A Solution Merging Heuristic for the Steiner Problem in Graphs Using Tree Decompositions

Fixed parameter tractable algorithms for bounded treewidth are known to exist for a wide class of graph optimization problems. While most research in this area has been focused on exact algorithms, it is hard to find decompositions of treewidth sufficiently small to make these al- gorithms fast enough for practical use. Consequently, tree decomposition based algorithms have limited applicability to large scale optimization. However, by first reducing the input graph so that a small width tree decomposition can be found, we can harness the power of tree decomposi- tion based techniques in a heuristic algorithm, usable on graphs of much larger treewidth than would be tractable to solve exactly. We propose a solution merging heuristic to the Steiner Tree Problem that applies this idea. Standard local search heuristics provide a natural way to generate subgraphs with lower treewidth than the original instance, and subse- quently we extract an improved solution by solving the instance induced by this subgraph. As such the fixed parameter tractable algorithm be- comes an efficient tool for our solution merging heuristic. For a large class of sparse benchmark instances the algorithm is able to find small width tree decompositions on the union of generated solutions. Subsequently it can often improve on the generated solutions fast

On the design and implementation of broadcast and global combine operations using the postal model

There are a number of models that were proposed in recent years for message passing parallel systems. Examples are the postal model and its generalization the LogP model. In the postal model a parameter λ is used to model the communication latency of the message-passing system. Each node during each round can send a fixed-size message and, simultaneously, receive a message of the same size. Furthermore, a message sent out during round r will incur a latency of hand will arrive at the receiving node at round r + λ - 1. Our goal in this paper is to bridge the gap between the theoretical modeling and the practical implementation. In particular, we investigate a number of practical issues related to the design and implementation of two collective communication operations, namely, the broadcast operation and the global combine operation. Those practical issues include, for example, 1) techniques for measurement of the value of λ on a given machine, 2) creating efficient broadcast algorithms that get the latency hand the number of nodes n as parameters and 3) creating efficient global combine algorithms for parallel machines with λ which is not an integer. We propose solutions that address those practical issues and present results of an experimental study of the new algorithms on the Intel Delta machine. Our main conclusion is that the postal model can help in performance prediction and tuning, for example, a properly tuned broadcast improves the known implementation by more than 20%

A second derivative SQP method: local convergence

In [19], we gave global convergence results for a second-derivative SQP method for minimizing the exact ℓ1-merit function for a fixed value of the penalty parameter. To establish this result, we used the properties of the so-called Cauchy step, which was itself computed from the so-called predictor step. In addition, we allowed for the computation of a variety of (optional) SQP steps that were intended to improve the efficiency of the algorithm. \ud \ud Although we established global convergence of the algorithm, we did not discuss certain aspects that are critical when developing software capable of solving general optimization problems. In particular, we must have strategies for updating the penalty parameter and better techniques for defining the positive-definite matrix Bk used in computing the predictor step. In this paper we address both of these issues. We consider two techniques for defining the positive-definite matrix Bk—a simple diagonal approximation and a more sophisticated limited-memory BFGS update. We also analyze a strategy for updating the penalty paramter based on approximately minimizing the ℓ1-penalty function over a sequence of increasing values of the penalty parameter.\ud \ud Algorithms based on exact penalty functions have certain desirable properties. To be practical, however, these algorithms must be guaranteed to avoid the so-called Maratos effect. We show that a nonmonotone varient of our algorithm avoids this phenomenon and, therefore, results in asymptotically superlinear local convergence; this is verified by preliminary numerical results on the Hock and Shittkowski test set

Theoretical and algorithmic approaches to field-programmable gate array partitioning

Many practical problems dealing with the design of Very Large Scale Integrated (VLSI) circuits can be modeled as graphs in which vertices represent components of the circuit and edges represent a relationship between these components. When expressed as graphs, these problems can then often be solved using graph theoretic methods. Unfortunately, many such problems are NP-complete, hence no practical exact solutions are known to exist. In this dissertation, we study NP-complete problems taken from the realm of partitioning for Field-Programmable Gate Arrays (FPGAs). We adopt a two-pronged approach of theory and practice, developing practical heuristics driven by theoretical study. The theoretical approach is motivated by well-quasi-order (WQO) theory, which can be used to show, among other things, that when some hard problems have fixed parameters, polynomial-time solutions exist. This is of significance in the area of FPGA partitioning, in which practical problems are often characterized by fixed parameter instances. WQO techniques are not generally practical, however, and we develop new methods to solve several important problems in VLSI that are not even amenable to WQO techniques. We begin with a representative partitioning problem, Min Degree Graph Partition (MDGP), the fixed-parameter version of which is closed under the immersion order. \Ve show that the obstruction set ( set of immersion minimal elements) for this problem is computable; we prove both upper and lower bounds on the obstruction set size; and we completely characterize all fixed-parameter MDGP simple tree obstructions. WQO theory tells us only that fixed-parameter MDGP is solvable in (high-degree) polynomial time. We attack the problem using what we refer to as kd-candidate subsets, culminating in linear-time decision and search algorithms. The kd-candidate subset method also paves the way for an efficient heuristic for the FPGA Minimization problem. We then broaden our scope to incorporate delay minimization into FPGA partitioning. We develop, analyze and test a novel method called critical path compression, inspired in part by compiler optimization techniques. We then look at a variety of generalizations of MDGP. Some of these problems are not immersion closed; others are not even defined in a way that WQO theory applies. However, almost all of them are efficiently solvable via the kd-candidate subset approach. Interspersed in these results are many refinements of what is known about the complexity of these problems. We also discuss a few other solution strategies, and present many open problems

