A Parallelizable Acceleration Framework for Packing Linear Programs

This paper presents an acceleration framework for packing linear programming problems where the amount of data available is limited, i.e., where the number of constraints m is small compared to the variable dimension n. The framework can be used as a black box to speed up linear programming solvers dramatically, by two orders of magnitude in our experiments. We present worst-case guarantees on the quality of the solution and the speedup provided by the algorithm, showing that the framework provides an approximately optimal solution while running the original solver on a much smaller problem. The framework can be used to accelerate exact solvers, approximate solvers, and parallel/distributed solvers. Further, it can be used for both linear programs and integer linear programs

Planar Ultrametric Rounding for Image Segmentation

We study the problem of hierarchical clustering on planar graphs. We formulate this in terms of an LP relaxation of ultrametric rounding. To solve this LP efficiently we introduce a dual cutting plane scheme that uses minimum cost perfect matching as a subroutine in order to efficiently explore the space of planar partitions. We apply our algorithm to the problem of hierarchical image segmentation

Heuristics with Performance Guarantees for the Minimum Number of Matches Problem in Heat Recovery Network Design

Heat exchanger network synthesis exploits excess heat by integrating process hot and cold streams and improves energy efficiency by reducing utility usage. Determining provably good solutions to the minimum number of matches is a bottleneck of designing a heat recovery network using the sequential method. This subproblem is an NP-hard mixed-integer linear program exhibiting combinatorial explosion in the possible hot and cold stream configurations. We explore this challenging optimization problem from a graph theoretic perspective and correlate it with other special optimization problems such as cost flow network and packing problems. In the case of a single temperature interval, we develop a new optimization formulation without problematic big-M parameters. We develop heuristic methods with performance guarantees using three approaches: (i) relaxation rounding, (ii) water filling, and (iii) greedy packing. Numerical results from a collection of 51 instances substantiate the strength of the methods

Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference

We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF inference problems. The core of our method is a very efficient bounding procedure, which combines scalable semidefinite programming (SDP) and a cutting-plane method for seeking violated constraints. In order to further speed up the computation, several strategies have been exploited, including model reduction, warm start and removal of inactive constraints. We analyze the performance of the proposed method under different settings, and demonstrate that our method either outperforms or performs on par with state-of-the-art approaches. Especially when the connectivities are dense or when the relative magnitudes of the unary costs are low, we achieve the best reported results. Experiments show that the proposed algorithm achieves better approximation than the state-of-the-art methods within a variety of time budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page

Motif Clustering and Overlapping Clustering for Social Network Analysis

Motivated by applications in social network community analysis, we introduce a new clustering paradigm termed motif clustering. Unlike classical clustering, motif clustering aims to minimize the number of clustering errors associated with both edges and certain higher order graph structures (motifs) that represent "atomic units" of social organizations. Our contributions are two-fold: We first introduce motif correlation clustering, in which the goal is to agnostically partition the vertices of a weighted complete graph so that certain predetermined "important" social subgraphs mostly lie within the same cluster, while "less relevant" social subgraphs are allowed to lie across clusters. We then proceed to introduce the notion of motif covers, in which the goal is to cover the vertices of motifs via the smallest number of (near) cliques in the graph. Motif cover algorithms provide a natural solution for overlapping clustering and they also play an important role in latent feature inference of networks. For both motif correlation clustering and its extension introduced via the covering problem, we provide hardness results, algorithmic solutions and community detection results for two well-studied social networks

Scalable Semidefinite Relaxation for Maximum A Posterior Estimation

Maximum a posteriori (MAP) inference over discrete Markov random fields is a fundamental task spanning a wide spectrum of real-world applications, which is known to be NP-hard for general graphs. In this paper, we propose a novel semidefinite relaxation formulation (referred to as SDR) to estimate the MAP assignment. Algorithmically, we develop an accelerated variant of the alternating direction method of multipliers (referred to as SDPAD-LR) that can effectively exploit the special structure of the new relaxation. Encouragingly, the proposed procedure allows solving SDR for large-scale problems, e.g., problems on a grid graph comprising hundreds of thousands of variables with multiple states per node. Compared with prior SDP solvers, SDPAD-LR is capable of attaining comparable accuracy while exhibiting remarkably improved scalability, in contrast to the commonly held belief that semidefinite relaxation can only been applied on small-scale MRF problems. We have evaluated the performance of SDR on various benchmark datasets including OPENGM2 and PIC in terms of both the quality of the solutions and computation time. Experimental results demonstrate that for a broad class of problems, SDPAD-LR outperforms state-of-the-art algorithms in producing better MAP assignment in an efficient manner.Comment: accepted to International Conference on Machine Learning (ICML 2014