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On Unique Independence Weighted Graphs
[article]

2009
*
arXiv
*
pre-print

A vertex-

arXiv:0907.0088v1
fatcat:i62xg7qpmrbnhdfmjhhbo72f2u
*weighted**graph*G is called a*unique**independence*vertex-*weighted**graph*if it has a*unique**independent*set with maximum sum of*weights*. ... ; we give, however, some combinatorial characterizations of*unique**independence*vertex-*weighted**graphs*. ... Neighborhood-based characterization of*unique**independence**weighted**graphs*In this section, we try to state some theorems*on*characterizations of*unique**weighted**graphs*mainly based*on*the neighborhood ...##
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Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs

2009
*
2009 24th Annual IEEE Conference on Computational Complexity
*

We study the inapproximability of Vertex Cover and

doi:10.1109/ccc.2009.38
dblp:conf/coco/AustrinKS09
fatcat:3mrjiec3pnbtpfoi5omaobh4vm
*Independent*Set*on*degree d*graphs*. ... . •*Independent*Set is*Unique*Games-hard to approximate to within a factor O( d log 2 d ). This improves the*Unique*Games hardness result of Samorodnitsky and Trevisan [21] . ... Then for all sufficiently small γ > 0, there is an algorithm which,*on*input a*Unique*Games instance Λ = (U,V, E, Π, L) outputs a*weighted**graph*G with the following properties: • Completeness: If IndOpt ...##
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Maximum Weight Independent Sets and Matchings in Sparse Random Graphs. Exact Results using the Local Weak Convergence Method
[article]

2003
*
arXiv
*
pre-print

*weight*of an

*independent*set. ... In this paper we assume in addition that the nodes of the

*graph*are equipped with non-negative

*weights*,

*independently*generated according to some common distribution, and we consider instead the maximum ... Clearly it is the

*unique*fixed point of T since any fixed point of T is also a fixed point of T 2 . 2 We now switch to analyzing the maximum

*weight*

*independent*set problem

*on*a tree. ...

##
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Page 648 of Mathematical Reviews Vol. , Issue 91B
[page]

1991
*
Mathematical Reviews
*

A tree is also said to be a

*unique**independence*tree if it has a largest*independent*set of vertices J; if moreover the complement of / is also*independent*then the tree is called a strong*unique**independence*... It is also shown that no bivariegated tree is a*unique**independence*tree and also that if a pseudobivariegated tree is a*unique**independence*tree then it is a strong*unique*inde- pendence tree. J. ...##
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Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method

2005
*
Random structures & algorithms (Print)
*

*weight*of an

*independent*set. ... In this paper we assume in addition that the nodes of the

*graph*are equipped with nonnegative

*weights*,

*independently*generated according to some common distribution, and we consider instead the maximum ... The authors also wish to thank Antar Bandyopadhyay for correcting an earlier version of the manuscript, and Nick Wormald for informing the authors

*on*the state of the art results

*on*largest

*independent*...

##
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Matching is as easy as matrix inversion

1987
*
Proceedings of the nineteenth annual ACM conference on Theory of computing - STOC '87
*

A new algorithm for finding a maximum matching in a general

doi:10.1145/28395.383347
dblp:conf/stoc/MulmuleyVV87
fatcat:jupztmizqfdzpjsvcatu37roqi
*graph*is presented; its special feature being that the only computationally non-trivial step required in its execution is the inversion of a ... The CLIQUE problem is 'Given a*graph*G(V, E) and an integer k, is there a clique of size k in the*graph*?'*On*the other hand,*UNIQUE*CLIQUE asks if there is exactly*one*clique of size k. ... Notice that the proof of the isolating lemma relies*on*the*independence*of the*weights*of the elements. It would be interesting to study how crucial the role of*independence*is. ...##
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Matching is as easy as matrix inversion

1987
*
Combinatorica
*

A new algorithm for finding a maximum matching in a general

doi:10.1007/bf02579206
fatcat:a6qd6xuqcbhufgqnk2sn4vyesa
*graph*is presented; its special feature being that the only computationally non-trivial step required in its execution is the inversion of a ... The CLIQUE problem is 'Given a*graph*G(V, E) and an integer k, is there a clique of size k in the*graph*?'*On*the other hand,*UNIQUE*CLIQUE asks if there is exactly*one*clique of size k. ... Notice that the proof of the isolating lemma relies*on*the*independence*of the*weights*of the elements. It would be interesting to study how crucial the role of*independence*is. ...##
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Decomposition by clique separators

