We study the problem of hierarchical clustering on planar graphs. We formulate this in terms of finding the closest ultrametric to a specified set of distances and solve it using an LP relaxation that leverages minimum cost perfect matching as a subroutine to efficiently explore the space of planar partitions. We apply our algorithm to the problem of hierarchical image segmentation.

We study the problem of hierarchical clustering on planar graphs. We formulate this in terms of finding the closest ultrametric to a specified set of distances and solve it using an LP relaxation that leverages minimum cost perfect matching as a subroutine to efficiently explore the space of planar partitions. We apply our algorithm to the problem of hierarchical image segmentation.

We propose a graph clustering formulation based on multicut (a.k.a. weighted correlation clustering) on the complete graph. Our formulation does not need specification of the graph topology as in the original sparse formulation of multicut, making our approach simpler and potentially better performing. In contrast to unweighted correlation clustering we allow for a more expressive weighted cost structure. In dense multicut, the clustering objective is given in a factorized form as inner products of node feature vectors. This allows for an efficient formulation and inference in contrast to multicut/weighted correlation clustering, which has at least quadratic representation and computation complexity when working on the complete graph. We show how to rewrite classical greedy algorithms for multicut in our dense setting and how to modify them for greater efficiency and solution quality. In particular, our algorithms scale to graphs with tens of thousands of nodes. Empirical evidence on instance segmentation on Cityscapes and clustering of ImageNet datasets shows the merits of our approach.

The minimum cost lifted multicut approach has proven practically good performance in a wide range of applications such as image decomposition, mesh segmentation, multiple object tracking, and motion segmentation. It addresses such problems in a graph-based model, where real-valued costs are assigned to the edges between entities such that the minimum cut decomposes the graph into an optimal number of segments. Driven by a probabilistic formulation of minimum cost multicuts, we provide a measure for the uncertainties of the decisions made during the optimization. We argue that access to such uncertainties is crucial for many practical applications and conduct an evaluation by means of sparsifications on three different, widely used datasets in the context of image decomposition (BSDS-500) and motion segmentation (DAVIS2016 and FBMS59) in terms of variation of information (VI) and Rand index (RI).

Image partitioning, or segmentation without semantics, is the task of decomposing an image into distinct segments, or equivalently to detect closed contours. Most prior work either requires seeds, one per segment; or a threshold; or formulates the task as multicut / correlation clustering, an NP-hard problem. Here, we propose a greedy algorithm for signed graph partitioning, the "Mutex Watershed". Unlike seeded watershed, the algorithm can accommodate not only attractive but also repulsive cues, allowing it to find a previously unspecified number of segments without the need for explicit seeds or a tunable threshold. We also prove that this simple algorithm solves to global optimality an objective function that is intimately related to the multicut / correlation clustering integer linear programming formulation. The algorithm is deterministic, very simple to implement, and has empirically linearithmic complexity. When presented with short-range attractive and long-range repulsive cues from a deep neural network, the Mutex Watershed gives the best results currently known for the competitive ISBI 2012 EM segmentation benchmark.

We study the problem of hierarchical clustering on planar graphs. We formulate this in terms of finding the closest ultrametric to a specified set of distances and solve it using an LP relaxation that leverages minimum cost perfect matching as a subroutine to efficiently explore the space of planar partitions. We apply our algorithm to the problem of hierarchical image segmentation.

We consider optimizing the coalition structure in Coalitional Skill Games (CSGs), a succinct representation of coalitional games. In CSGs, the value of a coalition depends on the tasks its members can achieve. The tasks require various skills to complete them, and agents may have different skill sets. The optimal coalition structure is a partition of the agents to coalitions, that maximizes the sum of utilities obtained by the coalitions. We show that CSGs can represent any characteristic function, and consider optimal coalition structure generation in this representation. We provide hardness results, showing that in general CSGs, as well as in very restricted versions of them, computing the optimal coalition structure is hard. On the positive side, we show that the problem can be reformulated as constraint satisfaction on a hyper graph, and present an algorithm that finds the optimal coalition structure in polynomial time for instances with bounded tree-width and number of tasks.