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We consider a class of optimization problems, where the input is an undirected graph with two w eight functions dened for each node, namely the node's pro t and its cost. The goal is to nd a connected set of nodes of low cost and high pro t. We present approximation algorithms for three natural optimization criteria that arise in this context, all of which are NP-hard. The budget problem asks for maximizing the pro t of the set subject to a budget constraint on its cost. The quota problem requires minimizing the cost of the set subject to a quota constraint on its pro t. Finally, the prize collecting problem calls for minimizing the cost of the set plus the pro t (here interpreted as a penalty) of the complement set. For all three problems, our algorithms give a n a p p r o ximation guarantee of O(log n), where n is the number of nodes. To the best of our knowledge, these are the rst approximation results for the quota problem and for the prize collecting problem, both of which a r e at least as hard to approximate as set cover. For the budget problem, our results improve on a previous O(log 2 n) r e s u l t of Guha, Moss, Naor, and Schieber. Our methods involve new theorems relating tree packings to (node) cut conditions. We also show similar theorems (with better bounds) using edge cut conditions. These imply bounds for the analogous budget and quota problems with edge costs which a r e comparable to known (constant factor) bounds.

We study problems that are motivated by a real-life pratkm of efficient recovery from a wide scale electric power outage caused by a major disaster such as a hurricane or an equipment failure. In most of these cases an optimized scheduling of the workforce is required, since the work crews on hand cannot handle the immediate recovery of the whole network.We model two variants of this problem: the budgeted problem, and the minimum weighted latency problem. We consider the problems for the general case as well as for two special cases: trees, and bipartite networks. All, but one of the problems (the budgeted tree problem), are NP-Hard and the algorithms given for them are approximation algorithms. For the budgeted tree problem we give an optimal solution. Interestingly, the budgeted problem for bipartite networks is exactly the budgeted maximum set cover problem, for which we give the best ratio approximation algorithm.

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