scholarly journals A probabilistic counting lemma for complete graphs

2007 ◽  
Vol 31 (4) ◽  
pp. 517-534 ◽  
Author(s):  
Stefanie Gerke ◽  
Martin Marciniszyn ◽  
Angelika Steger
Keyword(s):  
Complete Graphs ◽  
Counting Lemma

scholarly journals A Probabilistic Counting Lemma for Complete Graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Stefanie Gerke ◽  
Martin Marciniszyn ◽  
Angelika Steger
Keyword(s):  
Random Graphs ◽  
Regular Graph ◽  
Complete Graphs ◽  
Regularity Lemma ◽  
Expected Number ◽  
International Audience ◽  
Partition Class ◽  
Szemeredi's Regularity Lemma ◽  
Almost All ◽  
Counting Lemma

International audience We prove the existence of many complete graphs in almost all sufficiently dense partitions obtained by an application of Szemerédi's Regularity Lemma. More precisely, we consider the number of complete graphs $K_{\ell}$ on $\ell$ vertices in $\ell$-partite graphs where each partition class consists of $n$ vertices and there is an $\varepsilon$-regular graph on $m$ edges between any two partition classes. We show that for all $\beta > $0, at most a $\beta^m$-fraction of graphs in this family contain less than the expected number of copies of $K_{\ell}$ provided $\varepsilon$ is sufficiently small and $m \geq Cn^{2-1/(\ell-1)}$ for a constant $C > 0$ and $n$ sufficiently large. This result is a counting version of a restricted version of a conjecture by Kohayakawa, Łuczak and Rödl and has several implications for random graphs.

Roulette Wheel Selection based Heuristic Algorithm for the Orienteering Problem

2014 ◽  
Vol 13 (1) ◽  
pp. 4127-4145
Author(s):  
Madhushi Verma ◽  
Mukul Gupta ◽  
Bijeeta Pal ◽  
Prof. K. K. Shukla
Keyword(s):  
Triangle Inequality ◽  
Time Budget ◽  
Control Point ◽  
Hamiltonian Path ◽  
Real Life ◽  
Tourism Industry ◽  
Orienteering Problem ◽  
Complete Graphs ◽  
Roulette Wheel Selection ◽  
Roulette Wheel

Orienteering problem (OP) is an NP-Hard graph problem. The nodes of the graph are associated with scores or rewards and the edges with time delays. The goal is to obtain a Hamiltonian path connecting the two necessary check points, i.e. the source and the target along with a set of control points such that the total collected score is maximized within a specified time limit. OP finds application in several fields like logistics, transportation networks, tourism industry, etc. Most of the existing algorithms for OP can only be applied on complete graphs that satisfy the triangle inequality. Real-life scenario does not guarantee that there exists a direct link between all control point pairs or the triangle inequality is satisfied. To provide a more practical solution, we propose a stochastic greedy algorithm (RWS_OP) that uses the roulette wheel selectionmethod, does not require that the triangle inequality condition is satisfied and is capable of handling both complete as well as incomplete graphs. Based on several experiments on standard benchmark data we show that RWS_OP is faster, more efficient in terms of time budget utilization and achieves a better performance in terms of the total collected score ascompared to a recently reported algorithm for incomplete graphs.

scholarly journals Improving the Smoothed Complexity of FLIP for Max Cut Problems

2021 ◽  
Vol 17 (3) ◽  
pp. 1-38
Author(s):  
Ali Bibak ◽  
Charles Carlson ◽  
Karthekeyan Chandrasekaran
Keyword(s):  
General Framework ◽  
Edge Weight ◽  
Complete Graphs ◽  
Worst Case ◽  
Weight Density ◽  
Complete Problems ◽  
Substantial Progress ◽  
Run Time ◽  
Optimum Solution ◽  
Arbitrary Graphs

Finding locally optimal solutions for MAX-CUT and MAX- k -CUT are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Röglin (ACM Transactions on Algorithms, 2017) showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres, and Wei (STOC, 2017) showed that the smoothed complexity of FLIP for max-cut in complete graphs is ( O Φ 5 n 15.1 ), where Φ is an upper bound on the random edge-weight density and Φ is the number of vertices in the input graph. While Angel, Bubeck, Peres, and Wei’s result showed the first polynomial smoothed complexity, they also conjectured that their run-time bound is far from optimal. In this work, we make substantial progress toward improving the run-time bound. We prove that the smoothed complexity of FLIP for max-cut in complete graphs is O (Φ n 7.83 ). Our results are based on a carefully chosen matrix whose rank captures the run-time of the method along with improved rank bounds for this matrix and an improved union bound based on this matrix. In addition, our techniques provide a general framework for analyzing FLIP in the smoothed framework. We illustrate this general framework by showing that the smoothed complexity of FLIP for MAX-3-CUT in complete graphs is polynomial and for MAX - k - CUT in arbitrary graphs is quasi-polynomial. We believe that our techniques should also be of interest toward showing smoothed polynomial complexity of FLIP for MAX - k - CUT in complete graphs for larger constants k .

scholarly journals True complexity of polynomial progressions in finite fields

2021 ◽  
pp. 1-53
Author(s):  
Borys Kuca
Keyword(s):  
Finite Fields ◽  
Linear Forms ◽  
Gowers Norm ◽  
Counting Lemma

Abstract The true complexity of a polynomial progression in finite fields corresponds to the smallest-degree Gowers norm that controls the counting operator of the progression over finite fields of large characteristic. We give a conjecture that relates true complexity to algebraic relations between the terms of the progression, and we prove it for a number of progressions, including $x, x+y, x+y^{2}, x+y+y^{2}$ and $x, x+y, x+2y, x+y^{2}$ . As a corollary, we prove an asymptotic for the count of certain progressions of complexity 1 in subsets of finite fields. In the process, we obtain an equidistribution result for certain polynomial progressions, analogous to the counting lemma for systems of linear forms proved by Green and Tao.

scholarly journals Packing trees in complete graphs

1987 ◽  
Vol 67 (1) ◽  
pp. 27-42 ◽  
Author(s):  
Arthur M Hobbs ◽  
Brian A Bourgeois ◽  
Jothi Kasiraj
Keyword(s):  
Complete Graphs
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