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Documents authored by Curticapean, Radu


Document
On the VNP-Hardness of Some Monomial Symmetric Polynomials

Authors: Radu Curticapean, Nutan Limaye, and Srikanth Srinivasan

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)


Abstract
A polynomial P ∈ 𝔽[x_1,…,x_n] is said to be symmetric if it is invariant under any permutation of its input variables. The study of symmetric polynomials is a classical topic in mathematics, specifically in algebraic combinatorics and representation theory. More recently, they have been studied in several works in computer science, especially in algebraic complexity theory. In this paper, we prove the computational hardness of one of the most basic kinds of symmetric polynomials: the monomial symmetric polynomials, which are obtained by summing all distinct permutations of a single monomial. This family of symmetric functions is a natural basis for the space of symmetric polynomials (over any field), and generalizes many well-studied families such as the elementary symmetric polynomials and the power-sum symmetric polynomials. We show that certain families of monomial symmetric polynomials are VNP-complete with respect to oracle reductions. This stands in stark contrast to the case of elementary and power symmetric polynomials, both of which have constant-depth circuits of polynomial size.

Cite as

Radu Curticapean, Nutan Limaye, and Srikanth Srinivasan. On the VNP-Hardness of Some Monomial Symmetric Polynomials. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 16:1-16:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{curticapean_et_al:LIPIcs.FSTTCS.2022.16,
  author =	{Curticapean, Radu and Limaye, Nutan and Srinivasan, Srikanth},
  title =	{{On the VNP-Hardness of Some Monomial Symmetric Polynomials}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{16:1--16:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.16},
  URN =		{urn:nbn:de:0030-drops-174081},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.16},
  annote =	{Keywords: algebraic complexity, symmetric polynomial, permanent, Sidon set}
}
Document
Determinants from Homomorphisms

Authors: Radu Curticapean

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)


Abstract
We give a new combinatorial explanation for well-known relations between determinants and traces of matrix powers. Such relations can be used to obtain polynomial-time and poly-logarithmic space algorithms for the determinant. Our new explanation avoids linear-algebraic arguments and instead exploits a classical connection between subgraph and homomorphism counts.

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Radu Curticapean. Determinants from Homomorphisms. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 38:1-38:7, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{curticapean:LIPIcs.ESA.2022.38,
  author =	{Curticapean, Radu},
  title =	{{Determinants from Homomorphisms}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{38:1--38:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.38},
  URN =		{urn:nbn:de:0030-drops-169767},
  doi =		{10.4230/LIPIcs.ESA.2022.38},
  annote =	{Keywords: determinant, homomorphisms, matrix trace, Newton identities}
}
Document
Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths

Authors: Radu Curticapean, Holger Dell, and Thore Husfeldt

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)


Abstract
We systematically investigate the complexity of counting subgraph patterns modulo fixed integers. For example, it is known that the parity of the number of k-matchings can be determined in polynomial time by a simple reduction to the determinant. We generalize this to an n^{f(t,s)}-time algorithm to compute modulo 2^t the number of subgraph occurrences of patterns that are s vertices away from being matchings. This shows that the known polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry over into the setting of counting modulo 2^t. Complementing our algorithm, we also give a simple and self-contained proof that counting k-matchings modulo odd integers q is {Mod}_q W[1]-complete and prove that counting k-paths modulo 2 is ⊕W[1]-complete, answering an open question by Björklund, Dell, and Husfeldt (ICALP 2015).

Cite as

Radu Curticapean, Holger Dell, and Thore Husfeldt. Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 34:1-34:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{curticapean_et_al:LIPIcs.ESA.2021.34,
  author =	{Curticapean, Radu and Dell, Holger and Husfeldt, Thore},
  title =	{{Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{34:1--34:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.34},
  URN =		{urn:nbn:de:0030-drops-146154},
  doi =		{10.4230/LIPIcs.ESA.2021.34},
  annote =	{Keywords: Counting complexity, matchings, paths, subgraphs, parameterized complexity}
}
Document
Parameterized Streaming Algorithms for Min-Ones d-SAT

Authors: Akanksha Agrawal, Arindam Biswas, Édouard Bonnet, Nick Brettell, Radu Curticapean, Dániel Marx, Tillmann Miltzow, Venkatesh Raman, and Saket Saurabh

Published in: LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)


