Efficient Flow-based Approximation Algorithms for Submodular Hypergraph Partitioning via a Generalized Cut-Matching Game

In the past 20 years, increasing complexity in real world data has lead to the study of higher-order data models based on partitioning hypergraphs. However, hypergraph partitioning admits multiple formulations as hyperedges can be cut in multiple ways. Building upon a class of hypergraph partitioning problems introduced by Li & Milenkovic, we study the problem of minimizing ratio-cut objectives over hypergraphs given by a new class of cut functions, monotone submodular cut functions (mscf's), which captures hypergraph expansion and conductance as special cases. We first define the ratio-cut improvement problem, a family of local relaxations of the minimum ratio-cut problem. This problem is a natural extension of the Andersen & Lang cut improvement problem to the hypergraph setting. We demonstrate the existence of efficient algorithms for approximately solving this problem. These algorithms run in almost-linear time for the case of hypergraph expansion, and when the hypergraph rank is at most $O(1)$. Next, we provide an efficient $O(\log n)$-approximation algorithm for finding the minimum ratio-cut of $G$. We generalize the cut-matching game framework of Khandekar et. al. to allow for the cut player to play unbalanced cuts, and matching player to route approximate single-commodity flows. Using this framework, we bootstrap our algorithms for the ratio-cut improvement problem to obtain approximation algorithms for minimum ratio-cut problem for all mscf's. This also yields the first almost-linear time $O(\log n)$-approximation algorithms for hypergraph expansion, and constant hypergraph rank. Finally, we extend a result of Louis & Makarychev to a broader set of objective functions by giving a polynomial time $O\big(\sqrt{\log n}\big)$-approximation algorithm for the minimum ratio-cut problem based on rounding $\ell_2^2$-metric embeddings.Comment: Comments and feedback welcom

Analyzing Massive Graphs in the Semi-streaming Model

Massive graphs arise in a many scenarios, for example, traffic data analysis in large networks, large scale scientific experiments, and clustering of large data sets. The semi-streaming model was proposed for processing massive graphs. In the semi-streaming model, we have a random accessible memory which is near-linear in the number of vertices. The input graph (or equivalently, edges in the graph) is presented as a sequential list of edges (insertion-only model) or edge insertions and deletions (dynamic model). The list is read-only but we may make multiple passes over the list. There has been a few results in the insertion-only model such as computing distance spanners and approximating the maximum matching. In this thesis, we present some algorithms and techniques for (i) solving more complex problems in the semi-streaming model, (for example, problems in the dynamic model) and (ii) having better solutions for the problems which have been studied (for example, the maximum matching problem). In course of both of these, we develop new techniques with broad applications and explore the rich trade-offs between the complexity of models (insertion-only streams vs. dynamic streams), the number of passes, space, accuracy, and running time. 1. We initiate the study of dynamic graph streams. We start with basic problems such as the connectivity problem and computing the minimum spanning tree. These problems are trivial in the insertion-only model. However, they require non-trivial (and multiple passes for computing the exact minimum spanning tree) algorithms in the dynamic model. 2. Second, we present a graph sparsification algorithm in the semi-streaming model. A graph sparsification is a sparse graph that approximately preserves all the cut values of a graph. Such a graph acts as an oracle for solving cut-related problems, for example, the minimum cut problem and the multicut problem. Our algorithm produce a graph sparsification with high probability in one pass. 3. Third, we use the primal-dual algorithms to develop the semi-streaming algorithms. The primal-dual algorithms have been widely accepted as a framework for solving linear programs and semidefinite programs faster. In contrast, we apply the method for reducing space and number of passes in addition to reducing the running time. We also present some examples that arise in applications and show how to apply the techniques: the multicut problem, the correlation clustering problem, and the maximum matching problem. As a consequence, we also develop near-linear time algorithms for the $b$-matching problems which were not known before

Efficient Algorithms for Moral Lineage Tracing

Lineage tracing, the joint segmentation and tracking of living cells as they move and divide in a sequence of light microscopy images, is a challenging task. Jug et al. have proposed a mathematical abstraction of this task, the moral lineage tracing problem (MLTP), whose feasible solutions define both a segmentation of every image and a lineage forest of cells. Their branch-and-cut algorithm, however, is prone to many cuts and slow convergence for large instances. To address this problem, we make three contributions: (i) we devise the first efficient primal feasible local search algorithms for the MLTP, (ii) we improve the branch-and-cut algorithm by separating tighter cutting planes and by incorporating our primal algorithms, (iii) we show in experiments that our algorithms find accurate solutions on the problem instances of Jug et al. and scale to larger instances, leveraging moral lineage tracing to practical significance.Comment: Accepted at ICCV 201

Joint Cuts and Matching of Partitions in One Graph

As two fundamental problems, graph cuts and graph matching have been investigated over decades, resulting in vast literature in these two topics respectively. However the way of jointly applying and solving graph cuts and matching receives few attention. In this paper, we first formalize the problem of simultaneously cutting a graph into two partitions i.e. graph cuts and establishing their correspondence i.e. graph matching. Then we develop an optimization algorithm by updating matching and cutting alternatively, provided with theoretical analysis. The efficacy of our algorithm is verified on both synthetic dataset and real-world images containing similar regions or structures

JigsawNet: Shredded Image Reassembly using Convolutional Neural Network and Loop-based Composition

This paper proposes a novel algorithm to reassemble an arbitrarily shredded image to its original status. Existing reassembly pipelines commonly consist of a local matching stage and a global compositions stage. In the local stage, a key challenge in fragment reassembly is to reliably compute and identify correct pairwise matching, for which most existing algorithms use handcrafted features, and hence, cannot reliably handle complicated puzzles. We build a deep convolutional neural network to detect the compatibility of a pairwise stitching, and use it to prune computed pairwise matches. To improve the network efficiency and accuracy, we transfer the calculation of CNN to the stitching region and apply a boost training strategy. In the global composition stage, we modify the commonly adopted greedy edge selection strategies to two new loop closure based searching algorithms. Extensive experiments show that our algorithm significantly outperforms existing methods on solving various puzzles, especially those challenging ones with many fragment pieces