Partial multicovering and the d-consecutive ones property

A d-interval is the union of d disjoint intervals on the real line. In the d-interval stabbing problem (d-is) we are given a set of d-intervals and a set of points, each d-interval I has a stabbing requirement r(I) and each point has a weight, and the goal is to find a minimum weight multiset of points that stabs each d-interval I at least r(I) times. In practice there is a trade-off between fulfilling requirements and cost, and therefore it is interesting to study problems in which one is required to fulfill only a subset of the requirements. In this paper we study variants of dis in which a feasible solution is a multiset of points that may satisfy only a subset of the stabbing requirements. In partial d-is we are given an integer t, and the sum of requirements satisfied by the computed solution must be at least t. In prize collecting d-is each d-interval has a penalty that must be paid for every unit of unsatisfied requirement. We also consider a maximization version of prize collecting d-is in which each d-interval has a prize that is gained for every time, up to r(I), it is stabbed. Our study is motivated by several resource allocation and geometric facility location problems. We present a ( ρ )-approximation algorithm for prize collecting d-is, where ρ = minI r(I), and an O(d)-approximation algorithm for partial d-is. We obtain the latter result by designing a general framework for approximating partial multicovering problems that extends the framework for approximating partial covering problems from [21]. We also show that maximum prize collecting d-is is at least as hard to approximate as maximum independent set, even for d = 2, and present a d-approximation algorithm for maximum prize collecting d-dimensional rectangle stabbing.