6 Search Results for "Schrijver, Alexander"


Document
The Complexity of Finding Fair Independent Sets in Cycles

Authors: Ishay Haviv

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)


Abstract
Let G be a cycle graph and let V₁,…,V_m be a partition of its vertex set into m sets. An independent set S of G is said to fairly represent the partition if |S ∩ V_i| ≥ 1/2⋅|V_i| - 1 for all i ∈ [m]. It is known that for every cycle and every partition of its vertex set, there exists an independent set that fairly represents the partition (Aharoni et al., A Journey through Discrete Math., 2017). We prove that the problem of finding such an independent set is PPA-complete. As an application, we show that the problem of finding a monochromatic edge in a Schrijver graph, given a succinct representation of a coloring that uses fewer colors than its chromatic number, is PPA-complete as well. The work is motivated by the computational aspects of the "cycle plus triangles" problem and of its extensions.

Cite as

Ishay Haviv. The Complexity of Finding Fair Independent Sets in Cycles. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{haviv:LIPIcs.ITCS.2021.4,
  author =	{Haviv, Ishay},
  title =	{{The Complexity of Finding Fair Independent Sets in Cycles}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{4:1--4:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.4},
  URN =		{urn:nbn:de:0030-drops-135431},
  doi =		{10.4230/LIPIcs.ITCS.2021.4},
  annote =	{Keywords: Fair independent sets in cycles, the complexity class \{PPA\}, Schrijver graphs}
}
Document
On a Theorem of Lovász that hom(⋅, H) Determines the Isomorphism Type of H

Authors: Jin-Yi Cai and Artem Govorov

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)


Abstract
Graph homomorphism has been an important research topic since its introduction [László Lovász, 1967]. Stated in the language of binary relational structures in that paper [László Lovász, 1967], Lovász proved a fundamental theorem that the graph homomorphism function G ↦ hom(G, H) for 0-1 valued H (as the adjacency matrix of a graph) determines the isomorphism type of H. In the past 50 years various extensions have been proved by Lovász and others [László Lovász, 2006; Michael Freedman et al., 2007; Christian Borgs et al., 2008; Alexander Schrijver, 2009; László Lovász and Balázs Szegedy, 2009]. These extend the basic 0-1 case to admit vertex and edge weights; but always with some restrictions such as all vertex weights must be positive. In this paper we prove a general form of this theorem where H can have arbitrary vertex and edge weights. An innovative aspect is that we prove this by a surprisingly simple and unified argument. This bypasses various technical obstacles and unifies and extends all previous known versions of this theorem on graphs. The constructive proof of our theorem can be used to make various complexity dichotomy theorems for graph homomorphism effective, i.e., it provides an algorithm that for any H either outputs a P-time algorithm solving hom(⋅, H) or a P-time reduction from a canonical #P-hard problem to hom(⋅, H).

Cite as

Jin-Yi Cai and Artem Govorov. On a Theorem of Lovász that hom(⋅, H) Determines the Isomorphism Type of H. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{cai_et_al:LIPIcs.ITCS.2020.17,
  author =	{Cai, Jin-Yi and Govorov, Artem},
  title =	{{On a Theorem of Lov\'{a}sz that hom(⋅, H) Determines the Isomorphism Type of H}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{17:1--17:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.17},
  URN =		{urn:nbn:de:0030-drops-117022},
  doi =		{10.4230/LIPIcs.ITCS.2020.17},
  annote =	{Keywords: Graph homomorphism, Partition function, Complexity dichotomy, Connection matrices and tensors}
}
Document
Approximating the Orthogonality Dimension of Graphs and Hypergraphs

Authors: Ishay Haviv

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


Abstract
A t-dimensional orthogonal representation of a hypergraph is an assignment of nonzero vectors in R^t to its vertices, such that every hyperedge contains two vertices whose vectors are orthogonal. The orthogonality dimension of a hypergraph H, denoted by overline{xi}(H), is the smallest integer t for which there exists a t-dimensional orthogonal representation of H. In this paper we study computational aspects of the orthogonality dimension of graphs and hypergraphs. We prove that for every k >= 4, it is NP-hard (resp. quasi-NP-hard) to distinguish n-vertex k-uniform hypergraphs H with overline{xi}(H) <= 2 from those satisfying overline{xi}(H) >= Omega(log^delta n) for some constant delta>0 (resp. overline{xi}(H) >= Omega(log^{1-o(1)} n)). For graphs, we relate the NP-hardness of approximating the orthogonality dimension to a variant of a long-standing conjecture of Stahl. We also consider the algorithmic problem in which given a graph G with overline{xi}(G) <= 3 the goal is to find an orthogonal representation of G of as low dimension as possible, and provide a polynomial time approximation algorithm based on semidefinite programming.

