomorphism game, which generalises the well-studied graph isomorphism game. We show how the construction of the quantum monad can be refined to provide categorical semantics for quantum strategies in this game. This results in a category where morphisms coincide with quantum homomorphisms and isomorphisms coincide with quantum isomorphisms.
Cite as

Amin Karamlou. Quantum Relaxations of CSP and Structure Isomorphism. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 61:1-61:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{karamlou:LIPIcs.MFCS.2025.61,
  author =	{Karamlou, Amin},
  title =	{{Quantum Relaxations of CSP and Structure Isomorphism}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{61:1--61:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.61},
  URN =		{urn:nbn:de:0030-drops-241686},
  doi =		{10.4230/LIPIcs.MFCS.2025.61},
  annote =	{Keywords: CSP, graph isomorphism, quantum information, non-local game, quantum graph homomorphism, monad}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.105
NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability

Authors: Prem Nigam Kar, David E. Roberson, Tim Seppelt, and Peter Zeman

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
Mančinska and Roberson [FOCS'20] showed that two graphs are quantum isomorphic if and only if they are homomorphism indistinguishable over the class of planar graphs. Atserias et al. [JCTB'19] proved that quantum isomorphism is undecidable in general. The NPA hierarchy gives a sequence of semidefinite programming relaxations of quantum isomorphism. Recently, Roberson and Seppelt [ICALP'23] obtained a homomorphism indistinguishability characterization of the feasibility of each level of the Lasserre hierarchy of semidefinite programming relaxations of graph isomorphism. We prove a quantum analogue of this result by showing that each level of the NPA hierarchy of SDP relaxations for quantum isomorphism of graphs is equivalent to homomorphism indistinguishability over an appropriate class of planar graphs. By combining the convergence of the NPA hierarchy with the fact that the union of these graph classes is the set of all planar graphs, we are able to give a new proof of the result of Mančinska and Roberson [FOCS'20] that avoids the use of the theory of quantum groups. This homomorphism indistinguishability characterization also allows us to give a randomized polynomial-time algorithm deciding exact feasibility of each fixed level of the NPA hierarchy of SDP relaxations for quantum isomorphism.
Cite as

Prem Nigam Kar, David E. Roberson, Tim Seppelt, and Peter Zeman. NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 105:1-105:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kar_et_al:LIPIcs.ICALP.2025.105,
  author =	{Kar, Prem Nigam and Roberson, David E. and Seppelt, Tim and Zeman, Peter},
  title =	{{NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{105:1--105:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.105},
  URN =		{urn:nbn:de:0030-drops-234828},
  doi =		{10.4230/LIPIcs.ICALP.2025.105},
  annote =	{Keywords: Quantum isomorphism, NPA hierarchy, homomorphism indistinguishability}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.129
An Upper Bound on the Weisfeiler-Leman Dimension

Authors: Thomas Schneider and Pascal Schweitzer

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
The Weisfeiler-Leman (WL) algorithms form a family of incomplete approaches to the graph isomorphism problem. They recently found various applications in algorithmic group theory and machine learning. In fact, the algorithms form a parameterized family: for each k ∈ ℕ there is a corresponding k-dimensional algorithm WLk. The algorithms become increasingly powerful with increasing dimension, but at the same time the running time increases. The WL-dimension of a graph G is the smallest k ∈ ℕ for which WLk correctly decides isomorphism between G and every other graph. In some sense, the WL-dimension measures how difficult it is to test isomorphism of one graph to others using a fairly general class of combinatorial algorithms. Nowadays, it is a standard measure in descriptive complexity theory for the structural complexity of a graph. We prove that the WL-dimension of a graph on n vertices is at most 3/20 ⋅ n + o(n) = 0.15 ⋅ n + o(n). Reducing the question to coherent configurations, the proof develops various techniques to analyze their structure. This includes sufficient conditions under which a fiber can be restored uniquely up to isomorphism if it is removed, a recursive proof exploiting a degree reduction and treewidth bounds, as well as an exhaustive analysis of interspaces involving small fibers. As a base case, we also analyze the dimension of coherent configurations with small fiber size and thereby graphs with small color class size.
Cite as

Thomas Schneider and Pascal Schweitzer. An Upper Bound on the Weisfeiler-Leman Dimension. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 129:1-129:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{schneider_et_al:LIPIcs.ICALP.2025.129,
  author =	{Schneider, Thomas and Schweitzer, Pascal},
  title =	{{An Upper Bound on the Weisfeiler-Leman Dimension}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{129:1--129:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.129},
  URN =		{urn:nbn:de:0030-drops-235065},
  doi =		{10.4230/LIPIcs.ICALP.2025.129},
  annote =	{Keywords: Weisfeiler-Leman dimension, descriptive complexity, coherent configurations}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2017.76
Relaxations of Graph Isomorphism

Authors: Laura Mancinska, David E. Roberson, Robert Samal, Simone Severini, and Antonios Varvitsiotis

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Abstract
We introduce a nonlocal game that captures and extends the notion of graph isomorphism. This game can be won in the classical case if and only if the two input graphs are isomorphic. Thus, by considering quantum strategies we are able to define the notion of quantum isomorphism. We also consider the case of more general non-signalling strategies, and show that such a strategy exists if and only if the graphs are fractionally isomorphic. We prove several necessary conditions for quantum isomorphism, including cospectrality, and provide a construction for producing pairs of non-isomorphic graphs that are quantum isomorphic. We then show that both classical and quantum isomorphism can be reformulated as feasibility programs over the completely positive and completely positive semidefinite cones respectively. This leads us to considering relaxations of (quantum) isomorphism arrived at by relaxing the cone to either the doubly nonnegative (DNN) or positive semidefinite (PSD) cones. We show that DNN-isomorphism is equivalent to the previous defined notion of graph equivalence, a polynomial-time decidable relation that is related to coherent algebras. We also show that PSD-isomorphism implies several types of cospectrality, and that it is equivalent to cospectrality for connected 1-walk-regular graphs. Finally, we show that all of the above mentioned relations form a strict hierarchy of weaker and weaker relations, with non-singalling/fractional isomorphism being the weakest. The techniques used are an interesting mix of algebra, combinatorics, and quantum information.
Cite as

Laura Mancinska, David E. Roberson, Robert Samal, Simone Severini, and Antonios Varvitsiotis. Relaxations of Graph Isomorphism. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 76:1-76:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{mancinska_et_al:LIPIcs.ICALP.2017.76,
  author =	{Mancinska, Laura and Roberson, David E. and Samal, Robert and Severini, Simone and Varvitsiotis, Antonios},
  title =	{{Relaxations of Graph Isomorphism}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{76:1--76:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.76},
  URN =		{urn:nbn:de:0030-drops-74697},
  doi =		{10.4230/LIPIcs.ICALP.2017.76},
  annote =	{Keywords: graph isomorphism, quantum information, semidefinite programming}
}
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