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
2098322943 1576317141 2046709864 2118953734
We prove that any graph excluding Kr as a minor has can be partitioned into clusters of diameter at most Δ while removing at most O(r Δ) fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to remove O(r2 Δ) fraction of the edges. Our result is obtained by a new approach that relates the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops and robbers game on graphs excluding a fixed minor, can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter O(r) and strong-diameter partitions with padding parameter O(r2) for Kr-free graphs, O(k) for treewidth-k graphs, and O(log g) for graphs with genus g. Given an undirected graph with two different nonnegative costs associated with every edge e (say, w e for the weight and l e for the length of edge e) and a budget L, consider the problem of finding a spanning tree of total edge length at most L and minimum total weight under this restriction. This constrained minimum spanning tree problem is weakly NP-hard. We present a polynomial-time approximation scheme for this problem. This algorithm always produces a spanning tree of total length at most (1 + e)L and of total weight at most that of any spanning tree of total length at most L, for any fixed e >0. The algorithm uses Lagrangean relaxation, and exploits adjacency relations for matroids. We study approaches for the exact solution of the NP-hard minimum spanning tree problem under conflict constraints. Given a graph (G(V,E) ) and a set (C E E ) of conflicting edge pairs, the problem consists of finding a conflict-free minimum spanning tree, i.e. feasible solutions are allowed to include at most one of the edges from each pair in (C ). The problem was introduced recently in the literature, with several results on its complexity and approximability. Some formulations and both exact and heuristic algorithms were also discussed, but computational results indicate considerably large duality gaps and a lack of optimality certificates for benchmark instances. In this paper, we build on the representation of conflict constraints using an auxiliary conflict graph ( G (E,C) ), where stable sets correspond to conflict-free subsets of (E ). We introduce a general preprocessing method and a branch and cut algorithm using an IP formulation with exponentially sized classes of valid inequalities for both the spanning tree and the stable set polytopes. Encouraging computational results indicate that the dual bounds of our approach are significantly stronger than those previously available, already in the initial LP relaxation, and we are able to provide new feasibility and optimality certificates. The graph partitioning problem is that of dividing the vertices of a graph into sets of specified sizes such that few edges cross between sets. This NP-complete problem arises in many important scientific and engineering problems. Prominent examples include the decomposition of data structures for parallel computation, the placement of circuit elements and the ordering of sparse matrix computations. We present a multilevel algorithm for graph partitioning in which the graph is approximated by a sequence of increasingly smaller graphs. The smallest graph is then partitioned using a spectral method, and this partition is propagated back through the hierarchy of graphs. A variant of the Kernighan-Lin algorithm is applied periodically to refine the partition. The entire algorithm can be implemented to execute in time proportional to the size of the original graph. Experiments indicate that, relative to other advanced methods, the multilevel algorithm produces high quality partitions at low cost.