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278
Index Coding with Side Information
, 2006
"... Motivated by a problem of transmitting supplemental data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R1,..., Rn. He holds an input x ∈ {0, 1} n and wishes to broadcast a single message so that each receiver Ri c ..."
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Cited by 105 (0 self)
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Motivated by a problem of transmitting supplemental data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R1,..., Rn. He holds an input x ∈ {0, 1} n and wishes to broadcast a single message so that each receiver Ri can recover the bit xi. Each Ri has prior side information about x, induced by a directed graph G on n nodes; Ri knows the bits of x in the positions {j  (i, j) is an edge of G}. G is known to the sender and to the receivers. We call encoding schemes that achieve this goal INDEX codes for {0, 1} n with side information graph G. In this paper we identify a measure on graphs, the minrank, which exactly characterizes the minimum length of linear and certain types of nonlinear INDEX codes. We show that for natural classes of side information graphs, including directed acyclic graphs, perfect graphs, odd holes, and odd antiholes, minrank is the optimal length of arbitrary INDEX codes. For arbitrary INDEX codes and arbitrary graphs, we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph. This bound holds even for randomized codes, but is shown not to be tight.
Some properties of unitary Cayley graphs
, 2007
"... The unitary Cayley graph Xn has vertex set Zn = {0, 1,..., n−1}. Vertices a, b ..."
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Cited by 37 (5 self)
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The unitary Cayley graph Xn has vertex set Zn = {0, 1,..., n−1}. Vertices a, b
Combinatorial secant varieties
 Quart. J. Pure Applied Math
"... Abstract. The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Per ..."
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Cited by 37 (2 self)
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Abstract. The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal of the secant ideal coincides with the secant ideal of the initial ideal. For toric varieties, this leads to the notion of delightful triangulations of convex polytopes. 1.
Certifying Algorithms
, 2010
"... A certifying algorithm is an algorithm that produces, with each output, a certificate or witness (easytoverify proof) that the particular output has not been compromised by a bug. A user of a certifying algorithm inputs x, receives the output y and the certificate w, and then checks, either manual ..."
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Cited by 24 (6 self)
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A certifying algorithm is an algorithm that produces, with each output, a certificate or witness (easytoverify proof) that the particular output has not been compromised by a bug. A user of a certifying algorithm inputs x, receives the output y and the certificate w, and then checks, either manually or by use of a program, that w proves that y is a correct output for input x. In this way, he/she can be sure of the correctness of the output without having to trust the algorithm. We put forward the thesis that certifying algorithms are much superior to noncertifying algorithms, and that for complex algorithmic tasks, only certifying algorithms are satisfactory. Acceptance of this thesis would lead to a change of how algorithms are taught and how algorithms are researched. The widespread use of certifying algorithms would greatly enhance the reliability of algorithmic software. We survey the state of the art in certifying algorithms and add to it. In particular, we start a
Graph Minor Theory
 BULLETIN (NEW SERIES) OF THE AMERICAN MATHEMATICAL SOCIETY
, 2005
"... A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a farreaching ..."
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Cited by 22 (0 self)
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A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a farreaching generalization of this, conjectured by Wagner: If a class of graphs is minorclosed (i.e., it is closed under deleting and contracting edges), then it can be characterized by a finite number of excluded minors. The proof of this conjecture is based on a very general theorem about the structure of large graphs: If a minorclosed class of graphs does not contain all graphs, then every graph in it is glued together in a treelike fashion from graphs that can almost be embedded in a fixed surface. We describe the precise formulation of the main results and survey some of its applications to algorithmic and structural problems in graph theory.
Problems and results in extremal combinatorics  II
 DISCRETE MATHEMATICS
, 2003
"... Extremal Combinatorics is one of the central areas in Discrete Mathematics. It deals with problems that are often motivated by questions arising in other areas, including Theoretical Computer Science, Geometry and Game Theory. This paper contains a collection of problems and results in the area, inc ..."
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Extremal Combinatorics is one of the central areas in Discrete Mathematics. It deals with problems that are often motivated by questions arising in other areas, including Theoretical Computer Science, Geometry and Game Theory. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers. The topics considered here include questions in Extremal Graph Theory, Polyhedral Combinatorics and Probabilistic Combinatorics. This is not meant to be a comprehensive survey of the area, it is merely a collection of various extremal problems, which are hopefully interesting. The choice of the problems is inevitably biased, and as the title of the paper suggests, it is a sequel to a previous paper [2] of the same flavour, and hopefully a predecessor of another related future paper. Each section of this paper is essentially self contained, and can be read separately.
Combinatorial symbolic powers
 J. Algebra
"... Abstract. Symbolic powers of ideals are studied in the combinatorial context of monomial ideals. When the ideals are generated by quadratic squarefree monomials, the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blowups. As a result, perfect g ..."
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Cited by 21 (2 self)
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Abstract. Symbolic powers of ideals are studied in the combinatorial context of monomial ideals. When the ideals are generated by quadratic squarefree monomials, the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blowups. As a result, perfect graphs play an important role in the theory, dual to the role played by perfect graphs in the theory of secants of monomial ideals. Among the applications are a new, unified approach to the Gröbner bases of symbolic powers of determinantal and Pfaffian ideals.