192 research outputs found
Matching Is as Easy as the Decision Problem, in the NC Model
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for
it? This has been an outstanding open question in TCS for over three decades,
ever since the discovery of randomized NC matching algorithms [KUW85, MVV87].
Over the last five years, the theoretical computer science community has
launched a relentless attack on this question, leading to the discovery of
several powerful ideas. We give what appears to be the culmination of this line
of work: An NC algorithm for finding a minimum-weight perfect matching in a
general graph with polynomially bounded edge weights, provided it is given an
oracle for the decision problem. Consequently, for settling the main open
problem, it suffices to obtain an NC algorithm for the decision problem. We
believe this new fact has qualitatively changed the nature of this open
problem.
All known efficient matching algorithms for general graphs follow one of two
approaches: given by Edmonds [Edm65] and Lov\'asz [Lov79]. Our oracle-based
algorithm follows a new approach and uses many of the ideas discovered in the
last five years.
The difficulty of obtaining an NC perfect matching algorithm led researchers
to study matching vis-a-vis clever relaxations of the class NC. In this vein,
recently Goldwasser and Grossman [GG15] gave a pseudo-deterministic RNC
algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC
algorithm with the additional requirement that on the same graph, it should
return the same (i.e., unique) perfect matching for almost all choices of
random bits. A corollary of our reduction is an analogous algorithm for general
graphs.Comment: Appeared in ITCS 202
The Matching Problem in General Graphs is in Quasi-NC
We show that the perfect matching problem in general graphs is in Quasi-NC.
That is, we give a deterministic parallel algorithm which runs in
time on processors. The result is obtained by a
derandomization of the Isolation Lemma for perfect matchings, which was
introduced in the classic paper by Mulmuley, Vazirani and Vazirani [1987] to
obtain a Randomized NC algorithm.
Our proof extends the framework of Fenner, Gurjar and Thierauf [2016], who
proved the analogous result in the special case of bipartite graphs. Compared
to that setting, several new ingredients are needed due to the significantly
more complex structure of perfect matchings in general graphs. In particular,
our proof heavily relies on the laminar structure of the faces of the perfect
matching polytope.Comment: Accepted to FOCS 2017 (58th Annual IEEE Symposium on Foundations of
Computer Science
Algorithmic Graph Theory
The main focus of this workshop was on mathematical techniques needed for the development of eïŹcient solutions and algorithms for computationally diïŹcult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions
The simple, little and slow things count : on parameterized counting complexity
In this thesis, we study the parameterized complexity of counting problems, as introduced by Flum and Grohe. This area mainly involves questions of the following kind: On inputs x with a parameter k, can we solve a given counting problem in time f(k)*|x|^c for a function f that depends only on k? In the positive case, we call the problem fixed-parameter tractable (fpt). Otherwise, we try to prove its #W[1]-hardness, which is the parameterized analogue of #P-hardness.
We introduce a general technique that bridges parameterized counting complexity and the so-called Holant framework. We then apply this technique to the problem of counting perfect matchings (or equivalently, the permanent) subject to structural parameters of the input graph G: On the algorithmic side, we introduce a new tractable structural parameter, namely, the minimal size of an excluded single-crossing minor of G. We complement this by showing that counting perfect matchings is #W[1]-hard when parameterized by the size of an arbitrary excluded minor.
Then we turn our attention to counting general subgraphs H other than perfect matchings in a host graph G. Instead of imposing structural parameters on G, we parameterize by the size of H, giving rise to the problems #Sub(C) for fixed graph classes C: For inputs H and G with H in C, we wish to count H-copies in G. Here, C could be the class of matchings, cycles, paths, or any other recursively enumerable class. We give a full dichotomy for these problems: Either #Sub(C) has a polynomial-time algorithm or it is #W[1]-complete. Assuming that FPT and #W[1] do not coincide, we can thus precisely identify the graph classes C for which the subgraph counting problem #Sub(C) admits polynomial-time algorithms.
Furthermore, we obtain an unexpected application of our extensions to the Holant framework: We show that, given two unweighted graphs, it is C=P-complete to decide whether they have the same number of perfect matchings.
