Partitioning problems in dense hypergraphs

AbstractWe study the general partitioning problem and the discrepancy problem in dense hypergraphs. Using the regularity lemma (Szemerédi, Problemes Combinatories et Theorie des Graphes (1978), pp. 399–402) and its algorithmic version proved in Czygrinow and Rödl (SIAM J. Comput., to appear), we give polynomial-time approximation schemes for the general partitioning problem and for the discrepancy problem

Tiling in bipartite graphs with asymmetric minimum degrees

The problem of determining the optimal minimum degree condition for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of K_{s,s} was solved by Zhao. Later Hladk\'y and Schacht, and Czygrinow and DeBiasio determined the optimal minimum degree condition for a balanced bipartite graph on 2m(s+t) vertices to contain m vertex disjoint copies of K_{s,t} for fixed positive integers s<t. For a balanced bipartite graph G[U,V], let \delta_U be the minimum degree over all vertices in U and \delta_V be the minimum degree over all vertices in V. We consider the problem of determining the optimal value of \delta_U+\delta_V which guarantees that G can be tiled with K_{s,s}. We show that the optimal value depends on D:=|\delta_V-\delta_U|. When D is small, we show that \delta_U+\delta_V\geq n+3s-5 is best possible. As D becomes larger, we show that \delta_U+\delta_V can be made smaller, but no smaller than n+2s-2s^{1/2}. However, when D=n-C for some constant C, we show that there exist graphs with \delta_U+\delta_V\geq n+s^{s^{1/3}} which cannot be tiled with K_{s,s}.Comment: 34 pages, 4 figures. This is the unabridged version of the paper, containing the full proof of Theorem 1.7. The case when |\delta_U-\delta_V| is small and s>2 involves a lengthy case analysis, spanning pages 20-32; this section is not included in the "journal version

Pebbling in Dense Graphs

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. The pebbling number of a graph G is the minimum number pi(G) so that every configuration of pi(G) pebbles is solvable. A graph is Class 0 if its pebbling number equals its number of vertices. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. Here we prove that graphs on n>=9 vertices having minimum degree at least floor(n/2) are Class 0, as are bipartite graphs with m>=336 vertices in each part having minimum degree at least floor(m/2)+1. Both bounds are best possible. In addition, we prove that the pebbling threshold of graphs with minimum degree d, with sqrt{n} << d, is O(n^{3/2}/d), which is tight when d is proportional to n.Comment: 10 page

Distributed Approximation of Maximum Independent Set and Maximum Matching

We present a simple distributed $\Delta$-approximation algorithm for maximum weight independent set (MaxIS) in the $\mathsf{CONGEST}$ model which completes in $O(\texttt{MIS}(G)\cdot \log W)$ rounds, where $\Delta$ is the maximum degree, $\texttt{MIS}(G)$ is the number of rounds needed to compute a maximal independent set (MIS) on $G$, and $W$ is the maximum weight of a node. %Whether our algorithm is randomized or deterministic depends on the \texttt{MIS} algorithm used as a black-box. Plugging in the best known algorithm for MIS gives a randomized solution in $O(\log n \log W)$ rounds, where $n$ is the number of nodes. We also present a deterministic $O(\Delta +\log^* n)$-round algorithm based on coloring. We then show how to use our MaxIS approximation algorithms to compute a $2$-approximation for maximum weight matching without incurring any additional round penalty in the $\mathsf{CONGEST}$ model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of \emph{local aggregation algorithms} for which we describe a mechanism that allows the simulation to run in the $\mathsf{CONGEST}$ model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to ($2+\varepsilon$) allows us to devise a distributed algorithm requiring $O(\frac{\log \Delta}{\log\log\Delta})$ rounds for any constant $\varepsilon>0$. For the unweighted case, we can even obtain a $(1+\varepsilon)$-approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on $\Delta$

Distributed Approximations of f-Matchings and b-Matchings in Graphs of Sub-Logarithmic Expansion

We give a distributed algorithm which given ? > 0 finds a (1-?)-factor approximation of a maximum f-matching in graphs G = (V,E) of sub-logarithmic expansion. Using a similar approach we also give a distributed approximation of a maximum b-matching in the same class of graphs provided the function b: V ? ?^+ is L-Lipschitz for some constant L. Both algorithms run in O(log^* n) rounds in the LOCAL model, which is optimal