There has been a long-standing interest in computing diverse solutions to optimization problems. Motivated by reallocation of governmental institutions in Sweden, in 1995 J. Krarup posed the problem of finding $k$ edge-disjoint Hamiltonian Circuits of minimum total weight, called the peripatetic salesman problem (PSP). Since then researchers have investigated the complexity of finding diverse solutions to spanning trees, paths, vertex covers, matchings, and more. Unlike the PSP that has a constraint on the total weight of the solutions, recent work has involved finding diverse solutions that are all optimal. However, sometimes the space of exact solutions may be too small to achieve sufficient diversity. Motivated by this, we initiate the study of obtaining sufficiently-diverse, yet approximately-optimal solutions to optimization problems. Formally, given an integer $k$, an approximation factor $c$, and an instance $I$ of an optimization problem, we aim to obtain a set of $k$ solutions to $I$ that a) are all $c$ approximately-optimal for $I$ and b) maximize the diversity of the $k$ solutions. Finding such solutions, therefore, requires a better understanding of the global landscape of the optimization function. We show that, given any metric on the space of solutions, and the diversity measure as the sum of pairwise distances between solutions, this problem can be solved by combining ideas from dispersion and multicriteria optimization. We first provide a general reduction to an associated budget-constrained optimization (BCO) problem, where one objective function is to be maximized (minimized) subject to a bound on the second objective function. We then prove that bi-approximations to the BCO can be used to give bi-approximations to the diverse approximately optimal solutions problem with a little overhead.
翻译:长期以来,人们一直关注如何计算优化问题的多种解决办法。在瑞典政府机构的重新分配的推动下,1995年,J.Krarup在瑞典的政府机构的重新分配下,提出了找到最低总重量(称为超光速销售员问题(PSP))的硬度偏差拉密尔顿电路问题。自那时起,研究人员一直在调查寻找跨越树木、道路、顶层覆盖、匹配等多种解决办法的复杂性。与限制解决方案总重的PSP不同的是,最近的工作涉及寻找各种最优化的解决方案。然而,有时精确解决方案的空间可能太小,无法实现足够的多样性。为此,我们开始研究如何获得足够分散的、但大致最优化的优化问题解决方案。形式上,考虑到一个整数美元、近似于美元、比重的汇率问题,我们的目标是获得一套以美元为基数的双倍的双价解决方案,然后以美元为美元为美元,然后以美元为美元为最优化的比值,然后以美元和b) 精确的解决方案的空间空间的篇幅可能太小。 最优化的解决方案的多样化的多样化的多样化,因此,我们需要一种更好的预算的汇率的汇率的功能。