Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

Topics in Packing and Scheduling

Packing and scheduling models include some of the most fundamental problems in operations research and computer science. These broad classes include a wide range of models with applications including logistics, production planning, wireless network design, circuit design, and cloud computing, to name a few. In this thesis we study three such models: dynamic node packing, interval scheduling with economies of scale, and temporal bin packing with half-capacity jobs; each extends on a well-known problem in packing and scheduling. While the problems are generally distinct, this research was broadly inspired by applications to cloud computing. Specifically, this thesis is motivated by problems cloud service providers face when servicing requests for virtual machines. In Chapter 2, we propose a dynamic version of the node packing problem. In this model, instead of being given the edges upfront, we model them as Bernoulli random variables. At each step, the decision maker selects an available node and then observes edges adjacent to this node. The goal is a policy that maximizes the expected value of the resulting packing. We model the problem as a Markov decision problem and conduct a polyhedral study of the problem's achievable probabilities polytope. We develop a variety of valid inequalities based on paths, cycles, and cliques. In Chapter 3, we study interval scheduling problems exhibiting economies of scale. An instance is given by a set of interval jobs and a cost function. Specifically, we focus on the max-weight function and non-negative, non-decreasing concave functions of total schedule weight. The goal is a partition of the jobs minimizing the total cost with the constraint that jobs within the same schedule cannot overlap. We propose a set covering formulation and a column generation algorithm to solve its linear relaxation, providing efficient pricing algorithms for the studied cases. To obtain integer solutions, we extend the column generation approach using branch-and-price. In Chapter 4, we study a different model with interval jobs. In this problem, interval jobs are partitioned into bins such that at most two jobs in a bin overlap at once. The decision maker is tasked with minimizing the time-average number of bins required to pack all jobs. We call this problem temporal bin packing with half-capacity jobs; it is a special case of the general temporal bin packing problem with bounded parallelism. We study the worst-case performance of a well-known static lower bound, and, motivated by this analysis, we introduce a novel lower bound and integer programming formulation based on formulating the problem as a series of matching problems. We provide theoretical guarantees on the relative strengths of the static bound, the matching-based bound, and various linear programming bounds.Ph.D

Capacitated max-Batching with Interval Graph Compatibilities

We consider the problem of partitioning interval graphs into cliques of bounded size. Each interval has a weight, and the cost of a clique is the maximum weight of any interval in the clique. This natural graph problem can be interpreted as a batch scheduling problem. Solving an open question from [7, 4, 5], we show NP-hardness, even if the bound on the clique sizes is constant. Moreover, we give a PTAS based on a novel dynamic programming technique for this case.

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