We study the problem of learning to partition users into groups, where one must learn the compatibilities between the users to achieve optimal groupings. We define four natural objectives that optimize for average and worst case compatibilities and propose new algorithms for adaptively learning optimal groupings. When we do not impose any structure on the compatibilities, we show that the group formation objectives considered are @math hard to solve and we either give approximation guarantees or prove inapproximability results. We then introduce an elegant structure, namely that of , that makes many of these problems polynomial time solvable. We explicitly characterize the optimal groupings under this structure and show that the optimal solutions are related to and partitions, well-studied in the psychology literature. For one of the four objectives, we show @math hardness under the score structure and give a @math approximation algorithm for which no constant approximation was known thus far. Finally, under the score structure, we propose an online low sample complexity PAC algorithm for learning the optimal partition. We demonstrate the efficacy of the proposed algorithm on synthetic and real world datasets.
There is one class of graph partitioning problems that come with size restrictions, namely, the Graph Tree Partitioning Problem @cite_10 , where the objective is to partition a graph into equal size subsets, such that the weight of the spanning tree on each subset is either as high as possible (max-min) or as low as possible (min-max), or, the sum of the weights of the spanning trees are either as high (max-sum), or, as low (min-sum) as possible. Though these objectives are closely related to ours, they are not exactly the same problems.
@cite_10
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For a graph G and a collection of vertex pairs (s"1,t"1),...,(s"k,t"k) , the k disjoint paths problem is to find k vertex-disjoint paths P"1,...,P"k, where P"i is a path from s"i to t"i for each i=1,...,k. In the corresponding optimization problem, the shortest disjoint paths problem, the vertex-disjoint paths P"i have to be chosen such that a given objective function is minimized. We consider two different objectives, namely minimizing the total path length (minimum sum, or short: Min-Sum), and minimizing the length of the longest path (Min-Max), for k=2,3. Min-Sum: We extend recent results by Colin de Verdiere and Schrijver to prove that, for a planar graph and for terminals adjacent to at most two faces, the Min-Sum 2 Disjoint Paths Problem can be solved in polynomial time. We also prove that, for six terminals adjacent to one face in any order, the Min-Sum 3 Disjoint Paths Problem can be solved in polynomial time. Min-Max: The Min-Max 2 Disjoint Paths Problem is known to be NP-hard for general graphs. We present an algorithm that solves the problem for graphs with tree-width 2 in polynomial time. We thus close the gap between easy and hard instances, since the problem is weakly NP-hard for graphs with tree-width 3. In this paper the shortest path and the minimum spanning tree problems in a graph with @math nodes and @math cost scenarios (objectives) are discussed. In order to choose a solution the min-max criterion is applied. The minmax versions of both problems are hard to approximate within @math for any @math . The best approximation algorithm for the min-max shortest path problem, known to date, has approximation ratio of @math . On the other hand, for the min-max spanning tree, there is a randomized algorithm with approximation ratio of @math . In this paper a deterministic @math -approximation algorithm for min-max shortest path is constructed. For min-max spanning tree a deterministic @math -approximation algorithm is proposed, which works for a large class of graphs and a randomized @math -approximation algorithm, which can be applied to all graphs, is constructed. It is also shown that the approximation ratios obtained are close to the integrality gaps of the corresponding LP relaxations. While the complexity of min-max and min-max regret versions of most classical combinatorial optimization problems has been thoroughly investigated, there are very few studies about their approximation. For a bounded number of scenarios, we establish general approximation schemes which can be used for min-max and min-max regret versions of some polynomial or pseudo-polynomial problems. Applying these schemes to shortest path, minimum spanning tree, minimum weighted perfect matching on planar graphs, and knapsack problems, we obtain fully polynomial-time approximation schemes with better running times than the ones previously presented in the literature. This paper investigates, for the first time in the literature, the approximation of min–max (regret) versions of classical problems like shortest path, minimum spanning tree, and knapsack. For a constant number of scenarios, we establish fully polynomial-time approximation schemes for the min–max versions of these problems, using relationships between multi-objective and min–max optimization. Using dynamic programming and classical trimming techniques, we construct a fully polynomial-time approximation scheme for min–max regret shortest path. We also establish a fully polynomial-time approximation scheme for min–max regret spanning tree and prove that min–max regret knapsack is not at all approximable. For a non-constant number of scenarios, in which case min–max and min–max regret versions of polynomial-time solvable problems usually become strongly NP-hard, non-approximability results are provided for min–max (regret) versions of shortest path and spanning tree.