273 research outputs found
Distributed Symmetry Breaking in Hypergraphs
Fundamental local symmetry breaking problems such as Maximal Independent Set
(MIS) and coloring have been recognized as important by the community, and
studied extensively in (standard) graphs. In particular, fast (i.e.,
logarithmic run time) randomized algorithms are well-established for MIS and
-coloring in both the LOCAL and CONGEST distributed computing
models. On the other hand, comparatively much less is known on the complexity
of distributed symmetry breaking in {\em hypergraphs}. In particular, a key
question is whether a fast (randomized) algorithm for MIS exists for
hypergraphs.
In this paper, we study the distributed complexity of symmetry breaking in
hypergraphs by presenting distributed randomized algorithms for a variety of
fundamental problems under a natural distributed computing model for
hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can
be solved in rounds ( is the number of nodes of the
hypergraph) in the LOCAL model. We then present a key result of this paper ---
an -round hypergraph MIS algorithm in
the CONGEST model where is the maximum node degree of the hypergraph
and is any arbitrarily small constant.
To demonstrate the usefulness of hypergraph MIS, we present applications of
our hypergraph algorithm to solving problems in (standard) graphs. In
particular, the hypergraph MIS yields fast distributed algorithms for the {\em
balanced minimal dominating set} problem (left open in Harris et al. [ICALP
2013]) and the {\em minimal connected dominating set problem}. We also present
distributed algorithms for coloring, maximal matching, and maximal clique in
hypergraphs.Comment: Changes from the previous version: More references adde
Generalisation : graphs and colourings
The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-HĂŒbner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro PezzĂ©, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
MƱszaki informatikai problémåkhoz kapcsolódó diszkrét matematikai modellek vizsgålata = Discrete mathematical models related to problems in informatics
DiszkrĂ©t matematikai mĂłdszerekkel strukturĂĄlis Ă©s kvantitatĂv összefĂŒggĂ©seket bizonyĂtottunk; algoritmusokat terveztĂŒnk, komplexitĂĄsukat elemeztĂŒk. Az eredmĂ©nyek a grĂĄfok Ă©s hipergrĂĄfok elmĂ©letĂ©hez, valamint on-line ĂŒtemezĂ©shez kapcsolĂłdnak. NĂ©hĂĄny kiemelĂ©s: - Pontosan leĂrtuk azokat a szerkezeti feltĂ©teleket, amelyeknek teljesĂŒlni kell ahhoz, hogy egy kommunikĂĄciĂłs hĂĄlĂłzatban Ă©s annak minden összefĂŒggĆ rĂ©szĂ©ben legyen olyan, megadott tĂpusĂș összefĂŒggĆ rĂ©szhĂĄlĂłzat, ahonnan az összes többi elem közvetlenĂŒl elĂ©rhetĆ. (A problĂ©ma kĂ©t Ă©vtizeden ĂĄt megoldatlan volt.) - Aszimptotikusan pontos becslĂ©st adtunk egy n-elemƱ alaphalmaz olyan, k-asokbĂłl ĂĄllĂł halmazrendszereinek minimĂĄlis mĂ©retĂ©re, amelyekben minden k-osztĂĄlyĂș partĂciĂłhoz van olyan halmaz, ami az összes partĂciĂł-osztĂĄlyt metszi. (Nyitott problĂ©ma volt 1973 Ăłta, több szerzĆ egymĂĄstĂłl fĂŒggetlenĂŒl is felvetette.) - Halmazrendszerek partĂciĂłira az eddigieknĂ©l ĂĄltalĂĄnosabb modellt vezettĂŒnk be, megvizsgĂĄltuk rĂ©szosztĂĄlyainak hierarchikus szerkezetĂ©t Ă©s hatĂ©kony algoritmusokat adtunk. (Sok alkalmazĂĄs vĂĄrhatĂł az erĆforrĂĄs-allokĂĄciĂł terĂŒletĂ©n.) - Kidolgoztunk egy mĂłdszert, amellyel lokĂĄlisan vĂ©ges pozĂciĂłs jĂĄtĂ©kok nyerĆ stratĂ©giĂĄja megtalĂĄlhatĂł mindössze lineĂĄris mĂ©retƱ memĂłria hasznĂĄlatĂĄval. - FĂ©lig on-line ĂŒtemezĂ©si algoritmusokat terveztĂŒnk (kĂ©tgĂ©pes feladatra, nem azonos sebessĂ©gƱ processzorokra), amelyeknek versenykĂ©pessĂ©gi arĂĄnya bizonyĂtottan jobb, mint ami a legjobb teljesen on-line mĂłdszerekkel elĂ©rhetĆ. | Applying discrete mathematical methods, we proved structural and quantitative relations, designed algorithms and analyzed their complexity. The results deal with graph and hypergraph theory and on-line scheduling. Some selected ones are: - We described the exact structural conditions which have to hold in order that an intercommunication network and each of its connected parts contain a connected subnetwork of prescribed type, from which all the