2 Search Results for "Lagodzinski, J.A. Gregor"

Document
DOI: 10.4230/LIPIcs.STACS.2026.22
Modular Counting over 3-Element and Conservative Domains

Authors: Andrei A. Bulatov and Amirhossein Kazeminia

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
In the Constraint Satisfaction Problem (CSP for short) the goal is to decide the existence of a homomorphism from a given relational structure {G} to a given relational structure {H}. If the structure {H} is fixed and {G} is the only input, the problem is denoted CSP({H}). In its counting version, #CSP({H}), the task is to find the number of such homomorphisms. The CSP and #CSP have been used to model a wide variety of combinatorial problems and have received a tremendous amount of attention from researchers from multiple disciplines. In this paper we consider the modular version of the counting CSPs, that is, problems of the form #_pCSP({H}) of counting the number of homomorphisms to {H} modulo a fixed prime number p. Modular counting has been intensively studied during the last decade, although mainly in the case of graph homomorphisms. Here we continue the program of systematic research of modular counting of homomorphisms to general relational structures. The main results of the paper include a new way of reducing modular counting problems to smaller domains and a study of the complexity of such problems over 3-element domains and over conservative domains, that is, relational structures that allow to express (in a certain exact way) every possible unary predicate.
Cite as

Andrei A. Bulatov and Amirhossein Kazeminia. Modular Counting over 3-Element and Conservative Domains. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bulatov_et_al:LIPIcs.STACS.2026.22,
  author =	{Bulatov, Andrei A. and Kazeminia, Amirhossein},
  title =	{{Modular Counting over 3-Element and Conservative Domains}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{22:1--22:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.22},
  URN =		{urn:nbn:de:0030-drops-255114},
  doi =		{10.4230/LIPIcs.STACS.2026.22},
  annote =	{Keywords: Constraint Satisfaction Problem, Modular Counting}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.23
Fixed-Parameter Sensitivity Oracles

Authors: Davide Bilò, Katrin Casel, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, J.A. Gregor Lagodzinski, Martin Schirneck, and Simon Wietheger

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
We combine ideas from distance sensitivity oracles (DSOs) and fixed-parameter tractability (FPT) to design sensitivity oracles for FPT graph problems. An oracle with sensitivity f for an FPT problem Π on a graph G with parameter k preprocesses G in time O(g(f,k) ⋅ poly(n)). When queried with a set F of at most f edges of G, the oracle reports the answer to the Π - with the same parameter k - on the graph G-F, i.e., G deprived of F. The oracle should answer queries in a time that is significantly faster than merely running the best-known FPT algorithm on G-F from scratch. We design sensitivity oracles for the k-Path and the k-Vertex Cover problem. Our first oracle for k-Path has size O(k^{f+1}) and query time O(f min{f, log(f) + k}). We use a technique inspired by the work of Weimann and Yuster [FOCS 2010, TALG 2013] on distance sensitivity problems to reduce the space to O(({f+k}/f)^f ({f+k}/k)^k fk⋅log(n)) at the expense of increasing the query time to O(({f+k}/f)^f ({f+k}/k)^k f min{f,k}⋅log(n)). Both oracles can be modified to handle vertex-failures, but we need to replace k with 2k in all the claimed bounds. Regarding k-Vertex Cover, we design three oracles offering different trade-offs between the size and the query time. The first oracle takes O(3^{f+k}) space and has O(2^f) query time, the second one has a size of O(2^{f+k²+k}) and a query time of O(f+k²); finally, the third one takes O(fk+k²) space and can be queried in time O(1.2738^k + f). All our oracles are computable in time (at most) proportional to their size and the time needed to detect a k-path or k-vertex cover, respectively. We also provide an interesting connection between k-Vertex Cover and the fault-tolerant shortest path problem, by giving a DSO of size O(poly(f,k) ⋅ n) with query time in O(poly(f,k)), where k is the size of a vertex cover. Following our line of research connecting fault-tolerant FPT and shortest paths problems, we introduce parameterization to the computation of distance preservers. We study the problem, given a directed unweighted graph with a fixed source s and parameters f and k, to construct a polynomial-sized oracle that efficiently reports, for any target vertex v and set F of at most f edges, whether the distance from s to v increases at most by an additive term of k in G-F. The oracle size is O(2^k k²⋅n), while the time needed to answer a query is O(2^k f^ω k^ω), where ω < 2.373 is the matrix multiplication exponent. The second problem we study is about the construction of bounded-stretch fault-tolerant preservers. We construct a subgraph with O(2^{fk+f+k} k ⋅ n) edges that preserves those s-v-distances that do not increase by more than k upon failure of F. This improves significantly over the Õ(f n^{2-1/(2^f)}) bound in the unparameterized case by Bodwin et al. [ICALP 2017].
Cite as

Davide Bilò, Katrin Casel, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, J.A. Gregor Lagodzinski, Martin Schirneck, and Simon Wietheger. Fixed-Parameter Sensitivity Oracles. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.ITCS.2022.23,
  author =	{Bil\`{o}, Davide and Casel, Katrin and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Lagodzinski, J.A. Gregor and Schirneck, Martin and Wietheger, Simon},
  title =	{{Fixed-Parameter Sensitivity Oracles}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.23},
  URN =		{urn:nbn:de:0030-drops-156196},
  doi =		{10.4230/LIPIcs.ITCS.2022.23},
  annote =	{Keywords: data structures, distance preservers, distance sensitivity oracles, fault tolerance, fixed-parameter tractability, k-path, vertex cover}
}
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