With each separable optimization problem over a given set of vectors is associated its multichoice counterpart which involves choosing $n$ rather than one solutions from the set so as to maximize the given separable function over the sum of the chosen solutions. Such problems have been studied in various contexts under various names, such as load balancing in machine scheduling, congestion routing, minimum shared and vulnerable edge problems, and shifted optimization. Separable multichoice optimization has a very broad expressive power and can be hard already for explicitly given sets of binary points. In this article we consider the problem over monotone systems, also called independence systems. Typically such a system has exponential size, and we assume that it is presented implicitly by a linear optimization oracle. Our main results for separable multichoice optimization are the following. First, the problem over any monotone system with any separable concave function can be approximated in polynomial time with a constant approximation ratio which is independent of $n$. Second, the problem over any monotone system with an arbitrary separable function can be approximated in polynomial time with an approximation ratio of $1∕\left(O(logn)\right)$.

与给定向量集上的每个可分离优化问题相关联，其多选对应项涉及选择$ñ$而不是集合中的一个解，以便在所选解的总和上最大化给定的可分离函数。已经在各种情况下以各种名称研究了此类问题，例如机器调度中的负载平衡，拥塞路由，最小共享和易受攻击的边缘问题以及转移优化。可分离的多选优化具有非常广泛的表达能力，对于明确给定的二进制点集可能已经很难。在本文中，我们考虑单调系统的问题，也称为独立系统。通常，这样的系统具有指数大小，并且我们假设它是由线性优化预言隐式表示的。以下是可分离的多项选择优化的主要结果。首先，具有任意可分离凹函数的任何单调系统的问题都可以在多项式时间内以恒定的近似比来近似，该近似比与$ñ$。其次，在任意单调系统中具有任意可分函数的问题都可以在多项式时间内以近似比为$\mathrm{1个}∕（Ø（日志ñ））$。