Algorithms to Approximate Column-Sparse Packing Problems

Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go "half the remaining distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM Transactions of Algorithm

Three ways to cover a graph

We consider the problem of covering an input graph $H$ with graphs from a fixed covering class $G$. The classical covering number of $H$ with respect to $G$ is the minimum number of graphs from $G$ needed to cover the edges of $H$ without covering non-edges of $H$. We introduce a unifying notion of three covering parameters with respect to $G$, two of which are novel concepts only considered in special cases before: the local and the folded covering number. Each parameter measures "how far'' $H$ is from $G$ in a different way. Whereas the folded covering number has been investigated thoroughly for some covering classes, e.g., interval graphs and planar graphs, the local covering number has received little attention. We provide new bounds on each covering number with respect to the following covering classes: linear forests, star forests, caterpillar forests, and interval graphs. The classical graph parameters that result this way are interval number, track number, linear arboricity, star arboricity, and caterpillar arboricity. As input graphs we consider graphs of bounded degeneracy, bounded degree, bounded tree-width or bounded simple tree-width, as well as outerplanar, planar bipartite, and planar graphs. For several pairs of an input class and a covering class we determine exactly the maximum ordinary, local, and folded covering number of an input graph with respect to that covering class.Comment: 20 pages, 4 figure

On the Locality of Nash-Williams Forest Decomposition and Star-Forest Decomposition

Given a graph $G=(V,E)$ with arboricity $\alpha$, we study the problem of decomposing the edges of $G$ into $(1+\epsilon)\alpha$ disjoint forests in the distributed LOCAL model. Barenboim and Elkin [PODC `08] gave a LOCAL algorithm that computes a $(2+\epsilon)\alpha$-forest decomposition using $O(\frac{\log n}{\epsilon})$ rounds. Ghaffari and Su [SODA `17] made further progress by computing a $(1+\epsilon) \alpha$-forest decomposition in $O(\frac{\log^3 n}{\epsilon^4})$ rounds when $\epsilon \alpha = \Omega(\sqrt{\alpha \log n})$, i.e. the limit of their algorithm is an $(\alpha+ \Omega(\sqrt{\alpha \log n}))$-forest decomposition. This algorithm, based on a combinatorial construction of Alon, McDiarmid \& Reed [Combinatorica `92], in fact provides a decomposition of the graph into \emph{star-forests}, i.e. each forest is a collection of stars. Our main result in this paper is to reduce the threshold of $\epsilon \alpha$ in $(1+\epsilon)\alpha$-forest decomposition and star-forest decomposition. This further answers the $10^{\text{th}}$ open question from Barenboim and Elkin's "Distributed Graph Algorithms" book. Moreover, it gives the first $(1+\epsilon)\alpha$-orientation algorithms with {\it linear dependencies} on $\epsilon^{-1}$. At a high level, our results for forest-decomposition are based on a combination of network decomposition, load balancing, and a new structural result on local augmenting sequences. Our result for star-forest decomposition uses a more careful probabilistic analysis for the construction of Alon, McDiarmid, \& Reed; the bounds on star-arboricity here were not previously known, even non-constructively

Finding a cycle with maximum profit-to-time ratio : an application to optimum deployment of containerships

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1994.Includes bibliographical references (leaves 80-82).by Ronald W.Y. Chu.M.S