Finally, we prove conditional lower bounds for counting problems under the counting exponential-time hypothesis #ETH. This hypothesis, introduced by Dell et al., asserts that the satisfying assignments to n-variable formulas in 3-CNF cannot be counted in time 2^o(n). Building upon this, we introduce a general technique that allows to derive tight lower bounds for other counting problems, such as counting perfect matchings, the Tutte polynomial, and the matching polynomial.Die vorliegende Arbeit befasst sich mit der parametrisierten KomplexitĂ€t von ZĂ€hlproblemen, einem von Flum und Grohe gegrĂŒndeten Gebiet, in welchem Fragen der folgenden Art betrachtet werden: Können gegebene Probleme auf Eingaben x mit Parameter k in Zeit f(k)*|x|^c gelöst werden, wobei f eine Funktion ist, die nur von k abhĂ€ngt? Im positiven Falle bezeichnen wir das Problem als parametrisierbar (FPT). Andernfalls versuchen wir typischerweise, dessen #W[1]-HĂ€rte zu beweisen - diese lĂ€sst sich vereinfachend als ein parametrisiertes Ăquivalent der #P-HĂ€rte auffassen.
Wir fĂŒhren zunĂ€chst eine allgemeine Technik ein, welche die parametrisierte ZĂ€hlkomplexitĂ€t mit dem sogenannten Holant-Rahmenwerk verbindet. AnschlieĂend setzen wir diese zum ZĂ€hlen perfekter Paarungen (oder Ă€quivalent, zur Auswertung der Permanente) unter strukturellen Parametern des Eingabegraphens G ein: Wir zeigen, dass das ZĂ€hlen perfekter Paarungen parametrisierbar ist durch die minimale GröĂe eines ausgeschlossenen Minors von G, der höchstens eine Kreuzung besitzt. Dieses algorithmische Resultat komplementieren wir durch die #W[1]-HĂ€rte des ZĂ€hlens perfekter Paarungen, wenn die minimale GröĂe eines beliebigen ausgeschlossenen Minors als Parameter betrachtet wird.
AnschlieĂend widmen wir uns dem ZĂ€hlen beliebiger Subgraphen H in Graphen G. Anstelle von strukturellen Parametern betrachten wir die GröĂe von H als Parameter und erhalten hierdurch die Probleme #Sub(C) fĂŒr feste Graphklassen C: Auf Eingaben H und G mit H in C gilt es, die H-Kopien in G zu zĂ€hlen. Hierbei kann C die Klasse der Paarungen, Zyklen, Pfade, oder eine beliebige andere Klasse von Graphen darstellen. Wir zeigen eine vollstĂ€ndige Dichotomie fĂŒr diese Probleme: Das Problem #Sub(C) ist entweder in P oder #W[1]-hart. Unter der gĂ€ngigen Annahme, dass FPT und #W[1] nicht zusammenfallen, erhalten wir somit eine vollstĂ€ndige Klassifikation der Polynomialzeit-lösbaren Probleme #Sub(C).
Weiterhin erhalten wir eine unerwartete Anwendung unserer Erweiterungen des Holant-Rahmenwerks: Wir zeigen die C=P-VollstĂ€ndigkeit der Frage, ob die Anzahlen perfekter Paarungen in zwei gegebenen ungewichteten Graphen ĂŒbereinstimmen.
Schlussendlich zeigen wir bedingte untere Schranken fĂŒr ZĂ€hlprobleme unter der ZĂ€hlversion der Exponentialzeithypothese #ETH, eingefĂŒhrt durch Dell et al. Diese postuliert, dass die erfĂŒllenden Belegungen in 3-KNF-Formeln mit n Variablen nicht in Zeit 2^o(n) gezĂ€hlt werden können. Darauf aufbauend fĂŒhren wir eine allgemeine Technik ein, die es ermöglicht, scharfe untere Schranken fĂŒr andere ZĂ€hlprobleme zu erhalten: Dies umfasst das ZĂ€hlen perfekter Paarungen, das Tutte-Polynom und das Paarungs-Polynom
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