Knapsack with Vertex Cover, Set Cover, and Hitting Set

In the Vertex Cover Knapsack problem, we are given an undirected graph G = (V, E), with weights (w(u))u∈V and values (α(u))u∈V of the vertices, the size s of the knapsack, a target value p, and the goal is to compute if there exists a vertex cover U ⊆ V with total weight at most s, and total value at least p. This problem simultaneously generalizes the classical vertex cover and knapsack problems. We show that this problem is strongly NP-complete. However, it admits a pseudo-polynomial time algorithm for trees. In fact, we show that there is an algorithm that runs in time O ^{(2tw · n · min{s2, p2})} where tw is the treewidth of G. Moreover, we can compute a (1 − ε)approximate solution for maximizing the value of the solution given the knapsack size as input in time O ^{(2tw · poly(n, 1/ε, log (Ʃ _v∈V α(v))))} and a (1 + ε)-approximate solution to minimize the size of the solution given a target value as input, in time O ^{(2tw · poly(n, 1/ε, log (Ʃ _v∈V w(v))))} for every ε > 0. Restricting our attention to polynomial-time algorithms only, we then consider polynomial-time algorithms and present a 2 factor polynomial-time approximation algorithm for this problem for minimizing the total weight of the solution, which is optimal up to additive o(1) assuming Unique Games Conjecture (UGC). On the other hand, we show that there is no ρ factor polynomial-time approximation algorithm for maximizing the total value of the solution given a knapsack size for any ρ > 1 unless P = NP. Furthermore, we show similar results for the variants of the above problem when the solution U needs to be a minimal vertex cover, minimum vertex cover, and vertex cover of size at most k for some input integer k. Then, we consider set families (equivalently hypergraphs) and study the variants of the above problem when the solution needs to be a set cover and hitting set. We show that there are H_d and f factor polynomial-time approximation algorithms for Set Cover Knapsack where d is the maximum cardinality of any set and f is the maximum number of sets in the family where any element can belongs in the input for minimizing the weight of the knapsack given a target value, and a d factor polynomial-time approximation algorithm for d-Hitting Set Knapsack which are optimal up to additive o(1) assuming UGC. On the other hand, we show that there is no ρ factor polynomial-time approximation algorithm for maximizing the total value of the solution given a knapsack size for any ρ > 1 unless P = NP for both Set Cover Knapsack and d-Hitting Set Knapsack.