Cite as

Ishay Haviv. Approximating the Orthogonality Dimension of Graphs and Hypergraphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{haviv:LIPIcs.MFCS.2019.39,
  author =	{Haviv, Ishay},
  title =	{{Approximating the Orthogonality Dimension of Graphs and Hypergraphs}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{39:1--39:15},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.39},
  URN =		{urn:nbn:de:0030-drops-109836},
  doi =		{10.4230/LIPIcs.MFCS.2019.39},
  annote =	{Keywords: orthogonal representations of hypergraphs, orthogonality dimension, hardness of approximation, Kneser and Schrijver graphs, semidefinite programming}
}
Document
Shortest k-Disjoint Paths via Determinants

Authors: Samir Datta, Siddharth Iyer, Raghav Kulkarni, and Anish Mukherjee

Published in: LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)


Abstract
The well-known k-disjoint path problem (k-DPP) asks for pairwise vertex-disjoint paths between k specified pairs of vertices (s_i, t_i) in a given graph, if they exist. The decision version of the shortest k-DPP asks for the length of the shortest (in terms of total length) such paths. Similarly, the search and counting versions ask for one such and the number of such shortest set of paths, respectively. We restrict attention to the shortest k-DPP instances on undirected planar graphs where all sources and sinks lie on a single face or on a pair of faces. We provide efficient sequential and parallel algorithms for the search versions of the problem answering one of the main open questions raised by Colin de Verdière and Schrijver [Éric Colin de Verdière and Alexander Schrijver, 2011] for the general one-face problem. We do so by providing a randomised NC^2 algorithm along with an O(n^{omega/2}) time randomised sequential algorithm, for any fixed k. We also obtain deterministic algorithms with similar resource bounds for the counting and search versions. In contrast, previously, only the sequential complexity of decision and search versions of the "well-ordered" case has been studied. For the one-face case, sequential versions of our routines have better running times for constantly many terminals. The algorithms are based on a bijection between a shortest k-tuple of disjoint paths in the given graph and cycle covers in a related digraph. This allows us to non-trivially modify established techniques relating counting cycle covers to the determinant. We further need to do a controlled inclusion-exclusion to produce a polynomial sum of determinants such that all "bad" cycle covers cancel out in the sum allowing us to count "pure" cycle covers.

Cite as

Samir Datta, Siddharth Iyer, Raghav Kulkarni, and Anish Mukherjee. Shortest k-Disjoint Paths via Determinants. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 19:1-19:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{datta_et_al:LIPIcs.FSTTCS.2018.19,
  author =	{Datta, Samir and Iyer, Siddharth and Kulkarni, Raghav and Mukherjee, Anish},
  title =	{{Shortest k-Disjoint Paths via Determinants}},
  booktitle =	{38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)},
  pages =	{19:1--19:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-093-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{122},
  editor =	{Ganguly, Sumit and Pandya, Paritosh},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.19},
  URN =		{urn:nbn:de:0030-drops-99183},
  doi =		{10.4230/LIPIcs.FSTTCS.2018.19},
  annote =	{Keywords: disjoint paths, planar graph, parallel algorithm, cycle cover, determinant, inclusion-exclusion}
}
Document
Analysis of multi-stage open shop processing systems

Authors: Christian E.J. Eggermont, Alexander Schrijver, and Gerhard J. Woeginger

Published in: LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)


Abstract
We study algorithmic problems in multi-stage open shop processing systems that are centered around reachability and deadlock detection questions. We characterize safe and unsafe system states. We show that it is easy to recognize system states that can be reached from the initial state (where the system is empty), but that in general it is hard to decide whether one given system state is reachable from another given system state. We show that the problem of identifying reachable deadlock states is hard in general open shop systems, but is easy in the special case where no job needs processing on more than two machines (by linear programming and matching theory), and in the special case where all machines have capacity one (by graph-theoretic arguments).

Cite as

Christian E.J. Eggermont, Alexander Schrijver, and Gerhard J. Woeginger. Analysis of multi-stage open shop processing systems. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 484-494, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)


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@InProceedings{eggermont_et_al:LIPIcs.STACS.2011.484,
  author =	{Eggermont, Christian E.J. and Schrijver, Alexander and Woeginger, Gerhard J.},
  title =	{{Analysis of multi-stage open shop processing systems}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)},
  pages =	{484--494},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Schwentick, Thomas and D\"{u}rr, Christoph},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.484},
  URN =		{urn:nbn:de:0030-drops-30373},
  doi =		{10.4230/LIPIcs.STACS.2011.484},
  annote =	{Keywords: scheduling, resource allocation, deadlock, computational complexity}
}
Document
Shortest Vertex-Disjoint Two-Face Paths in Planar Graphs

Authors: Éric Colin de Verdiére and Alexander Schrijver

Published in: LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)


Abstract
Let $G$ be a directed planar graph of complexity~$n$, each arc having a nonnegative length. Let $s$ and~$t$ be two distinct faces of~$G$; let $s_1,ldots,s_k$ be vertices incident with~$s$; let $t_1,ldots,t_k$ be vertices incident with~$t$. We give an algorithm to compute $k$ pairwise vertex-disjoint paths connecting the pairs $(s_i,t_i)$ in~$G$, with minimal total length, in $O(knlog n)$ time.

Cite as

Éric Colin de Verdiére and Alexander Schrijver. Shortest Vertex-Disjoint Two-Face Paths in Planar Graphs. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 181-192, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)


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@InProceedings{colindeverdiere_et_al:LIPIcs.STACS.2008.1347,
  author =	{Colin de Verdi\'{e}re, \'{E}ric and Schrijver, Alexander},
  title =	{{Shortest Vertex-Disjoint Two-Face Paths in Planar Graphs}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{181--192},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Albers, Susanne and Weil, Pascal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1347},
  URN =		{urn:nbn:de:0030-drops-13474},
  doi =		{10.4230/LIPIcs.STACS.2008.1347},
  annote =	{Keywords: Algorithm, planar graph, disjoint paths, shortest path}
}
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