Characterizing Demand Graphs for (Fixed-Parameter) Shallow-Light Steiner Network

We consider the Shallow-Light Steiner Network problem from a fixed-parameter perspective. Given a graph G, a distance bound L, and p pairs of vertices (s_1,t_1),...,(s_p,t_p), the objective is to find a minimum-cost subgraph G\u27 such that s_i and t_i have distance at most L in G\u27 (for every i in [p]). Our main result is on the fixed-parameter tractability of this problem for parameter p. We exactly characterize the demand structures that make the problem "easy", and give FPT algorithms for those cases. In all other cases, we show that the problem is W[1]-hard. We also extend our results to handle general edge lengths and costs, precisely characterizing which demands allow for good FPT approximation algorithms and which demands remain W[1]-hard even to approximate

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

Algorithms and Hardness Results for Compressing Graphs with Distance Constraints

Graphs have been widely utilized in network design and other applications. A natural question is, can we keep as few edges of the original graph as possible, but still make sure that the vertices are connected within certain distance constraints. In this thesis, we will consider different versions of graph compression problems, including graph spanners, approximate distance oracles, and Steiner networks. Since these problems are all NP-hard problems, we will mostly focus on designing approximation algorithms and proving inapproximability results

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

Optimizing resource allocation in computational sustainability: Models, algorithms and tools

The 17 Sustainable Development Goals laid out by the United Nations include numerous targets as well as indicators of progress towards sustainable development. Decision-makers tasked with meeting these targets must frequently propose upfront plans or policies made up of many discrete actions, such as choosing a subset of locations where management actions must be taken to maximize the utility of the actions. These types of resource allocation problems involve combinatorial choices and tradeoffs between multiple outcomes of interest, all in the context of complex, dynamic systems and environments. The computational requirements for solving these problems bring together elements of discrete optimization, large-scale spatiotemporal modeling and prediction, and stochastic models. This dissertation leverages network models as a flexible family of computational tools for building prediction and optimization models in three sustainability-related domain areas: 1) minimizing stochastic network cascades in the context of invasive species management; 2) maximizing deterministic demand-weighted pairwise reachability in the context of flood resilient road infrastructure planning; and 3) maximizing vertex-weighted and edge-weighted connectivity in wildlife reserve design. We use spatially explicit network models to capture the underlying system dynamics of interest in each setting, and contribute discrete optimization problem formulations for maximizing sustainability objectives with finite resources. While there is a long history of research on optimizing flows, cascades and connectivity in networks, these decision problems in the emerging field of computational sustainability involve novel objectives, new combinatorial structure, or new types of intervention actions. In particular, we formulate a new type of discrete intervention in stochastic network cascades modeled with multivariate Hawkes processes. In conjunction, we derive an exact optimization approach for the proposed intervention based on closed-form expressions of the objective functions, which is applicable in a broad swath of domains beyond invasive species, such as social networks and disease contagion. We also formulate a new variant of Steiner Forest network design, called the budget-constrained prize-collecting Steiner forest, and prove that this optimization problem possesses a specific combinatorial structure, restricted supermodularity, that allows us to design highly effective algorithms. In each of the domains, the optimization problem is defined over aspects that need to be predicted, hence we also demonstrate improved machine learning approaches for each.Ph.D

Shortest Paths and Steiner Trees in VLSI Routing

Routing is one of the major steps in very-large-scale integration (VLSI) design. Its task is to find disjoint wire connections between sets of points on a chip, subject to numerous constraints. This problem is solved in a two-stage approach, which consists of so-called global and detailed routing steps. For each set of metal components to be connected, global routing reduces the search space by computing corridors in which detailed routing sequentially determines the desired connections as shortest paths. In this thesis, we present new theoretical results on Steiner trees and shortest paths, the two main mathematical concepts in routing. In the practical part, we give computational results of BonnRoute, a VLSI routing tool developed at the Research Institute for Discrete Mathematics at the University of Bonn. Interconnect signal delays are becoming increasingly important in modern chip designs. Therefore, the length of paths or direct delay measures should be taken into account when constructing rectilinear Steiner trees. We consider the problem of finding a rectilinear Steiner minimum tree (RSMT) that --- as a secondary objective --- minimizes a signal delay related objective. Given a source we derive some structural properties of RSMTs for which the weighted sum of path lengths from the source to the other terminals is minimized. Also, we present an exact algorithm for constructing RSMTs with weighted sum of path lengths as secondary objective, and a heuristic for various secondary objectives. Computational results for industrial designs are presented. We further consider the problem of finding a shortest rectilinear Steiner tree in the plane in the presence of rectilinear obstacles. The Steiner tree is allowed to run over obstacles; however, if it intersects an obstacle, then no connected component of the induced subtree must be longer than a given fixed length. This kind of length restriction is motivated by its application in VLSI routing where a large Steiner tree requires the insertion of repeaters which must not be placed on top of obstacles. We show that there are optimal length-restricted Steiner trees with a special structure. In particular, we prove that a certain graph (called augmented Hanan grid) always contains an optimal solution. Based on this structural result, we give an approximation scheme for the special case that all obstacles are of rectangular shape or are represented by at most a constant number of edges. Turning to the shortest paths problem, we present a new generic framework for Dijkstra's algorithm for finding shortest paths in digraphs with non-negative integral edge lengths. Instead of labeling individual vertices, we label subgraphs which partition the given graph. Much better running times can be achieved if the number of involved subgraphs is small compared to the order of the original graph and the shortest path problems restricted to these subgraphs is computationally easy. As an application we consider the VLSI routing problem, where we need to find millions of shortest paths in partial grid graphs with billions of vertices. Here, the algorithm can be applied twice, once in a coarse abstraction (where the labeled subgraphs are rectangles), and once in a detailed model (where the labeled subgraphs are intervals). Using the result of the first algorithm to speed up the second one via goal-oriented techniques leads to considerably reduced running time. We illustrate this with the routing program BonnRoute on leading-edge industrial chips. Finally, we present computational results of BonnRoute obtained on real-world VLSI chips. BonnRoute fulfills all requirements of modern VLSI routing and has been used by IBM and its customers over many years to produce more than one thousand different chips. To demonstrate the strength of BonnRoute as a state-of-the-art industrial routing tool, we show that it performs excellently on all traditional quality measures such as wire length and number of vias, but also on further criteria of equal importance in the every-day work of the designer