Knapsack: Connectedness, Path, and Shortest-Path

We study the KNAPSACK problem with graph-theoretic constraints. That is, there exists a graph structure on the input set of items of KNAPSACK and the solution also needs to satisfy certain graph theoretic properties on top of the KNAPSACK constraints. In particular, we study CONNECTED KNAPSACK where the solution must be a connected subset of items which has maximum value and satisfies the size constraint of the knapsack. We show that this problem is strongly NP-complete even for graphs of maximum degree four and NP-complete even for star graphs. On the other hand, we develop an algorithm running in time O (2(O(tw log tw)) . poly(n) min{s(2), d(2)}) where tw, s, d, n are respectively treewidth of the graph, the size of the knapsack, the target value of the knapsack, and the number of items. We also exhibit a (1- epsilon) factor approximation algorithm running in time O (2(O(tw log tw)) . poly( n, 1/epsilon)) for every epsilon > 0. We show similar results for PATH KNAPSACK and Shortest PATH KNAPSACK, where the solution must also induce a path and shortest path, respectively. Our results suggest that CONNECTED KNAPSACK is computationally the hardest, followed by PATH KNAPSACK and then SHORTEST PATH KNAPSACK.