other nodes of the network can be reached via direct link. (This problem was open for two decades.) - We gave asymptotically tight estimates on the minimum size of set systems of k-element sets over an n-element set such that, for each k-partition of the set, the set system contains a k-set meeting all classes of the partition. (This was an open problem since 1973, raised by several authors independently.) - We introduced a new model, more general than the previous ones, for partitions of set systems. We studied the hierarchic structure of its subclasses, and designed efficient algorithms. (Many applications are expected in the area of resource allocation.) - We developed a method to learn winning strategies in locally finite positional games, which requires linear-size memory only. - We designed semi-online scheduling algorithms (for two uniform processors of unequal speed), whose competitive ratio provably beats the best possible one achievable in the purely on-line setting
Derandomizing Concentration Inequalities with dependencies and their combinatorial applications
Both in combinatorics and design and analysis of randomized algorithms for combinatorial optimization problems, we often use the famous bounded differences inequality by C. McDiarmid (1989), which is based on the martingale inequality by K. Azuma (1967), to show positive probability of success. In the case of sum of independent random variables, the inequalities of Chernoff (1952) and Hoeffding (1964) can be used and can be efficiently derandomized, i.e. we can construct the required event in deterministic, polynomial time (Srivastav and Stangier 1996). With such an algorithm one can construct the sought combinatorial structure or design an efficient deterministic algorithm from the probabilistic existentce result or the randomized algorithm.
The derandomization of C. McDiarmid's bounded differences inequality was an open problem. The main result in Chapter 3 is an efficient derandomization of the bounded differences inequality, with the time required to compute the conditional expectation of the objective function being part of the complexity.
The following chapters 4 through 7 demonstrate the generality and power of the derandomization framework developed in Chapter 3.
In Chapter 5, we derandomize the Maker's random strategy in the Maker-Breaker subgraph game given by Bednarska and Luczak (2000), which is fundamental for the field, and analyzed with the concentration inequality of Janson, Luczak and Rucinski. But since we use the bounded differences inequality, it is necessary to give a new proof of the existence of subgraphs in G(n,M)-random graphs (Chapter 4).
In Chapter 6, we derandomize the two-stage randomized algorithm for the set-multicover problem by El Ouali, Munstermann and Srivastav (2014).
In Chapter 7, we show that the algorithm of Bansal, Caprara and Sviridenko (2009) for the multidimensional bin packing problem can be elegantly derandomized with our derandomization framework of bounded differences inequality, while the authors use a potential function based approach, leading to a rather complex analysis.
In Chapter 8, we analyze the constrained hypergraph coloring problem given in Ahuja and Srivastav (2002), which generalizes both the property B problem for the non-monochromatic 2-coloring of hypergraphs and the multidimensional bin packing problem using the bounded differences inequality instead of the Lovasz local lemma. We also derandomize the algorithm using our framework.
In Chapter 9, we turn to the generalization of the well-known concentration inequality of Hoeffding (1964) by Janson (1994), to sums of random variables, that are not independent, but are partially dependent, or in other words, are independent in certain groups. Assuming the same dependency structure as in Janson (1994), we generalize the well-known concentration inequality of Alon and Spencer (1991).
In Chapter 10, we derandomize the inequality of Alon and Spencer. The derandomization of our generalized Alon-Spencer inequality under partial dependencies remains an interesting, open problem
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