放射治疗计划中的最近向量问题
A closest vector problem arising in radiation therapy planning
摘要：
在本文中，我们考虑的问题，找到一个向量，可以写为一个非负整数线性组合的给定0-1向量，生成元，使l_1-距离之间的这个向量和一个给定的目标向量是最小的。我们证明了这个最接近的向量问题是NP-困难的近似在一个O（d）的附加误差，其中d是周围的向量空间的维数。我们表明，该问题可以近似在一个O（d^{3/2}）的附加误差在多项式时间内，通过舍入的自然LP松弛的问题的最优解。我们还观察到，在特定的情况下，目标向量是整数和生成元形成一个完全么模矩阵，该问题可以解决在多项式时间。最近向量问题出现在放射治疗计划的制定中。在这种情况下，目标是一个非负整数矩阵，生成元是某些0-1矩阵，其行满足连续1属性。在这里，我们主要考虑的版本的问题，其中的一组发电机包括所有这些矩阵，在每一个非零行的数量，至少是一个特定的常数。这组生成器对所谓的最小分离约束进行编码。
英文摘要：
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.
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