Abstract
In this work, we initiate the study of the Min-Ones d-SAT problem in the parameterized streaming model. An instance of the problem consists of a d-CNF formula F and an integer k, and the objective is to determine if F has a satisfying assignment which sets at most k variables to 1. In the parameterized streaming model, input is provided as a stream, just as in the usual streaming model. A key difference is that the bound on the read-write memory available to the algorithm is O(f(k) log n) (f: N -> N, a computable function) as opposed to the O(log n) bound of the usual streaming model. The other important difference is that the number of passes the algorithm makes over its input must be a (preferably small) function of k. We design a (k + 1)-pass parameterized streaming algorithm that solves Min-Ones d-SAT (d >= 2) using space O((kd^(ck) + k^d)log n) (c > 0, a constant) and a (d + 1)^k-pass algorithm that uses space O(k log n). We also design a streaming kernelization for Min-Ones 2-SAT that makes (k + 2) passes and uses space O(k^6 log n) to produce a kernel with O(k^6) clauses. To complement these positive results, we show that any k-pass algorithm for or Min-Ones d-SAT (d >= 2) requires space Omega(max{n^(1/k) / 2^k, log(n / k)}) on instances (F, k). This is achieved via a reduction from the streaming problem POT Pointer Chasing (Guha and McGregor [ICALP 2008]), which might be of independent interest. Given this, our (k + 1)-pass parameterized streaming algorithm is the best possible, inasmuch as the number of passes is concerned. In contrast to the results of Fafianie and Kratsch [MFCS 2014] and Chitnis et al. [SODA 2015], who independently showed that there are 1-pass parameterized streaming algorithms for Vertex Cover (a restriction of Min-Ones 2-SAT), we show using lower bounds from Communication Complexity that for any d >= 1, a 1-pass streaming algorithm for Min-Ones d-SAT requires space Omega(n). This excludes the possibility of a 1-pass parameterized streaming algorithm for the problem. Additionally, we show that any p-pass algorithm for the problem requires space Omega(n/p).

Cite as

Akanksha Agrawal, Arindam Biswas, Édouard Bonnet, Nick Brettell, Radu Curticapean, Dániel Marx, Tillmann Miltzow, Venkatesh Raman, and Saket Saurabh. Parameterized Streaming Algorithms for Min-Ones d-SAT. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 8:1-8:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{agrawal_et_al:LIPIcs.FSTTCS.2019.8,
  author =	{Agrawal, Akanksha and Biswas, Arindam and Bonnet, \'{E}douard and Brettell, Nick and Curticapean, Radu and Marx, D\'{a}niel and Miltzow, Tillmann and Raman, Venkatesh and Saurabh, Saket},
  title =	{{Parameterized Streaming Algorithms for Min-Ones d-SAT}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Chattopadhyay, Arkadev and Gastin, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.8},
  URN =		{urn:nbn:de:0030-drops-115708},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.8},
  annote =	{Keywords: min, ones, sat, d-sat, parameterized, kernelization, streaming, space, efficient, algorithm, parameter}
}
Document
Counting Problems in Parameterized Complexity

Authors: Radu Curticapean

Published in: LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)


Abstract
This survey is an invitation to parameterized counting problems for readers with a background in parameterized algorithms and complexity. After an introduction to the peculiarities of counting complexity, we survey the parameterized approach to counting problems, with a focus on two topics of recent interest: Counting small patterns in large graphs, and counting perfect matchings and Hamiltonian cycles in well-structured graphs. While this survey presupposes familiarity with parameterized algorithms and complexity, we aim at explaining all relevant notions from counting complexity in a self-contained way.

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Radu Curticapean. Counting Problems in Parameterized Complexity. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 1:1-1:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{curticapean:LIPIcs.IPEC.2018.1,
  author =	{Curticapean, Radu},
  title =	{{Counting Problems in Parameterized Complexity}},
  booktitle =	{13th International Symposium on Parameterized and Exact Computation (IPEC 2018)},
  pages =	{1:1--1:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-084-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{115},
  editor =	{Paul, Christophe and Pilipczuk, Michal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.1},
  URN =		{urn:nbn:de:0030-drops-102026},
  doi =		{10.4230/LIPIcs.IPEC.2018.1},
  annote =	{Keywords: counting complexity, parameterized complexity, graph motifs, perfect matchings, graph minor theory, Hamiltonian cycles}
}
Document
A Fixed-Parameter Perspective on #BIS

Authors: Radu Curticapean, Holger Dell, Fedor V. Fomin, Leslie Ann Goldberg, and John Lapinskas

Published in: LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)


