Embedding Graphs under Centrality Constraints for Network Visualization

Visual rendering of graphs is a key task in the mapping of complex network data. Although most graph drawing algorithms emphasize aesthetic appeal, certain applications such as travel-time maps place more importance on visualization of structural network properties. The present paper advocates two graph embedding approaches with centrality considerations to comply with node hierarchy. The problem is formulated first as one of constrained multi-dimensional scaling (MDS), and it is solved via block coordinate descent iterations with successive approximations and guaranteed convergence to a KKT point. In addition, a regularization term enforcing graph smoothness is incorporated with the goal of reducing edge crossings. A second approach leverages the locally-linear embedding (LLE) algorithm which assumes that the graph encodes data sampled from a low-dimensional manifold. Closed-form solutions to the resulting centrality-constrained optimization problems are determined yielding meaningful embeddings. Experimental results demonstrate the efficacy of both approaches, especially for visualizing large networks on the order of thousands of nodes.Comment: Submitted to IEEE Transactions on Visualization and Computer Graphic

Service in Your Neighborhood: Fairness in Center Location

When selecting locations for a set of centers, standard clustering algorithms may place unfair burden on some individuals and neighborhoods. We formulate a fairness concept that takes local population densities into account. In particular, given k centers to locate and a population of size n, we define the "neighborhood radius" of an individual i as the minimum radius of a ball centered at i that contains at least n/k individuals. Our objective is to ensure that each individual has a center that is within at most a small constant factor of her neighborhood radius. We present several theoretical results: We show that optimizing this factor is NP-hard; we give an approximation algorithm that guarantees a factor of at most 2 in all metric spaces; and we prove matching lower bounds in some metric spaces. We apply a variant of this algorithm to real-world address data, showing that it is quite different from standard clustering algorithms and outperforms them on our objective function and balances the load between centers more evenly

Proportionally Representative Clustering

In recent years, there has been a surge in effort to formalize notions of fairness in machine learning. We focus on clustering -- one of the fundamental tasks in unsupervised machine learning. We propose a new axiom ``proportional representation fairness'' (PRF) that is designed for clustering problems where the selection of centroids reflects the distribution of data points and how tightly they are clustered together. Our fairness concept is not satisfied by existing fair clustering algorithms. We design efficient algorithms to achieve PRF both for unconstrained and discrete clustering problems. Our algorithm for the unconstrained setting is also the first known polynomial-time approximation algorithm for the well-studied Proportional Fairness (PF) axiom (Chen, Fain, Lyu, and Munagala, ICML, 2019). Our algorithm for the discrete setting also matches the best known approximation factor for PF.Comment: Revised version includes a new author (Jeremy Vollen) and new results: Our algorithm for the unconstrained setting is also the first known polynomial-time approximation algorithm for the well-studied Proportional Fairness (PF) axiom (Chen, Fain, Lyu, and Munagala, ICML, 2019). Our algorithm for the discrete setting also matches the best known approximation factor for P

Mean asymptotic behaviour of radix-rational sequences and dilation equations (Extended version)

The generating series of a radix-rational sequence is a rational formal power series from formal language theory viewed through a fixed radix numeration system. For each radix-rational sequence with complex values we provide an asymptotic expansion for the sequence of its Ces\`aro means. The precision of the asymptotic expansion depends on the joint spectral radius of the linear representation of the sequence; the coefficients are obtained through some dilation equations. The proofs are based on elementary linear algebra

A General Class of Heuristics for Minimum Weight Perfect Matching and Fast Special Cases with Doubly and Triply Logarithmic Errors

We give a class of heuristic algorithms for minimum weight perfect matching on a complete edgeweighted graph K(V) satisfying the triangle inequality, where V is a set of an even number, n, of vertices.This class is a generalization of the Onethird heuristics, the hypergreedy heuristic, and it possibly employs any given exact or approximate perfect matching algorithm as an auxiliary heuristic to an appropriate subgraph of K(V)

An examination of heuristic algorithms for the travelling salesman problem

The role of heuristics in combinatorial optimization is discussed. Published heuristics for the Travelling Salesman Problem (TSP) were reviewed and morphological boxes were used to develop new heuristics for the TSP. New and published heuristics were programmed for symmetric TSPs where the triangle inequality holds, and were tested on micro computer. The best of the quickest heuristics was the furthest insertion heuristic, finding tours 3 to 9% above the best known solutions (2 minutes for 100 nodes). Better results were found by longer running heuristics, e.g. the cheapest angle heuristic (CCAO), 0-6% above best (80 minutes for 100 nodes). The savings heuristic found the best results overall, but took more than 2 hours to complete. Of the new heuristics, the MST path algorithm at times improved on the results of the furthest insertion heuristic while taking the same time as the CCAO. The study indicated that there is little likelihood of improving on present methods unless a fundamental new approach is discovered. Finally a case study using TSP heuristics to aid the planning of grid surveys was described

The Plant Propagation Algorithm for Discrete Optimisation

The thesis is concerned with novel Nature-Inspired heuristics for the so called NP-hard problems of optimisation. A particular algorithm which has been recently introduced and shown to be effective in continuous optimisation is the Plant Propagation Algorithm or PPA. Here, we intend to extend it to cope with combinatorial optimisation. In order to show that our extension is viable and effective, we consider three types of problems which are good representatives of the whole topic. These are the Travelling Salesman Problem or TSP, the Knapsack Problem or KP and the scheduling problem of Berth Allocation as arises in container ports or BAP. Because PPA is a population-based search heuristic, we devote a chapter to the important issue of generating good and yet computationally relatively light initial populations of solutions to kick start the search process. In the case of the TSP we revisit and extend the Strip Algorithm (SA). We introduce the 2-Part SA and show that it is better than the classical SA. We also introduce new variants such as the Adaptive SA and the Spiral SA which cope with clustered cities and instances with cities concentrated around the center of the unit square, respectively. In the case of KP we adapt the Roulette Wheel selection approach to generate solutions to start with PPA. And in the case of BAP, we introduce a number of simple heuristics which consider a schedule as a flat box with one side being the processing time and the other the position of vessels on the wharf. The heuristics try to generate schedules by avoiding overlap as much as possible. All approaches and algorithms are implemented and tested against well established algorithms. The results are recorded and discussed extensively. The thesis ends with a conclusion and ideas for further research