Unifying Sparsest Cut, Cluster Deletion, and Modularity Clustering Objectives with Correlation Clustering

Graph clustering, or community detection, is the task of identifying groups of closely related objects in a large network. In this paper we introduce a new community-detection framework called LambdaCC that is based on a specially weighted version of correlation clustering. A key component in our methodology is a clustering resolution parameter, $\lambda$, which implicitly controls the size and structure of clusters formed by our framework. We show that, by increasing this parameter, our objective effectively interpolates between two different strategies in graph clustering: finding a sparse cut and forming dense subgraphs. Our methodology unifies and generalizes a number of other important clustering quality functions including modularity, sparsest cut, and cluster deletion, and places them all within the context of an optimization problem that has been well studied from the perspective of approximation algorithms. Our approach is particularly relevant in the regime of finding dense clusters, as it leads to a 2-approximation for the cluster deletion problem. We use our approach to cluster several graphs, including large collaboration networks and social networks

Algorithmic Solutions for Combinatorial Problems in Resource Management of Manufacturing Environments

This thesis studies the use of heuristic algorithms in a number of combinatorial problems that occur in various resource constrained environments. Such problems occur, for example, in manufacturing, where a restricted number of resources (tools, machines, feeder slots) are needed to perform some operations. Many of these problems turn out to be computationally intractable, and heuristic algorithms are used to provide efficient, yet sub-optimal solutions. The main goal of the present study is to build upon existing methods to create new heuristics that provide improved solutions for some of these problems. All of these problems occur in practice, and one of the motivations of our study was the request for improvements from industrial sources. We approach three different resource constrained problems. The first is the tool switching and loading problem, and occurs especially in the assembly of printed circuit boards. This problem has to be solved when an efficient, yet small primary storage is used to access resources (tools) from a less efficient (but unlimited) secondary storage area. We study various forms of the problem and provide improved heuristics for its solution. Second, the nozzle assignment problem is concerned with selecting a suitable set of vacuum nozzles for the arms of a robotic assembly machine. It turns out that this is a specialized formulation of the MINMAX resource allocation formulation of the apportionment problem and it can be solved efficiently and optimally. We construct an exact algorithm specialized for the nozzle selection and provide a proof of its optimality. Third, the problem of feeder assignment and component tape construction occurs when electronic components are inserted and certain component types cause tape movement delays that can significantly impact the efficiency of printed circuit board assembly. Here, careful selection of component slots in the feeder improves the tape movement speed. We provide a formal proof that this problem is of the same complexity as the turnpike problem (a well studied geometric optimization problem), and provide a heuristic algorithm for this problem.Siirretty Doriast

Dissection with the Fewest Pieces is Hard, Even to Approximate

We prove that it is NP-hard to dissect one simple orthogonal polygon into another using a given number of pieces, as is approximating the fewest pieces to within a factor of 1+1/1080−ε .National Science Foundation (U.S.) (Grant CCF-1217423)National Science Foundation (U.S.) (Grant CCF-1065125)National Science Foundation (U.S.) (Grant CCF-1420692

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

Algorithmic and Combinatorial Results in Selection and Computational Geometry

This dissertation investigates two sets of algorithmic and combinatorial problems. Thefirst part focuses on the selection problem under the pairwise comparison model. For the classic “median of medians” scheme, contrary to the popular belief that smaller group sizes cause superlinear behavior, several new linear time algorithms that utilize small groups are introduced. Then the exact number of comparisons needed for an optimal selection algorithm is studied. In particular, the implications of a long standing conjecture known as Yao’s hypothesis are explored. For the multiparty model, we designed low communication complexity protocols for selecting an exact or an approximate median of data that is distributed among multiple players. In the second part, three computational geometry problems are studied. For the longestspanning tree with neighborhoods, approximation algorithms are provided. For the stretch factor of polygonal chains, upper bounds are proved and almost matching lower bound constructions in \mathbb{R}^2 and higher dimensions are developed. For the piercing number τ and independence number ν of a family of axis-parallel rectangles in the plane, a lower bound construction for ν = 4 that matches Wegner’s conjecture is analyzed. The previous matching construction for ν = 3, due to Wegner himself, dates back to 1968