Compressing Sparse Sequences under Local Decodability Constraints

We consider a variable-length source coding problem subject to local decodability constraints. In particular, we investigate the blocklength scaling behavior attainable by encodings of $r$-sparse binary sequences, under the constraint that any source bit can be correctly decoded upon probing at most $d$ codeword bits. We consider both adaptive and non-adaptive access models, and derive upper and lower bounds that often coincide up to constant factors. Notably, such a characterization for the fixed-blocklength analog of our problem remains unknown, despite considerable research over the last three decades. Connections to communication complexity are also briefly discussed.Comment: 8 pages, 1 figure. First five pages to appear in 2015 International Symposium on Information Theory. This version contains supplementary materia

On the Complexity of Making a Distinguished Vertex Minimum or Maximum Degree by Vertex Deletion

In this paper, we investigate the approximability of two node deletion problems. Given a vertex weighted graph $G=(V,E)$ and a specified, or "distinguished" vertex $p \in V$, MDD(min) is the problem of finding a minimum weight vertex set $S \subseteq V\setminus \{p\}$ such that $p$ becomes the minimum degree vertex in $G[V \setminus S]$; and MDD(max) is the problem of finding a minimum weight vertex set $S \subseteq V\setminus \{p\}$ such that $p$ becomes the maximum degree vertex in $G[V \setminus S]$. These are known $NP$-complete problems and have been studied from the parameterized complexity point of view in previous work. Here, we prove that for any $\epsilon > 0$, both the problems cannot be approximated within a factor $(1 - \epsilon)\log n$, unless $NP \subseteq DTIME(n^{\log\log n})$. We also show that for any $\epsilon > 0$, MDD(min) cannot be approximated within a factor $(1 -\epsilon)\log n$ on bipartite graphs, unless $NP \subseteq DTIME(n^{\log\log n})$, and that for any $\epsilon > 0$, MDD(max) cannot be approximated within a factor $(1/2 - \epsilon)\log n$ on bipartite graphs, unless $NP \subseteq DTIME(n^{\log\log n})$. We give an $O(\log n)$ factor approximation algorithm for MDD(max) on general graphs, provided the degree of $p$ is $O(\log n)$. We then show that if the degree of $p$ is $n-O(\log n)$, a similar result holds for MDD(min). We prove that MDD(max) is $APX$-complete on 3-regular unweighted graphs and provide an approximation algorithm with ratio $1.583$ when $G$ is a 3-regular unweighted graph. In addition, we show that MDD(min) can be solved in polynomial time when $G$ is a regular graph of constant degree.Comment: 16 pages, 4 figures, submitted to Elsevier's Journal of Discrete Algorithm

Maximizing Utility Among Selfish Users in Social Groups

We consider the problem of a social group of users trying to obtain a "universe" of files, first from a server and then via exchange amongst themselves. We consider the selfish file-exchange paradigm of give-and-take, whereby two users can exchange files only if each has something unique to offer the other. We are interested in maximizing the number of users who can obtain the universe through a schedule of file-exchanges. We first present a practical paradigm of file acquisition. We then present an algorithm which ensures that at least half the users obtain the universe with high probability for $n$ files and $m=O(\log n)$ users when $n\rightarrow\infty$, thereby showing an approximation ratio of 2. Extending these ideas, we show a $1+\epsilon_1$ - approximation algorithm for $m=O(n)$, $\epsilon_1>0$ and a $(1+z)/2 +\epsilon_2$ - approximation algorithm for $m=O(n^z)$, $z>1$, $\epsilon_2>0$. Finally, we show that for any $m=O(e^{o(n)})$, there exists a schedule of file exchanges which ensures that at least half the users obtain the universe.Comment: 11 pages, 3 figures; submitted for review to the National Conference on Communications (NCC) 201

The Online Disjoint Set Cover Problem and its Applications

Given a universe $U$ of $n$ elements and a collection of subsets $\mathcal{S}$ of $U$, the maximum disjoint set cover problem (DSCP) is to partition $\mathcal{S}$ into as many set covers as possible, where a set cover is defined as a collection of subsets whose union is $U$. We consider the online DSCP, in which the subsets arrive one by one (possibly in an order chosen by an adversary), and must be irrevocably assigned to some partition on arrival with the objective of minimizing the competitive ratio. The competitive ratio of an online DSCP algorithm $A$ is defined as the maximum ratio of the number of disjoint set covers obtained by the optimal offline algorithm to the number of disjoint set covers obtained by $A$ across all inputs. We propose an online algorithm for solving the DSCP with competitive ratio $\ln n$. We then show a lower bound of $\Omega(\sqrt{\ln n})$ on the competitive ratio for any online DSCP algorithm. The online disjoint set cover problem has wide ranging applications in practice, including the online crowd-sourcing problem, the online coverage lifetime maximization problem in wireless sensor networks, and in online resource allocation problems.Comment: To appear in IEEE INFOCOM 201