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The Minrank of Random Graphs

2018
*
IEEE Transactions on Information Theory
*

We prove tight bounds on

doi:10.1109/tit.2018.2810384
fatcat:imnq4jcj5zg7fafql4v45znqcm
*the**minrank**of*directed Erdős-Rényi*random**graphs*G(n, p) for all regimes*of*p ∈ [0, 1]. ...*The**minrank**of*a directed*graph*G is*the*minimum rank*of*a matrix M that can be obtained from*the*adjacency matrix*of*G by switching some ones to zeros (i.e., deleting edges) and then setting all diagonal ... Since*the**minrank**of*a directed*graph*does not exceed*the**minrank**of*its undirected counterpart, a lower bound for a directed*random**graph*implies*the*same lower bound for an undirected*random**graph*. ...##
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The Minrank of Random Graphs
[article]

2017
*
arXiv
*
pre-print

We prove tight bounds on

arXiv:1607.04842v2
fatcat:qi67l4tiuvdwreihhfaui4a7ce
*the**minrank**of**random*Erdős-Rényi*graphs*G(n,p) for all regimes*of*p∈[0,1]. ...*The**minrank**of*a*graph*G is*the*minimum rank*of*a matrix M that can be obtained from*the*adjacency matrix*of*G by switching some ones to zeros (i.e., deleting edges) and then setting all diagonal entries ... Acknowledgements We would like to thank Ishay Haviv for his valuable comments on an earlier version*of*this work. ...##
###
The Minrank of Random Graphs over Arbitrary Fields
[article]

2019
*
arXiv
*
pre-print

We obtain tight bounds for

arXiv:1809.01873v2
fatcat:oiru65ffjrfgfb4u7icws3fmje
*the*typical*minrank**of**the*binomial*random**graph*G(n,p) over any finite or infinite field, showing that for every field F= F(n) and every p=p(n) satisfying n^-1≤ p ≤ 1-n^-0.99 ...*The**minrank**of*a*graph*G on*the*set*of*vertices [n] over a field F is*the*minimum possible rank*of*a matrix M∈F^n× n with nonzero diagonal entries such that M_i,j=0 whenever i and j are distinct nonadjacent ...*The*research on this project was initiated during a joint research workshop*of*Tel Aviv University and*the*Free University*of*Berlin on*Graph*and Hypergraph Coloring Problems, held in Berlin in August ...##
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A Field-Size Independent Code Construction for Groupcast Index Coding Problems
[article]

2019
*
arXiv
*
pre-print

In this paper, we define

arXiv:1901.00164v1
fatcat:vhubobl7xraibk4hxk3yh64uyq
*the*notion*of**minrank*-critical edges in a side-information*graph*and derive some properties*of**minrank*, which identifies*minrank*-non-critical edges. ... However, both*the*methods are NP-hard.*The*number*of*computations required to find*the**minrank*depends on*the*number*of*edges present in*the*side-information*graph*. ... ACKNOWLEDGEMENT This work was supported partly by*the*Science and Engineering Research Board (SERB)*of*Department*of*Science and Technology (DST), Government*of*India, through J.C. ...##
###
On Conceptually Simple Algorithms for Variants of Online Bipartite Matching
[article]

2017
*
arXiv
*
pre-print

Finally, following

arXiv:1706.09966v1
fatcat:vtkzsgn6tvcopokn6xspw2jyka
*the*work in Besser and Poloczek, we depart from an adversarial or stochastic ordering and investigate a natural*randomized*algorithm (*MinRanking*) in*the*priority model. ...*The*convergence is extremely fast ---*the*5-pass Category-Advice algorithm is already within 0.01%*of**the*inverse*of**the*golden ratio. ... As defined,*MinRanking*is a fully*randomized*priority algorithm, (i.e. in terms*of*both*the*ordering and decisions being*randomized*) but it can also be modified to be a*randomized*priority algorithm in ...##
###
Linear Index Coding via Semidefinite Programming

2013
*
Combinatorics, probability & computing
*

Namely, we show an exact expression for

doi:10.1017/s0963548313000564
fatcat:hcrr67t5wra5nk4vgjak72ylgq
*the*maximum possible value*of**the*Lovász ϑ-function*of*a*graph*with*minrank*k. ... Since*the*SDP we use is not a relaxation*of**the*minimization problem we consider, a crucial component*of*our analysis is an upper bound on*the*objective value*of**the*SDP in terms*of**the**minrank*. ...*Minrank*versus Lovász ϑ-function Now we are ready to derive our tight bound on*the*ϑ-function*of**graphs*with*minrank*k. ...##
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Linear Index Coding via Semidefinite Programming
[article]

2011
*
arXiv
*
pre-print

Namely, we show an exact expression for

arXiv:1107.1958v1
fatcat:2prvgdt6vjh4hfloflll3bsupu
*the*maximum possible value*of**the*Lovasz theta-function*of*a*graph*with*minrank*k. ... Since*the*SDP we use is not a relaxation*of**the*minimization problem we consider, a crucial component*of*our analysis is an upper bound on*the*objective value*of**the*SDP in terms*of**the**minrank*. ... On*the*other hand, there is a long line*of*research on (*randomized*) polynomial time algorithms for*graphs*with bounded chromatic number. ...##
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On Minrank and the Lovász Theta Function

2018
*
International Workshop on Approximation Algorithms for Combinatorial Optimization
*