Topics in Graph Algorithms: Structural Results and Algorithmic Techniques, with Applications

Coping with computational intractability has inspired the development of a variety of algorithmic techniques. The main challenge has usually been the design of polynomial time algorithms for NP-complete problems in a way that guarantees some, often worst-case, satisfactory performance when compared to exact (optimal) solutions. We mainly study some emergent techniques that help to bridge the gap between computational intractability and practicality. We present results that lead to better exact and approximation algorithms and better implementations. The problems considered in this dissertation share much in common structurally, and have applications in several scientific domains, including circuit design, network reliability, and bioinformatics. We begin by considering the relationship between graph coloring and the immersion order, a well-quasi-order defined on the set of finite graphs. We establish several (structural) results and discuss their potential algorithmic consequences. We discuss graph metrics such as treewidth and pathwidth. Treewidth is well studied, mainly because many problems that are NP-hard in general have polynomial time algorithms when restricted to graphs of bounded treewidth. Pathwidth has many applications ranging from circuit layout to natural language processing. We present a linear time algorithm to approximate the pathwidth of planar graphs that have a fixed disk dimension. We consider the face cover problem, which has potential applications in facilities location and logistics. Being fixed-parameter tractable, we develop an algorithm that solves it in time O(5k + n2) where k is the input parameter. This is a notable improvement over the previous best known algorithm, which runs in O(8kn). In addition to the structural and algorithmic results, this text tries to illustrate the practicality of fixed-parameter algorithms. This is achieved by implementing some algorithms for the vertex cover problem, and conducting experiments on real data sets. Our experiments advocate the viewpoint that, for many practical purposes, exact solutions of some NP-complete problems are affordable

Parameter Synthesis for Markov Models

Markov chain analysis is a key technique in reliability engineering. A practical obstacle is that all probabilities in Markov models need to be known. However, system quantities such as failure rates or packet loss ratios, etc. are often not---or only partially---known. This motivates considering parametric models with transitions labeled with functions over parameters. Whereas traditional Markov chain analysis evaluates a reliability metric for a single, fixed set of probabilities, analysing parametric Markov models focuses on synthesising parameter values that establish a given reliability or performance specification $\varphi$. Examples are: what component failure rates ensure the probability of a system breakdown to be below 0.00000001?, or which failure rates maximise reliability? This paper presents various analysis algorithms for parametric Markov chains and Markov decision processes. We focus on three problems: (a) do all parameter values within a given region satisfy $\varphi$?, (b) which regions satisfy $\varphi$ and which ones do not?, and (c) an approximate version of (b) focusing on covering a large fraction of all possible parameter values. We give a detailed account of the various algorithms, present a software tool realising these techniques, and report on an extensive experimental evaluation on benchmarks that span a wide range of applications.Comment: 38 page

Analysis of Linkage-Friendly Genetic Algorithms

Evolutionary algorithms (EAs) are stochastic population-based algorithms inspired by the natural processes of selection, mutation, and recombination. EAs are often employed as optimum seeking techniques. A formal framework for EAs is proposed, in which evolutionary operators are viewed as mappings from parameter spaces to spaces of random functions. Formal definitions within this framework capture the distinguishing characteristics of the classes of recombination, mutation, and selection operators. EAs which use strictly invariant selection operators and order invariant representation schemes comprise the class of linkage-friendly genetic algorithms (lfGAs). Fast messy genetic algorithms (fmGAs) are lfGAs which use binary tournament selection (BTS) with thresholding, periodic filtering of a fixed number of randomly selected genes from each individual, and generalized single-point crossover. Probabilistic variants of thresholding and filtering are proposed. EAs using the probabilistic operators are generalized fmGAs (gfmGAs). A dynamical systems model of lfGAs is developed which permits prediction of expected effectiveness. BTS with probabilistic thresholding is modeled at various levels of abstraction as a Markov chain. Transitions at the most detailed level involve decisions between classes of individuals. The probability of correct decision making is related to appropriate maximal order statistics, the distributions of which are obtained. Existing filtering models are extended to include probabilistic individual lengths. Sensitivity of lfGA effectiveness to exogenous parameters limits practical applications. The lfGA parameter selection problem is formally posed as a constrained optimization problem in which the cost functional is related to expected effectiveness. Kuhn-Tucker conditions for the optimality of gfmGA parameters are derived

Cramer-Rao Bounds for Joint RSS/DoA-Based Primary-User Localization in Cognitive Radio Networks

Knowledge about the location of licensed primary-users (PU) could enable several key features in cognitive radio (CR) networks including improved spatio-temporal sensing, intelligent location-aware routing, as well as aiding spectrum policy enforcement. In this paper we consider the achievable accuracy of PU localization algorithms that jointly utilize received-signal-strength (RSS) and direction-of-arrival (DoA) measurements by evaluating the Cramer-Rao Bound (CRB). Previous works evaluate the CRB for RSS-only and DoA-only localization algorithms separately and assume DoA estimation error variance is a fixed constant or rather independent of RSS. We derive the CRB for joint RSS/DoA-based PU localization algorithms based on the mathematical model of DoA estimation error variance as a function of RSS, for a given CR placement. The bound is compared with practical localization algorithms and the impact of several key parameters, such as number of nodes, number of antennas and samples, channel shadowing variance and correlation distance, on the achievable accuracy are thoroughly analyzed and discussed. We also derive the closed-form asymptotic CRB for uniform random CR placement, and perform theoretical and numerical studies on the required number of CRs such that the asymptotic CRB tightly approximates the numerical integration of the CRB for a given placement.Comment: 20 pages, 11 figures, 1 table, submitted to IEEE Transactions on Wireless Communication