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### The Minrank of Random Graphs

Alexander Golovnev, Oded Regev, Omri Weinstein
2018 IEEE Transactions on Information Theory
We prove tight bounds on 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.  ...

### The Minrank of Random Graphs [article]

Alexander Golovnev, Oded Regev, Omri Weinstein
2017 arXiv   pre-print
We prove tight bounds on 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]

Noga Alon, Igor Balla, Lior Gishboliner, Adva Mond, Frank Mousset
2019 arXiv   pre-print
We obtain tight bounds for 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  ...

### A Field-Size Independent Code Construction for Groupcast Index Coding Problems [article]

Mahesh Babu Vaddi, B.Sundar Rajan
2019 arXiv   pre-print
In this paper, we define 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]

Allan Borodin, Denis Pankratov, Amirali Salehi-Abari
2017 arXiv   pre-print
Finally, following 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

EDEN CHLAMTÁČ, ISHAY HAVIV
2013 Combinatorics, probability & computing
Namely, we show an exact expression for 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.  ...

### Linear Index Coding via Semidefinite Programming [article]

Eden Chlamtac, Ishay Haviv
2011 arXiv   pre-print
Namely, we show an exact expression for 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.  ...

### On Minrank and the Lovász Theta Function

Ishay Haviv, Michael Wagner
2018 International Workshop on Approximation Algorithms for Combinatorial Optimization
Two classical upper bounds on 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]

Eden Chlamtáč, Ishay Haviv
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms
Namely, we show an exact expression for 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.  ...

### On Conceptually Simple Algorithms for Variants of Online Bipartite Matching [chapter]

Allan Borodin, Denis Pankratov, Amirali Salehi-Abari
2018 Lecture Notes in Computer Science
Finally, following 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.  ...  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.  ...

### On Minrank and the Lovász Theta Function [article]

Ishay Haviv
2018 arXiv   pre-print
Two classical upper bounds on 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

Ishay Haviv, Michael Wagner
2018 International Workshop on Approximation Algorithms for Combinatorial Optimization
For an integer n, a 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 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) .  ...

### Index Coding With Side Information

Ziv Bar-Yossef, Yitzhak Birk, T. S. Jayram, Tomer Kol
2011 IEEE Transactions on Information Theory
We resolve 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.  ...

### Index Coding with Side Information

Ziv Bar-Yossef, Yitzhak Birk, T. Jayram, Tomer Kol
2006 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)
We resolve 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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