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Correlation decay and partition function zeros: Algorithms and phase transitions [article]

Jingcheng Liu and Alistair Sinclair and Piyush Srivastava
2022 arXiv   pre-print
This extension allows us to give the first deterministic FPTAS for counting the number of q-colorings of a graph of maximum degree Δ provided only that q≥ 2Δ.  ...  This matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo.  ...  Acknowledgments We thank Guus Regts and anonymous reviewers for their various helpful comments. JL was a PhD student at UC Berkeley when this work was carried out.  ... 
arXiv:1906.01228v6 fatcat:6qeo7cwcangt7n2dto6rhxp2c4

Local Multicoloring Algorithms: Computing a Nearly-Optimal TDMA Schedule in Constant Time

Fabian Kuhn, Marc Herbstritt
2009 Symposium on Theoretical Aspects of Computer Science  
We prove that there is a deterministic algorithm that uses a total of |C| = O(∆ 2 log(N )/ε 2 ) colors such that for every node v of G (i.e., for every , Ω(∆ 2 + log log N ) colors are necessary even to  ...  For graphs with maximum degree ∆ = Ω(log |V |), Ω(∆ log |V |/ log log |V |) colors are needed in expectation, even to compute a valid coloring.  ...  A node v with degree δ chooses the colors of all instances for which 2 i ≥ δ.  ... 
doi:10.4230/lipics.stacs.2009.1852 dblp:conf/stacs/Kuhn09 fatcat:73jcuckk6necnfcskc2usc55wy

Balanced families of perfect hash functions and their applications

Noga Alon, Shai Gutner
2010 ACM Transactions on Algorithms  
Using the technique of color-coding we can apply our explicit constructions to devise approximation algorithms for various counting problems in graphs.  ...  Our main result is that for any constant δ > 1, a δbalanced (n, k)-family of perfect hash functions of size 2 O(k log log k) log n can be constructed in time 2 O(k log log k) n log n.  ...  We settle in the affirmative the open question they raise concerning the existence of a deterministic approximate counting algorithm for this problem.  ... 
doi:10.1145/1798596.1798607 fatcat:yxwtjntrbzgcfo44vxkyrm2xba

Local Multicoloring Algorithms: Computing a Nearly-Optimal TDMA Schedule in Constant Time [article]

Fabian Kuhn
2009 arXiv   pre-print
Many channel access schemes assign a fixed set of time slots, frequencies, or (orthogonal) codes to the nodes of a network such that nodes that interfere with each other receive disjoint sets of time slots  ...  Finding a valid assignment of time slots, frequencies, or codes hence directly corresponds to computing a multicoloring of a graph G.  ...  A node v with degree δ chooses the colors of all instances for which 2 i ≥ δ.  ... 
arXiv:0902.1868v1 fatcat:qmik7yjekbdbdkv4mkiw5jyany

Balanced Families of Perfect Hash Functions and Their Applications

Noga Alon, Shai Gutner
2008 arXiv   pre-print
Using the technique of color-coding we can apply our explicit constructions to devise approximation algorithms for various counting problems in graphs.  ...  Our main result is that for any constant δ > 1, a δ-balanced (n,k)-family of perfect hash functions of size 2^O(k k) n can be constructed in time 2^O(k k) n n.  ...  We settle in the affirmative the open question they raise concerning the existence of a deterministic approximate counting algorithm for this problem.  ... 
arXiv:0805.4300v1 fatcat:tmnirbqvgfba3lms6glnxebvku

Generalized Reordering Buffer Management

Yossi Azar, Matthias Englert, Iftah Gamzu, Eytan Kidron, Marc Herbstritt
2014 Symposium on Theoretical Aspects of Computer Science  
We further demonstrate that if our deterministic algorithm can employ k/(1 − δ) servers where δ ∈ (0, 1), then it achieves a competitive ratio of O(min{ln b/δ 2 , b/δ}) against an optimal offline adversary  ...  We complement our randomized approach by presenting a deterministic algorithm that attains a competitive ratio of O(min{k 2 ln b, kb}).  ...  A Deterministic Algorithm We develop two deterministic algorithms: the first has a competitive ratio of O(ln b/δ 2 ) in a δ-augmentation setting and a competitive ratio of O(k 2 ln b) when there is no  ... 
doi:10.4230/lipics.stacs.2014.87 dblp:conf/stacs/AzarEGK14 fatcat:4atds2ch2jcsfl3h7deyrnqnty

Distributed (Δ+1)-Coloring in Sublogarithmic Rounds [article]

David G. Harris, Johannes Schneider, Hsin-Hao Su
2018 arXiv   pre-print
We give a new randomized distributed algorithm for (Δ+1)-coloring in the LOCAL model, running in O(√(Δ))+ 2^O(√( n)) rounds in a graph of maximum degree Δ.  ...  Our algorithm also extends to list-coloring where the palette of each node contains Δ+1 colors.  ...  Acknowledgments: We would like to thank Seth Pettie for his valuable comments. We thank the anonymous reviewers for many helpful suggestions and comments.  ... 
arXiv:1603.01486v4 fatcat:esqg5746onfyjprhdl2lhoso6i

Optimization of data allocation in hierarchical memory for blocked shortest paths algorithms