Two classical upper bounds on

doi:10.4230/lipics.approx-random.2018.13
dblp:conf/approx/Haviv18
fatcat:5bkrrrn6gbedzhr3tizxr2vxja
*the*Shannon capacity*of**graphs*are*the*ϑ-function due to Lovász and*the**minrank*parameter due to Haemers. ... This implies a limitation on*the*ϑ-function-based algorithmic approach to approximating*the**minrank*parameter*of**graphs*. ... APPROX/*RANDOM*2018 ...##
###
Linear Index Coding via Semidefinite Programming
[chapter]

2012
*
Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms
*

Namely, we show an exact expression for

doi:10.1137/1.9781611973099.35
dblp:conf/soda/ChlamtacH12
fatcat:vodw2q2ilrgcnitag55ylbaemu
*the*maximum possible value*of**the*Lovász ϑ-function*of*a*graph*with*minrank*k. ... Since*the*SDP we use is not a relaxation*of**the*minimization problem we consider, a crucial component*of*our analysis is an upper bound on*the*objective value*of**the*SDP in terms*of**the**minrank*. ...*graph*) is not a relaxation*of**the**minrank*parameter. ...##
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On Conceptually Simple Algorithms for Variants of Online Bipartite Matching
[chapter]

2018
*
Lecture Notes in Computer Science
*

Finally, following

doi:10.1007/978-3-319-89441-6_19
fatcat:2koqimajsvhljb5p6zvcflwt54
*the*work in Besser and Poloczek [7], we depart from an adversarial or stochastic ordering and investigate a natural*randomized*algorithm (*MinRanking*) in*the*priority model. ...*The*convergence is extremely fast -*the*5-pass Category-Advice algorithm is already within 0.01%*of**the*inverse*of**the*golden ratio. ... Theorem 5. 4 .Figure 6 : 46*MinRanking*running on*the*family*of**graphs*{G b } has*the*asymptotic approxi-*The*Besser-Poloczek*graph*with parameter b = 3. ...##
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On Minrank and the Lovász Theta Function
[article]

2018
*
arXiv
*
pre-print

Two classical upper bounds on

arXiv:1802.03920v2
fatcat:65wcl64ulng6xjph3timvdrhja
*the*Shannon capacity*of**graphs*are*the*ϑ-function due to Lovász and*the**minrank*parameter due to Haemers. ... This implies a limitation on*the*ϑ-function-based algorithmic approach to approximating*the**minrank*parameter*of**graphs*. ... To obtain*the*lower bounds on*the**minrank*in Theorems 1.1, 1.2, and 1.3, we apply a known relation between*the**minrank**of*a*graph*and*the**minrank**of*its complement (see Lemma 2.2) . ...##
###
On Minrank and Forbidden Subgraphs

2018
*
International Workshop on Approximation Algorithms for Combinatorial Optimization
*

For an integer n, a

doi:10.4230/lipics.approx-random.2018.42
dblp:conf/approx/Haviv18a
fatcat:d732lo7pjfdqjofh7iu2aijlri
*graph*H, and a field F, let g(n, H, F) denote*the*maximum possible*minrank*over F*of*an n-vertex*graph*whose complement contains no copy*of*H. ...*The**minrank*over a field F*of*a*graph*G on*the*vertex set {1, 2, . . . , n} is*the*minimum possible rank*of*a matrix M ∈ F n×n such that M i,i = 0 for every i, and M i,j = 0 for every distinct nonadjacent ... Acknowledgements We are grateful to Alexander Golovnev and Pavel Pudlák for useful discussions and to*the*anonymous referees for their valuable suggestions. ...##
###
Unicast-Uniprior Index Coding Problems: Minrank and Criticality
[article]

2019
*
arXiv
*
pre-print

In this work, we give an algorithm to compute

arXiv:1901.04926v1
fatcat:5liwlqnnznc2jhmcplqvxiizxq
*the**minrank**of*a unicast-uniprior problem. ...*The*proposed algorithm greatly simplifies*the*computation*of**minrank*for unicast-uniprior problem settings, over*the*existing method whose complexity is exponential in*the*number*of*messages. ... Finding*the**minrank*requires computation*of*ranks*of*each*of**the*z fitting matrices over GF (2) . ...##
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Index Coding With Side Information

2011
*
IEEE Transactions on Information Theory
*

We resolve

doi:10.1109/tit.2010.2103753
fatcat:xyg4u4ylrzci7izhka5v3li534
*the*conjecture for certain natural classes*of**graphs*. For arbitrary*graphs*, we show that*the**minrank*bound is tight for both linear codes and certain classes*of*non-linear codes. ... In this paper we identify a measure on*graphs*,*the**minrank*, which we conjecture to exactly characterize*the*minimum length*of*INDEX codes. ... Let Sandwich property*of**minrank*We start with an observation relating*minrank*to other well-known*graph*measures. ...##
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Index Coding with Side Information

2006
*
2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)
*

We resolve

doi:10.1109/focs.2006.42
dblp:conf/focs/Bar-YossefBJK06
fatcat:uixxwebt6zbl3g4kufzt7z3xbi
*the*conjecture for certain natural classes*of**graphs*. For arbitrary*graphs*, we show that*the**minrank*bound is tight for both linear codes and certain classes*of*non-linear codes. ... In this paper we identify a measure on*graphs*,*the**minrank*, which we conjecture to exactly characterize*the*minimum length*of*INDEX codes. ... Let Sandwich property*of**minrank*We start with an observation relating*minrank*to other well-known*graph*measures. ...
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