Maximizing the number of nonnegative subsets

Given a set of $n$ real numbers, if the sum of elements of every subset of size larger than $k$ is negative, what is the maximum number of subsets of nonnegative sum? In this note we show that the answer is $\binom{n-1}{k-1} + \binom{n-1}{k-2} + \cdots + \binom{n-1}{0}+1$, settling a problem of Tsukerman. We provide two proofs, the first establishes and applies a weighted version of Hall's Theorem and the second is based on an extension of the nonuniform Erd\H{o}s-Ko-Rado Theorem

AZ-identities and Strict 2-part Sperner Properties of Product Posets

One of the central issues in extremal set theory is Sperner's theorem and its generalizations. Among such generalizations is the best-known BLYM inequality and the Ahlswede--Zhang (AZ) identity which surprisingly generalizes the BLYM inequality into an identity. Sperner's theorem and the BLYM inequality has been also generalized to a wide class of posets. Another direction in this research was the study of more part Sperner systems. In this paper we derive AZ type identities for regular posets. We also characterize all maximum 2-part Sperner systems for a wide class of product posets

On generic erasure correcting sets and related problems

Motivated by iterative decoding techniques for the binary erasure channel Hollmann and Tolhuizen introduced and studied the notion of generic erasure correcting sets for linear codes. A generic $(r,s)$--erasure correcting set generates for all codes of codimension $r$ a parity check matrix that allows iterative decoding of all correctable erasure patterns of size $s$ or less. The problem is to derive bounds on the minimum size $F(r,s)$ of generic erasure correcting sets and to find constructions for such sets. In this paper we continue the study of these sets. We derive better lower and upper bounds. Hollmann and Tolhuizen also introduced the stronger notion of $(r,s)$--sets and derived bounds for their minimum size $G(r,s)$. Here also we improve these bounds. We observe that these two conceps are closely related to so called $s$--wise intersecting codes, an area, in which $G(r,s)$ has been studied primarily with respect to ratewise performance. We derive connections. Finally, we observed that hypergraph covering can be used for both problems to derive good upper bounds.Comment: 9 pages, to appear in IEEE Transactions on Information Theor

A Combinatorial Model of Two-Sided Search

Aydinian H, Cicalese F, Deppe C, Lebedev V. A Combinatorial Model of Two-Sided Search. INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE. 2018;29(4):481-504.We study a new model of combinatorial group testing: the item to be found (a.k.a. the target) occupies an unknown node in a graph. At each time instant, we can test (or query) a subset of the nodes and learn whether the target occupies any of such nodes. Immediately after the result of the test is available, the target can move to any node adjacent to its present location. The search finishes when we are able to locate the object with some predefined accuracy s (a parameter fixed beforehand), i.e., to indicate a set of s nodes that includes the location of the object. In this paper we study two types of problems related to the above model: (i) what is the minimum value of the accuracy parameter for which a search strategy in the above sense exists; (ii) given the accuracy, what is the minimum number of tests that allow to locate the target. We study these questions on paths, cycles, and trees as underlying graphs and provide tight answers for the above questions. We also consider a restricted variant of the problem, where the number of moves of the target is bounded

A Heuristic Solution of a Cutting Problem Using Hypergraphs

Deppe C, Wischmann C. A Heuristic Solution of a Cutting Problem Using Hypergraphs. In: Aydinian H, Cicalese F, Deppe C, eds. Information Theory, Combinatorics, and Search Theory. Lecture Notes in Computer Science. Vol 7777. Berlin, Heidelberg: Springer Berlin Heidelberg; 2013: 658-676

Threshold and Majority Group Testing

Ahlswede R, Deppe C, Lebedev V. Threshold and Majority Group Testing. In: Aydinian H, Cicalese F, Deppe C, eds. Information Theory, Combinatorics, and Search Theory. Lecture Notes in Computer Science. Vol 7777. Berlin, Heidelberg: Springer Berlin Heidelberg; 2013: 488-508