A closest vector problem arising in radiation therapy planning

TL;DR

This paper investigates the computational complexity of a closest vector problem relevant to radiation therapy planning, establishing NP-hardness, approximation algorithms, and polynomial-time solutions under specific conditions.

Contribution

It proves NP-hardness of the problem, provides approximation algorithms, and identifies cases where the problem is solvable in polynomial time, advancing understanding in computational optimization for therapy planning.

Findings

NP-hard to approximate within O(d) additive error

Polynomial-time approximation within O(d^{3/2}) error

Polynomial solution when target is integer and generators are totally unimodular

Abstract

In this paper we consider the problem of finding a vector that can be written as a nonnegative integer linear combination of given 0-1 vectors, the generators, such that the l_1-distance between this vector and a given target vector is minimized. We prove that this closest vector problem is NP-hard to approximate within a O(d) additive error, where d is the dimension of the ambient vector space. We show that the problem can be approximated within a O(d^{3/2}) additive error in polynomial time, by rounding an optimal solution of a natural LP relaxation for the problem. We also observe that in the particular case where the target vector is integer and the generators form a totally unimodular matrix, the problem 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 0-1 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.