Combinatorial Algorithms for Packings, Coverings and Tilings of Hypercubes

Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto. fi Author Ashik Mathew Kizhakkepallathu Name of the doctoral dissertation Combinatorial Algorithms for Packings, Coverings and Tilings of Hypercubes Publisher School of Electrical Engineering Unit Department of Communications and Networking Series Aalto University publication series DOCTORAL DISSERTATIONS 112/2015 Field of research Information Theory Manuscript submitted 10 April 2015 Date of the defence 18 September 2015 Permission to publish granted (date) 15 June 2015 Language English Monograph Article dissertation (summary + original articles) Abstract Packing, covering and tiling problems not only form a fundamental theme of study in a number of diverse branches of mathematics such as combinatorics and geometry but several applications of them are well established in fields ranging from information theory and computer science to material sciences. Several existence, counting and classification problems related to packings, coverings and tilings of hypercubes in a torus form the main focus of study in this thesis. The Shannon capacity of a graphG is an important information theoretic parameter of the graph and is defined as c(G) = supd≥1(α(G )) 1 d , where α(G) is the independence number of G. Finding the Shannon capacity of odd cycles of length greater than 5 is a well known open problem and is in close connection with the problem of packing hypercubes in a torus. The packing and covering problems that we study are also respectively equivalent to clique and dominating set problems in certain dense graphs, problems which are known to be computationally hard to solve. New lower bounds for the Shannon capacity of odd cycles and triangular graphs are obtained using local and exhaustive search algorithms. We also study a problem related to holes in non-extensible cube packings and verify a conjecture related to holes for 5-dimensional packings. The classification problem of 5-dimensional cube tilings in the discrete torus of width 4 is settled as part of our research by representing this problem as an exact cover problem and using computational methods. Some results on the cubicity of interval graphs are also obtained.Packing, covering and tiling problems not only form a fundamental theme of study in a number of diverse branches of mathematics such as combinatorics and geometry but several applications of them are well established in fields ranging from information theory and computer science to material sciences. Several existence, counting and classification problems related to packings, coverings and tilings of hypercubes in a torus form the main focus of study in this thesis. The Shannon capacity of a graphG is an important information theoretic parameter of the graph and is defined as c(G) = supd≥1(α(G )) 1 d , where α(G) is the independence number of G. Finding the Shannon capacity of odd cycles of length greater than 5 is a well known open problem and is in close connection with the problem of packing hypercubes in a torus. The packing and covering problems that we study are also respectively equivalent to clique and dominating set problems in certain dense graphs, problems which are known to be computationally hard to solve. New lower bounds for the Shannon capacity of odd cycles and triangular graphs are obtained using local and exhaustive search algorithms. We also study a problem related to holes in non-extensible cube packings and verify a conjecture related to holes for 5-dimensional packings. The classification problem of 5-dimensional cube tilings in the discrete torus of width 4 is settled as part of our research by representing this problem as an exact cover problem and using computational methods. Some results on the cubicity of interval graphs are also obtained. 1