Abstract
The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class #RHPi_1. It is believed that #BIS does not have an efficient approximation algorithm but also that it is not NP-hard. We study the robustness of the intermediate complexity of #BIS by considering variants of the problem parameterised by the size of the independent set. We map the complexity landscape for three problems, with respect to exact computation and approximation and with respect to conventional and parameterised complexity. The three problems are counting independent sets of a given size, counting independent sets with a given number of vertices in one vertex class and counting maximum independent sets amongst those with a given number of vertices in one vertex class. Among other things, we show that all of these problems are NP-hard to approximate within any polynomial ratio. (This is surprising because the corresponding problems without the size parameter are complete in #RHPi_1, and hence are not believed to be NP-hard.) We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, whereas the other two are W[1]-hard to approximate even within any polynomial ratio. Finally, we show that, when restricted to graphs of bounded degree, all three problems have efficient exact fixed-parameter algorithms.

Cite as

Radu Curticapean, Holger Dell, Fedor V. Fomin, Leslie Ann Goldberg, and John Lapinskas. A Fixed-Parameter Perspective on #BIS. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 13:1-13:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{curticapean_et_al:LIPIcs.IPEC.2017.13,
  author =	{Curticapean, Radu and Dell, Holger and Fomin, Fedor V. and Goldberg, Leslie Ann and Lapinskas, John},
  title =	{{A Fixed-Parameter Perspective on #BIS}},
  booktitle =	{12th International Symposium on Parameterized and Exact Computation (IPEC 2017)},
  pages =	{13:1--13:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-051-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{89},
  editor =	{Lokshtanov, Daniel and Nishimura, Naomi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.13},
  URN =		{urn:nbn:de:0030-drops-85613},
  doi =		{10.4230/LIPIcs.IPEC.2017.13},
  annote =	{Keywords: Approximate counting, parameterised complexity, independent sets}
}
Document
Finding Detours is Fixed-Parameter Tractable

Authors: Ivona Bezáková, Radu Curticapean, Holger Dell, and Fedor V. Fomin

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


Abstract
We consider the following natural "above guarantee" parameterization of the classical longest path problem: For given vertices s and t of a graph G, and an integer k, the longest detour problem asks for an (s,t)-path in G that is at least k longer than a shortest (s,t)-path. Using insights into structural graph theory, we prove that the longest detour problem is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time exp(O(k)) * poly(n). This matches (up to the base of the exponential) the best algorithms for finding a path of length at least k. Furthermore, we study a related problem, exact detour, that asks whether a graph G contains an (s,t)-path that is exactly k longer than a shortest (s,t)-path. For this problem, we obtain a randomized algorithm with running time about 2.746^k * poly(n), and a deterministic algorithm with running time about 6.745^k * poly(n), showing that this problem is FPT as well. Our algorithms for the exact detour problem apply to both undirected and directed graphs.

Cite as

Ivona Bezáková, Radu Curticapean, Holger Dell, and Fedor V. Fomin. Finding Detours is Fixed-Parameter Tractable. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 54:1-54:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{bezakova_et_al:LIPIcs.ICALP.2017.54,
  author =	{Bez\'{a}kov\'{a}, Ivona and Curticapean, Radu and Dell, Holger and Fomin, Fedor V.},
  title =	{{Finding Detours is Fixed-Parameter Tractable}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{54:1--54:14},
  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.2017.54},
  URN =		{urn:nbn:de:0030-drops-74790},
  doi =		{10.4230/LIPIcs.ICALP.2017.54},
  annote =	{Keywords: longest path, fixed-parameter tractable algorithms, above-guarantee parameterization, graph minors}
}
Document
Counting Edge-Injective Homomorphisms and Matchings on Restricted Graph Classes

Authors: Radu Curticapean, Holger Dell, and Marc Roth

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)


Abstract
We consider the parameterized problem of counting all matchings with exactly k edges in a given input graph G. This problem is #W[1]-hard (Curticapean, ICALP 2013), so it is unlikely to admit f(k)poly(n) time algorithms. We show that #W[1]-hardness persists even when the input graph G comes from restricted graph classes, such as line graphs and bipartite graphs of arbitrary constant girth and maximum degree two on one side. To prove the result for line graphs, we observe that k-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of k paths of length two into (arbitrary) host graphs. Here, a homomorphism from H to G is edge-injective if it maps any two distinct edges of H to distinct edges in G. We show that edge-injective homomorphisms from a pattern graph H can be counted in polynomial time if H has bounded vertex-cover number after removing isolated edges. For hereditary classes H of pattern graphs, we obtain a full complexity dichotomy theorem by proving that counting edge-injective homomorphisms, restricted to patterns from H, is #W[1]-hard if no such bound exists. Our proofs rely on an edge-colored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.

