We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. In the all-neighbours knapsack problem, an item can be selected only if all its neighbours are also selected. We give approximation algorithms and hardness results when the nodes have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.
Abstract The budgeted maximum coverage problem is: given a collection S of sets with associated costs defined over a domain of weighted elements, and a budget L , find a subset of S ′⫅ S such that the total cost of sets in S ′ does not exceed L , and the total weight of elements covered by S ′ is maximized. This problem is NP-hard. For the special case of this problem, where each set has unit cost, a (1−1 e ) -approximation is known. Yet, prior to this work, no approximation results were known for the general cost version. The contribution of this paper is a (1−1 e ) -approximation algorithm for the budgeted maximum coverage problem. We also argue that this approximation factor is the best possible, unless NP ⫅ DTIME (n O ( log log n) ) .