An approximate version of the tree packing conjecture

AbstractA diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every ε > 0 there exists n0 such that every diregular bipartite tournament on 2n ≥ n0 vertices contains a collection of (1/2–ε)n cycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every c > 1/2 and ε > 0 there exists n0 such that every cn-regular bipartite digraph on 2n ≥ n0 vertices contains (1−ε)cn edge-disjoint Hamilton cycles.

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Common Randomness, Multiuser Secrecy and Tree Packing

2008 46th Annual Allerton Conference on Communication, Control, and Computing ◽

10.1109/allerton.2008.4797559 ◽

2008 ◽

Cited By ~ 4

Author(s):

Sirin Nitinawarat ◽

Chunxuan Ye ◽

Alexander Barg ◽

Prakash Narayan ◽

Alex Reznik

Keyword(s):

Tree Packing

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Fractional spanning tree packing, forest covering and eigenvalues

Discrete Applied Mathematics ◽

10.1016/j.dam.2016.04.027 ◽

2016 ◽

Vol 213 ◽

pp. 219-223 ◽

Cited By ~ 2

Author(s):

Yanmei Hong ◽

Xiaofeng Gu ◽

Hong-Jian Lai ◽

Qinghai Liu

Keyword(s):

Spanning Tree ◽

Tree Packing

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LP Relaxation and Tree Packing for Minimum $k$-Cut

SIAM Journal on Discrete Mathematics ◽

10.1137/19m1299359 ◽

2020 ◽

Vol 34 (2) ◽

pp. 1334-1353

Author(s):

Chandra Chekuri ◽

Kent Quanrud ◽

Chao Xu

Keyword(s):

Lp Relaxation ◽

Tree Packing

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On computational complexity of construction of c -optimal linear regression models over finite experimental domains

Tatra Mountains Mathematical Publications ◽

10.2478/v10127-012-0002-3 ◽

2012 ◽

Vol 51 (1) ◽

pp. 11-21

Author(s):

Jaromír Antoch ◽

Michal Černý ◽

Milan Hladík

Keyword(s):

Linear Programming ◽

Regression Models ◽

Parallel Implementation ◽

Optimal Designs ◽

Polynomial Algorithms ◽

Linear Regression Models ◽

Optimal Linear ◽

New Algorithms ◽

Approximate Version ◽

Strongly Polynomial

ABSTRACT Recent complexity-theoretic results on finding c-optimal designs over finite experimental domain X are discussed and their implications for the analysis of existing algorithms and for the construction of new algorithms are shown. Assuming some complexity-theoretic conjectures, we show that the approximate version of c-optimality does not have an efficient parallel implementation. Further, we study the question whether for finding the c-optimal designs over finite experimental domain X there exist a strongly polynomial algorithms and show relations between considered design problem and linear programming. Finally, we point out some complexity-theoretic properties of the SAC algorithm for c-optimality.

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