Cite as

Radu Curticapean, Holger Dell, and Marc Roth. Counting Edge-Injective Homomorphisms and Matchings on Restricted Graph Classes. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 25:1-25:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{curticapean_et_al:LIPIcs.STACS.2017.25,
  author =	{Curticapean, Radu and Dell, Holger and Roth, Marc},
  title =	{{Counting Edge-Injective Homomorphisms and Matchings on Restricted Graph Classes}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{25:1--25:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.25},
  URN =		{urn:nbn:de:0030-drops-70080},
  doi =		{10.4230/LIPIcs.STACS.2017.25},
  annote =	{Keywords: matchings, homomorphisms, line graphs, counting complexity, parameterized complexity}
}
Document
Parity Separation: A Scientifically Proven Method for Permanent Weight Loss

Authors: Radu Curticapean

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)


Abstract
Given an edge-weighted graph G, let PerfMatch(G) denote the weighted sum over all perfect matchings M in G, weighting each matching M by the product of weights of edges in M. If G is unweighted, this plainly counts the perfect matchings of G. In this paper, we introduce parity separation, a new method for reducing PerfMatch to unweighted instances: For graphs G with edge-weights 1 and -1, we construct two unweighted graphs G1 and G2 such that PerfMatch(G) = PerfMatch(G1) - PerfMatch(G2). This yields a novel weight removal technique for counting perfect matchings, in addition to those known from classical #P-hardness proofs. Our technique is based upon the Holant framework and matchgates. We derive the following applications: Firstly, an alternative #P-completeness proof for counting unweighted perfect matchings. Secondly, C=P-completeness for deciding whether two given unweighted graphs have the same number of perfect matchings. To the best of our knowledge, this is the first C=P-completeness result for the “equality-testing version” of any natural counting problem that is not already #P-hard under parsimonious reductions. Thirdly, an alternative tight lower bound for counting unweighted perfect matchings under the counting exponential-time hypothesis #ETH.

Cite as

Radu Curticapean. Parity Separation: A Scientifically Proven Method for Permanent Weight Loss. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 47:1-47:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{curticapean:LIPIcs.ICALP.2016.47,
  author =	{Curticapean, Radu},
  title =	{{Parity Separation: A Scientifically Proven Method for Permanent Weight Loss}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{47:1--47:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.47},
  URN =		{urn:nbn:de:0030-drops-63279},
  doi =		{10.4230/LIPIcs.ICALP.2016.47},
  annote =	{Keywords: perfect matchings, counting complexity, structural complexity, exponentialtime hypothesis}
}
Document
Counting Matchings with k Unmatched Vertices in Planar Graphs

Authors: Radu Curticapean

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)


Abstract
We consider the problem of counting matchings in planar graphs. While perfect matchings in planar graphs can be counted by a classical polynomial-time algorithm [Kasteleyn 1961], the problem of counting all matchings (possibly containing unmatched vertices, also known as defects) is known to be #P-complete on planar graphs [Jerrum 1987]. To interpolate between matchings and perfect matchings, we study the parameterized problem of counting matchings with k unmatched vertices in a planar graph G, on input G and k. This setting has a natural interpretation in statistical physics, and it is a special case of counting perfect matchings in k-apex graphs (graphs that become planar after removing k vertices). Starting from a recent #W[1]-hardness proof for counting perfect matchings on k-apex graphs [Curtican and Xia 2015], we obtain: - Counting matchings with k unmatched vertices in planar graphs is #W[1]-hard. - In contrast, given a plane graph G with s distinguished faces, there is an O(2^s n^3) time algorithm for counting those matchings with k unmatched vertices such that all unmatched vertices lie on the distinguished faces. This implies an f(k,s)n^O(1) time algorithm for counting perfect matchings in k-apex graphs whose apex neighborhood is covered by s faces.

Cite as

Radu Curticapean. Counting Matchings with k Unmatched Vertices in Planar Graphs. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 33:1-33:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{curticapean:LIPIcs.ESA.2016.33,
  author =	{Curticapean, Radu},
  title =	{{Counting Matchings with k Unmatched Vertices in Planar Graphs}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{33:1--33:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.33},
  URN =		{urn:nbn:de:0030-drops-63847},
  doi =		{10.4230/LIPIcs.ESA.2016.33},
  annote =	{Keywords: counting complexity, parameterized complexity, matchings, planar graphs}
}
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