A closest vector problem arising in radiation therapy planning

In this paper we consider the problem of finding a vector that can be written as a nonnegative, integer and linear combination of given 0-1 vectors, the generators, such that the l1-distance between this vector and a given target vector is minimized. We prove that this closest vector problem is NP-hard to approximate within an additive error of (ln 2− ǫ)d ≈ (0, 693 − ǫ) d for all ǫ > 0 where d is the dimension of the ambient vector space. We show that the problem can be approximated within an additive error of ( e 4 + ln 2 2 ) d3/2 ≈ 1.026 d3/2 in polynomial time, by rounding an optimal solution of a natural LP relaxation for the problem. We also give a proof that in the particular case where the vectors satisfy the consecutive ones property, the problem can be formulated as a min-cost flow problem, hence can be solved in polynomial time. The closest vector problem arises in the elaboration of radiation therapy plans. In this context, the target is a nonnegative integer matrix and the generators are certain binary matrices whose rows satisfy the consecutive ones property. Here we mainly consider the version of the problem in which the set of generators comprises all those matrices that have on each nonzero row a number of ones that is at least a certain constant. This set of generators encodes the so-called minimum separation constraint.