Deterministic Constructions for Large Girth Protograph LDPC Codes

The bit-error threshold of the standard ensemble of Low Density Parity Check (LDPC) codes is known to be close to capacity, if there is a non-zero fraction of degree-two bit nodes. However, the degree-two bit nodes preclude the possibility of a block-error threshold. Interestingly, LDPC codes constructed using protographs allow the possibility of having both degree-two bit nodes and a block-error threshold. In this paper, we analyze density evolution for protograph LDPC codes over the binary erasure channel and show that their bit-error probability decreases double exponentially with the number of iterations when the erasure probability is below the bit-error threshold and long chain of degree-two variable nodes are avoided in the protograph. We present deterministic constructions of such protograph LDPC codes with girth logarithmic in blocklength, resulting in an exponential fall in bit-error probability below the threshold. We provide optimized protographs, whose block-error thresholds are better than that of the standard ensemble with minimum bit-node degree three. These protograph LDPC codes are theoretically of great interest, and have applications, for instance, in coding with strong secrecy over wiretap channels.Comment: 5 pages, 2 figures; To appear in ISIT 2013; Minor changes in presentatio

Product Dimension of Forests and Bounded Treewidth Graphs

The product dimension of a graph G is defined as the minimum natural number l such that G is an induced subgraph of a direct product of l complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and k-degenerate graphs. We show that every forest on n vertices has a product dimension at most 1.441logn+3. This improves the best known upper bound of 3logn for the same due to Poljak and Pultr. The technique used in arriving at the above bound is extended and combined with a result on existence of orthogonal Latin squares to show that every graph on n vertices with a treewidth at most t has a product dimension at most (t+2)(logn+1). We also show that every k-degenerate graph on n vertices has a product dimension at most \ceil{8.317klogn}+1. This improves the upper bound of 32klogn for the same by Eaton and Rodl.Comment: 12 pages, 3 figure

Approximating the Regular Graphic TSP in near linear time

We present a randomized approximation algorithm for computing traveling salesperson tours in undirected regular graphs. Given an $n$-vertex, $k$-regular graph, the algorithm computes a tour of length at most $\left(1+\frac{7}{\ln k-O(1)}\right)n$, with high probability, in $O(nk \log k)$ time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS 2012) for the same problem, in terms of both approximation factor, and running time. The key ingredient of our algorithm is a technique that uses edge-coloring algorithms to sample a cycle cover with $O(n/\log k)$ cycles with high probability, in near linear time. Additionally, we also give a deterministic $\frac{3}{2}+O\left(\frac{1}{\sqrt{k}}\right)$ factor approximation algorithm running in time $O(nk)$.Comment: 12 page

Message passing for the coloring problem: Gallager meets Alon and Kahale

Message passing algorithms are popular in many combinatorial optimization problems. For example, experimental results show that {\em survey propagation} (a certain message passing algorithm) is effective in finding proper $k$-colorings of random graphs in the near-threshold regime. In 1962 Gallager introduced the concept of Low Density Parity Check (LDPC) codes, and suggested a simple decoding algorithm based on message passing. In 1994 Alon and Kahale exhibited a coloring algorithm and proved its usefulness for finding a $k$-coloring of graphs drawn from a certain planted-solution distribution over $k$-colorable graphs. In this work we show an interpretation of Alon and Kahale's coloring algorithm in light of Gallager's decoding algorithm, thus showing a connection between the two problems - coloring and decoding. This also provides a rigorous evidence for the usefulness of the message passing paradigm for the graph coloring problem. Our techniques can be applied to several other combinatorial optimization problems and networking-related issues.Comment: 11 page