1985
*
Discrete Mathematics
*

We describe how such a decomposition can be used in divide-and-conquer algorithms for various

doi:10.1016/0012-365x(85)90051-2
fatcat:en7gsgzgxvdz3ddba7ay75mbnu
*graph*problems, such as*graph*coloring and finding maximum*independent*sets. ... We consider the problem of decomposing a*graph*by means of clique separators, by which we mean finding cliques (complete*graphs*) whose removal disconnects the*graph*. ... Acknowledgment My thanks to Marty Golumbic, whose lecture*on*EPT*graphs*stimulated this research. ...##
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Distinguishing graphs by their left and right homomorphism profiles

2011
*
European journal of combinatorics (Print)
*

We introduce a new property of

doi:10.1016/j.ejc.2011.03.012
fatcat:l7iefsbarzggpnesrnrcjco5ia
*graphs*called 'q-state Potts*uniqueness*' and relate it to chromatic and Tutte*uniqueness*, and also to 'chromatic-flow*uniqueness*', recently studied by Duan, Wu and Yu. ... We establish for which edge-*weighted**graphs*H homomorphism functions from multigraphs G to H are specializations of the Tutte polynomial of G, in particular answering a question of Freedman, Lovász and ... A bivariate version of the*independence*polynomial is I(G; x, y) = −*independent*sets U⊆V x |U| y |V \U| = y |V | I(G; x/y).Twin vertices in an edge-*weighted**graph*H*on*vertex set [q] with adjacency matrix ...##
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A weighted graph problem from commutative algebra
[article]

2011
*
arXiv
*
pre-print

We give an especially simple proof of a theorem in

arXiv:0704.3696v3
fatcat:ftuiq6d3dfgrlgtd2t7ct6o6lu
*graph*theory that forms the key part of the solution to a problem in commutative algebra,*on*how to characterize the integral closure of a polynomial ... Our focus is*on*the problem as*one*that merely concerns*weighted**graphs*, but motivating background from commutative algebra will first be sketched. ... After the reduction via h, no vertex v of G f can lie*on*a*unique*edge. Thus the*graph*G f is a disjoint union of odd cycles, some perhaps loops. ...##
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A tolerance-based heuristic approach for the weighted independent set problem

2013
*
Journal of combinatorial optimization
*

Several computational experiments show that our heuristics works very well

doi:10.1007/s10878-013-9606-z
fatcat:kiohxzqrujeitnvejom3pnx6va
*on**graphs*of a small density. ... In this paper we suggest a tolerance-based polynomial heuristic algorithm for the*weighted**independent*set problem. ...*One*can easily verify that the found solution {x 1 , x 4 } is also the*unique*optimal*independent*set of the initial*graph*. ...##
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Ties in ranking scores can be treated as weighted samples
[article]

2022
*
arXiv
*
pre-print

Prior proposals for cumulative statistics suggest making tiny random perturbations to the scores (

arXiv:2207.13632v2
fatcat:6x5ugiwwhbeyvel4bivlrupzt4
*independent*variables in a regression) in order to ensure the scores'*uniqueness*. ... This reduces the problem of analyzing data with scores that may not be*unique*to the problem of analyzing a*weighted*data set with scores that are*unique*by construction. ... The*graphs*of the earlier analyses directly displayed all members of the original data set, omitting no*one*. ...##
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The Maximum Degree of an Exponentially Distributed Random Graph

2022
*
SinkrOn
*

Let G G (n, p) be a

doi:10.33395/sinkron.v7i3.11602
fatcat:7wpot5v7ynadbg4ltjuwkyybhi
*graph**on*n vertices where each pair of vertices is joined*independently*with probability p for 0 < p < 1 and q = 1 p. ... In this work, we introduce*weighted*random graf G with exponential distribution and investigate that the probability that every vertex of G has degree at most np + b√pqn is equal to 0.595656764. ... ., 2018) have studied the*weighted*random*graph*model for the case of large networks. ...##
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Forthcoming papers

1990
*
Discrete Mathematics
*

*weighted*degrees of the vertices become distinct. shows that every additive set is a set of

*uniqueness*. ... Shepp, Sets

*uniquely*determined by projections

*on*axes II. Discrete case. ...

##
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Randomized Techniques for Parameterized Algorithms
[chapter]

2012
*
Lecture Notes in Computer Science
*

The probability that there is a

doi:10.1007/978-3-642-33293-7_2
fatcat:knowy7egpbe2lcszijxvnlaba4
*unique*S ∈ F having minimum*weight*is at least 1 − |U| N . ... By setting N := |V (G )| O(1) , we can assume that there is a*unique*minimum*weight*Hamiltonian cycle. If N is polynomial in the input size, we can guess this minimum*weight*. ... If there is a*unique**weight*-C Hamiltonian cycle: the number of*weight*-C colored cycle covers is 2 mod 4. Assign random*weights*≤ 2|E (G )| to the edges. ...
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