Spectral Gap for Complete Graphs: Upper and Lower Estimates

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.

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The quadrangular genus of complete graphs

Problems of theoretical cybernetics ◽

10.29003/m69.ptk18.2017/11-13 ◽

2017 ◽

Author(s):

Serge Lawrencenko

Keyword(s):

Complete Graphs

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Improving the Smoothed Complexity of FLIP for Max Cut Problems

ACM Transactions on Algorithms ◽

10.1145/3454125 ◽

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 .

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Generalization of the Conway–Gordon Theorem and Intrinsic Linking on Complete Graphs

Annals of Combinatorics ◽

10.1007/s00026-021-00536-5 ◽

2021 ◽

Author(s):

Hiroko Morishita ◽

Ryo Nikkuni

Keyword(s):

Complete Graphs

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Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

Communications in Mathematics and Statistics ◽

10.1007/s40304-020-00222-7 ◽

2021 ◽

Author(s):

Jürgen Jost ◽

Raffaella Mulas ◽

Florentin Münch

Keyword(s):

Lower Bound ◽

Bipartite Graph ◽

Spectral Gap ◽

Complete Bipartite Graph ◽

Graph Laplacian ◽

New Method ◽

Single Edge ◽

Largest Eigenvalue ◽

Complement Graph ◽

The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

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Free partition functions and an averaged holographic duality

Journal of High Energy Physics ◽

10.1007/jhep01(2021)130 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Nima Afkhami-Jeddi ◽

Henry Cohn ◽

Thomas Hartman ◽

Amirhossein Tajdini

Keyword(s):

Partition Function ◽

Spectral Gap ◽

Central Charge ◽

Three Dimensional ◽

Two Dimensions ◽

Three Dimensions ◽

Partition Functions ◽

General Constraints ◽

Dimensional Gravity ◽

Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

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Persistence of the spectral gap for the Landau–Pekar equations

Letters in Mathematical Physics ◽

10.1007/s11005-020-01350-5 ◽

2021 ◽

Vol 111 (1) ◽

Author(s):

Dario Feliciangeli ◽

Simone Rademacher ◽

Robert Seiringer

Keyword(s):

Initial Data ◽

Spectral Gap ◽

Effective Hamiltonian ◽

Adiabatic Theorem ◽

Strongly Coupled

AbstractThe Landau–Pekar equations describe the dynamics of a strongly coupled polaron. Here, we provide a class of initial data for which the associated effective Hamiltonian has a uniform spectral gap for all times. For such initial data, this allows us to extend the results on the adiabatic theorem for the Landau–Pekar equations and their derivation from the Fröhlich model obtained in previous works to larger times.

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