Balanced Families of Perfect Hash Functions and Their Applications

The construction of perfect hash functions is a well-studied topic. In this paper, this concept is generalized with the following definition. We say that a family of functions from $[n]$ to $[k]$ is a $\delta$-balanced $(n,k)$-family of perfect hash functions if for every $S \subseteq [n]$, $|S|=k$, the number of functions that are 1-1 on $S$ is between $T/\delta$ and $\delta T$ for some constant $T>0$. The standard definition of a family of perfect hash functions requires that there will be at least one function that is 1-1 on $S$, for each $S$ of size $k$. In the new notion of balanced families, we require the number of 1-1 functions to be almost the same (taking $\delta$ to be close to 1) for every such $S$. Our main result is that for any constant $\delta > 1$, a $\delta$-balanced $(n,k)$-family of perfect hash functions of size $2^{O(k \log \log k)} \log n$ can be constructed in time $2^{O(k \log \log k)} n \log n$. Using the technique of color-coding we can apply our explicit constructions to devise approximation algorithms for various counting problems in graphs. In particular, we exhibit a deterministic polynomial time algorithm for approximating both the number of simple paths of length $k$ and the number of simple cycles of size $k$ for any $k \leq O(\frac{\log n}{\log \log \log n})$ in a graph with $n$ vertices. The approximation is up to any fixed desirable relative error

Some results on (a:b)-choosability

A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not necessarily $(c:d)$-choosable. Applying probabilistic methods, an upper bound for the $k^{th}$ choice number of a graph is given. We also prove that a directed graph with maximum outdegree $d$ and no odd directed cycle is $(k(d+1):k)$-choosable for every $k \geq 1$. Other results presented in this article are related to the strong choice number of graphs (a generalization of the strong chromatic number). We conclude with complexity analysis of some decision problems related to graph choosability

Admission Control to Minimize Rejections and Online Set Cover with Repetitions

We study the admission control problem in general networks. Communication requests arrive over time, and the online algorithm accepts or rejects each request while maintaining the capacity limitations of the network. The admission control problem has been usually analyzed as a benefit problem, where the goal is to devise an online algorithm that accepts the maximum number of requests possible. The problem with this objective function is that even algorithms with optimal competitive ratios may reject almost all of the requests, when it would have been possible to reject only a few. This could be inappropriate for settings in which rejections are intended to be rare events. In this paper, we consider preemptive online algorithms whose goal is to minimize the number of rejected requests. Each request arrives together with the path it should be routed on. We show an $O(\log^2 (mc))$-competitive randomized algorithm for the weighted case, where $m$ is the number of edges in the graph and $c$ is the maximum edge capacity. For the unweighted case, we give an $O(\log m \log c)$-competitive randomized algorithm. This settles an open question of Blum, Kalai and Kleinberg raised in \cite{BlKaKl01}. We note that allowing preemption and handling requests with given paths are essential for avoiding trivial lower bounds

The complexity of planar graph choosability

AbstractA graph G is k-choosable if for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). We consider the complexity of deciding whether a given graph is k-choosable for some constant k. In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem of deciding whether a given planar triangle-free graph is 3-choosable. We also obtain simple constructions of a planar graph which is not 4-choosable and a planar triangle-free graph which is not 3-choosable