Set covering with our eyes closed

Given a universe $U$ of $n$ elements and a weighted collection $\mathscr{S}$ of $m$ subsets of $U$, the universal set cover problem is to a priori map each element $u \in U$ to a set $S(u) \in \mathscr{S}$ containing $u$ such that any set $X{\subseteq U}$ is covered by S(X)=\cup_{u\in XS(u). The aim is to find a mapping such that the cost of $S(X)$ is as close as possible to the optimal set cover cost for $X$. (Such problems are also called oblivious or a priori optimization problems.) Unfortunately, for every universal mapping, the cost of $S(X)$ can be $\Omega(\sqrt{n})$ times larger than optimal if the set $X$ is adversarially chosen. In this paper we study the performance on average, when $X$ is a set of randomly chosen elements from the universe: we show how to efficiently find a universal map whose expected cost is $O(\log mn)$ times the expected optimal cost. In fact, we give a slightly improved analysis and show that this is the best possible. We generalize these ideas to weighted set cover and show similar guarantees to (nonmetric) facility location, where we have to balance the facility opening cost with the cost of connecting clients to the facilities. We show applications of our results to universal multicut and disc-covering problems and show how all these universal mappings give us algorithms for the stochastic online variants of the problems with the same competitive factors

Oblivious routing in directed graphs with random demands

Oblivious routing algorithms for general undirected networks were introduced by Räcke, and this work has led to many subsequent improvements and applications. More precisely, Räcke showed that there is an oblivious routing algorithm with polylogarithmic competitive ratio (w.r.t. edge congestion) for any undirected graph. Comparatively little positive results are known about oblivious routing in general directed networks. Using a novel approach, we present the first oblivious routing algorithm which is O(log 2 n)-competitive with high probability in directed graphs given that the demands are chosen randomly from a known demand-distribution. On the other hand, we show that no oblivious routing algorithm can be o( log n log log n) competitive even with constant probability in general directed graphs. Our routing algorithms are not oblivious in the traditional definition, but we add the concept of demand-dependence, i.e., the path chosen for an s-t pair may depend on the demand between s and t. This concept that still preserves that routing decisions are only based on local information proves very powerful in our randomized demand model. Finally, we show that our approach for designing competitive oblivious routing algorithms is quite general and has applications in other contexts like stochastic scheduling

Oblivious Routing in Directed Graphs with Random Demands

The concept of oblivious routing aims at developing routing algorithms that base their routing decisions only on local knowledge and that therefore can be implemented very efficiently in a distributed environment. Traditionally, for an oblivious routing algorithm the routing path chosen between a source s and a target t may onl

Online Network Design under Uncertainty

Today, computer and information networks play a significant role in the success of businesses, both large and small. Networks provide access to various services and resources to end users and devices. There has been extensive research on de- signing networks according to numerous criteria such as cost-efficiency, availability, adaptivity, survivability, among others. In this dissertation, we revisit some of the most fundamental network design problems in the presence of uncertainty. In most realistic models, we are forced to make decisions in the presence of an incomplete input, which is the source of uncertainty for an optimization algorithm. There are different types of uncertainty. For example, in stochastic settings, we may have some random variables derived from some known/unknown distributions. In online settings, the complete input is not known in a-priori and pieces of the input become available sequentially; leaving the algorithm to make decisions only with partial data. In this dissertation, we consider network design and network optimization problems with uncertainty. In particular, we study online bounded-degree Steiner network design, online survivable network design, and stochastic k-server. We analyze their complexity and design competitive algorithms for them