A. A. Prihozhy
2021 Sistemnyj Analiz i Prikladnaâ Informatika  
In second case, the paper formulates a problem of weighted defective color-count constrained coloring of the conflict graph and solves it by developing a random algorithm.  ...  a non-conflict matrix allocation is possible at δ = 0.5 for M = 4 and at δ = 0.1 for M = 20.  ...  The Hajnal-Szemerédi theorem [17] proves that any graph with maximum degree Δ has an equitable coloring with Δ + 1 colors.  ... 
doi:10.21122/2309-4923-2021-3-40-50 fatcat:pxf7pjf3azcwpobyxc5llcpvna

Distributed Edge Coloring in Time Polylogarithmic in Δ

Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Dennis Olivetti
2022 Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing  
This improves a result of Balliu, Kuhn, and Olivetti [PODC '20], who gave an algorithm with a quasi-polylogarithmic dependency on Δ.  ...  The best previous 𝑂 (Δ)-edge coloring algorithm that can be implemented in the CONGEST model is by Barenboim and Elkin [PODC '11] and it computes a 2 𝑂 (1/𝜀) Δedge coloring in time 𝑂 (Δ 𝜀 + log *  ...  While for MIS and matching, there is a linear-in-Δ lower bound, for (Δ + 1)-vertex coloring and (2Δ − 1)-edge coloring, there are in fact algorithms with a sublinear-in-Δ complexity.  ... 
doi:10.1145/3519270.3538440 fatcat:skc5omvjyre4df3hx6d2a4vcwa

Distributed local approximation algorithms for maximum matching in graphs and hypergraphs [article]

David G. Harris
2020 arXiv   pre-print
This gives an algorithm for (2 Δ - 1)-edge-list coloring in Õ(log^2 Δlog n) rounds deterministically or Õ( (loglog n)^3 ) rounds randomly.  ...  Our main result is a deterministic algorithm to generate a matching which is an O(r)-approximation to the maximum weight matching, running in Õ(r logΔ + log^2 Δ + log^* n) rounds.  ...  Let G be a graph with maximum degree ∆. (1) There is aÕ(log n log 2 ∆)-round deterministic algorithm for (2∆ − 1)-list-edge-coloring. (2) There is aÕ((log log n) 3 )-round randomized algorithm for (2∆  ... 
arXiv:1807.07645v7 fatcat:4yaui54cd5ahhhac7byids5k6u

Correlation decay and deterministic FPTAS for counting list-colorings of a graph [article]

David Gamarnik, Dmitriy Katz
2007 arXiv   pre-print
We propose a deterministic algorithm for approximately counting the number of list colorings of a graph.  ...  For the case when the size of the each list is a large constant, we show the existence of a deterministic FPTAS for computing the total number of list colorings.  ...  Acknowledgements The authors are very thankful to Devavrat Shah for contributing many important comments for this work.  ... 
arXiv:math/0606143v2 fatcat:bc54aenln5bk3d4fhq5ridsmxi

Correlation decay and deterministic FPTAS for counting colorings of a graph

David Gamarnik, Dmitriy Katz
2012 Journal of Discrete Algorithms  
We construct a deterministic algorithm for approximately counting the number of colorings of a graph.  ...  The same deterministic algorithm has complexity 2 O (log 2 n) , without any assumptions on the number of colors, where n is the instance size.  ...  Acknowledgements The authors are very thankful to Devavrat Shah for contributing many important comments for this work, and to the anonymous reviewers for a careful reading of the manuscript and providing  ... 
doi:10.1016/j.jda.2010.10.002 fatcat:jod2arkcsnfcrd3mp5oo2u4wh4

MIS on trees

Christoph Lenzen, Roger Wattenhofer
2011 Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing - PODC '11  
This basic symmetry breaking structure is vital for many distributed algorithms, which by now has been fueling the search for fast local algorithms to find such sets over several decades.  ...  Our algorithm is uniform and nodes need to exchange merely O(log n) many bits with high probability.  ...  This was changed by Barenboim and Elkin [3] , who devised a deterministic algorithm with running time o(log n) for graphs with arboricity 2 o( √ log n).  ... 
doi:10.1145/1993806.1993813 dblp:conf/podc/LenzenW11 fatcat:nqaqwrvtlncu5ju62yhg2x3qpq

Simple deterministic approximation algorithms for counting matchings

Mohsen Bayati, David Gamarnik, Dimitriy Katz, Chandra Nair, Prasad Tetali
2007 Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07  
Additionally, for an arbitrary graph, we construct a deterministic algorithm for computing approximately the number of matchings within running time exp(O( √ n log 2 n)), where n is the number of vertices  ...  We construct a deterministic fully polynomial time approximation scheme (FPTAS) for computing the total number of matchings in a bounded degree graph.  ...  Conclusions We have constructed a deterministic algorithm for counting approximately the number of matchings of a given graph.  ... 
doi:10.1145/1250790.1250809 dblp:conf/stoc/BayatiGKNT07 fatcat:ihuxkfheevgcdjmlyikmdfvtve

Approximating the Number of Network Motifs

Mira Gonen, Yuval Shavitt
2009 Internet Mathematics  
We present several algorithms with time complexity O e 2k k • n • |E| • log 1 δ / 2 that, for the first time, approximate for every vertex the number of non-induced occurrences of the motif the vertex  ...  Some of our algorithms use the color coding technique.  ...  Acknowledgment: We thank Dana Ron for many hours of fruitful discussions.  ... 
doi:10.1080/15427951.2009.10390645 fatcat:andmbsem3zanpke5rgriqkhhum
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