Submodular Welfare Maximization

An overview of different variants of the submodular welfare maximization problem in combinatorial auctions. In particular, I studied the existing algorithmic and game theoretic results for submodular welfare maximization problem and its applications in other areas such as social networks

Submodular Maximization Meets Streaming: Matchings, Matroids, and More

We study the problem of finding a maximum matching in a graph given by an input stream listing its edges in some arbitrary order, where the quantity to be maximized is given by a monotone submodular function on subsets of edges. This problem, which we call maximum submodular-function matching (MSM), is a natural generalization of maximum weight matching (MWM), which is in turn a generalization of maximum cardinality matching (MCM). We give two incomparable algorithms for this problem with space usage falling in the semi-streaming range---they store only $O(n)$ edges, using $O(n\log n)$ working memory---that achieve approximation ratios of $7.75$ in a single pass and $(3+\epsilon)$ in $O(\epsilon^{-3})$ passes respectively. The operations of these algorithms mimic those of Zelke's and McGregor's respective algorithms for MWM; the novelty lies in the analysis for the MSM setting. In fact we identify a general framework for MWM algorithms that allows this kind of adaptation to the broader setting of MSM. In the sequel, we give generalizations of these results where the maximization is over "independent sets" in a very general sense. This generalization captures hypermatchings in hypergraphs as well as independence in the intersection of multiple matroids.Comment: 18 page

Submodular Function Maximization for Group Elevator Scheduling

We propose a novel approach for group elevator scheduling by formulating it as the maximization of submodular function under a matroid constraint. In particular, we propose to model the total waiting time of passengers using a quadratic Boolean function. The unary and pairwise terms in the function denote the waiting time for single and pairwise allocation of passengers to elevators, respectively. We show that this objective function is submodular. The matroid constraints ensure that every passenger is allocated to exactly one elevator. We use a greedy algorithm to maximize the submodular objective function, and derive provable guarantees on the optimality of the solution. We tested our algorithm using Elevate 8, a commercial-grade elevator simulator that allows simulation with a wide range of elevator settings. We achieve significant improvement over the existing algorithms.Comment: 10 pages; 2017 International Conference on Automated Planning and Scheduling (ICAPS

Discrete Optimization Methods for Segmentation and Matching

This dissertation studies discrete optimization methods for several computer vision problems. In the first part, a new objective function for superpixel segmentation is proposed. This objective function consists of two components: entropy rate of a random walk on a graph and a balancing term. The entropy rate favors formation of compact and homogeneous clusters, while the balancing function encourages clusters with similar sizes. I present a new graph construction for images and show that this construction induces a matroid. The segmentation is then given by the graph topology which maximizes the objective function under the matroid constraint. By exploiting submodular and monotonic properties of the objective function, I develop an efficient algorithm with a worst-case performance bound of $\frac{1}{2}$ for the superpixel segmentation problem. Extensive experiments on the Berkeley segmentation benchmark show the proposed algorithm outperforms the state of the art in all the standard evaluation metrics. Next, I propose a video segmentation algorithm by maximizing a submodular objective function subject to a matroid constraint. This function is similar to the standard energy function in computer vision with unary terms, pairwise terms from the Potts model, and a novel higher-order term based on appearance histograms. I show that the standard Potts model prior, which becomes non-submodular for multi-label problems, still induces a submodular function in a maximization framework. A new higher-order prior further enforces consistency in the appearance histograms both spatially and temporally across the video. The matroid constraint leads to a simple algorithm with a performance bound of $\frac{1}{2}$. A branch and bound procedure is also presented to improve the solution computed by the algorithm. The last part of the dissertation studies the object localization problem in images given a single hand-drawn example or a gallery of shapes as the object model. Although many shape matching algorithms have been proposed for the problem, chamfer matching remains to be the preferred method when speed and robustness are considered. In this dissertation, I significantly improve the accuracy of chamfer matching while reducing the computational time from linear to sublinear (shown empirically). It is achieved by incorporating edge orientation information in the matching algorithm so the resulting cost function is piecewise smooth and the cost variation is tightly bounded. Moreover, I present a sublinear time algorithm for exact computation of the directional chamfer matching score using techniques from 3D distance transforms and directional integral images. In addition, the smooth cost function allows one to bound the cost distribution of large neighborhoods and skip the bad hypotheses. Experiments show that the proposed approach improves the speed of the original chamfer matching up to an order of 45 times, and it is much faster than many state of art techniques while the accuracy is comparable. I further demonstrate the application of the proposed algorithm in providing seamless operation for a robotic bin picking system

Approximate Inference for Determinantal Point Processes

In this thesis we explore a probabilistic model that is well-suited to a variety of subset selection tasks: the determinantal point process (DPP). DPPs were originally developed in the physics community to describe the repulsive interactions of fermions. More recently, they have been applied to machine learning problems such as search diversification and document summarization, which can be cast as subset selection tasks. A challenge, however, is scaling such DPP-based methods to the size of the datasets of interest to this community, and developing approximations for DPP inference tasks whose exact computation is prohibitively expensive. A DPP defines a probability distribution over all subsets of a ground set of items. Consider the inference tasks common to probabilistic models, which include normalizing, marginalizing, conditioning, sampling, estimating the mode, and maximizing likelihood. For DPPs, exactly computing the quantities necessary for the first four of these tasks requires time cubic in the number of items or features of the items. In this thesis, we propose a means of making these four tasks tractable even in the realm where the number of items and the number of features is large. Specifically, we analyze the impact of randomly projecting the features down to a lower-dimensional space and show that the variational distance between the resulting DPP and the original is bounded. In addition to expanding the circumstances in which these first four tasks are tractable, we also tackle the other two tasks, the first of which is known to be NP-hard (with no PTAS) and the second of which is conjectured to be NP-hard. For mode estimation, we build on submodular maximization techniques to develop an algorithm with a multiplicative approximation guarantee. For likelihood maximization, we exploit the generative process associated with DPP sampling to derive an expectation-maximization (EM) algorithm. We experimentally verify the practicality of all the techniques that we develop, testing them on applications such as news and research summarization, political candidate comparison, and product recommendation