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Graph Algorithms for Planning and Partitioning

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Graph Algorithms for Planning and Partitioning Graph Algorithms for Planning and Partitioning Shuchi Chawla CMU-CS-05-184 September 30, 2005 School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Thesis Committee: Avrim Blum, Chair Anupam Gupta R. Ravi Moses Charikar, Princeton University Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Copyright c 2005 by Shuchi Chawla This research was sponsored by the National Science Foundation under grant nos. IIS-0121678, CCR-0105488, CCR-0122581, CCR-9732705, CCR-0085982 and through a generous fellowship from the IBM Corporation. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity. Keywords: approximation algorithms, combinatorial optimization, robot navigation, vehicle routing, graph partitioning, hardness of approximation i Abstract In this thesis, we present new approximation algorithms as well as hardness of approximation results for several planning and partitioning problems. In planning problems, one is typically given a set of locations to visit, along with timing constraints, such as deadlines for visiting them; The goal is to visit a large number of locations as efficiently as possible. We give the first approximation al- gorithms for problems such as ORIENTEERING, DEADLINES-TSP, and TIME-WINDOWS-TSP, as well as results for planning in stochastic graphs (Markov decision processes). The goal in partition- ing problems is to partition a set of objects into clusters while satisfying “split” or “combine” con- straints on pairs of objects. We consider three kinds of partitioning problems, viz. CORRELATION- CLUSTERING, SPARSEST-CUT, and MULTICUT. We give approximation algorithms for the first two, and improved hardness of approximation results for SPARSEST-CUT and MULTICUT. ii To my grandmothers, (late) Smt. Sita Devi Chawla and Smt. Satyawati Malhotra iii iv Acknowledgements I am grateful to my advisor, Prof. Avrim Blum, for being understanding and patient, for being a great source of information and insight, and for giving me the freedom to work at my own pace on topics of my choice. Thanks also to Dr. Cynthia Dwork and Prof. Anupam Gupta for being great mentors and role models. Prof. Moses Charikar and Prof. R. Ravi helped shape a large part of this thesis through numerous discussions; I am grateful to them for their help and advice during the final year of my graduate study. This thesis would not have been possible without the cooperation of my numerous co-authors — thanks to you all! An enormous thanks to Aditya and Luis for maintaining my sanity (and sometimes making me lose it) during graduate school, and for being the best friends and officemates one could ask for. Finally, I would like to thank my parents and sister for their unconditional love and support through all my endeavors. v vi Contents 1 Introduction 1 1.1 Path-planning problems . 2 1.1.1 Overview of our results . 5 1.1.2 Planning under uncertainty . 6 1.2 Graph Partitioning . 8 1.2.1 Overview of our results . 10 2 Planning with Time Constraints 11 2.1 Introduction . 11 2.1.1 Related work . 12 2.1.2 Notation and definitions . 14 2.1.3 Summary of results . 16 2.2 MIN-EXCESS-PATH . 17 2.2.1 A segment partition . 18 2.2.2 A dynamic program . 20 2.2.3 An improved analysis . 21 2.3 ORIENTEERING . 22 2.3.1 A 3-approximation . 23 2.3.2 Difficulties in obtaining a 2 + δ approximation . 24 2.4 Incorporating timing constraints . 24 2.4.1 Using ORIENTEERING to approximate DEADLINES-TSP . 25 2.4.2 Minimal vertices . 26 2.4.3 A partition into sets . 27 2.4.4 Finding the best collection . 29 2.4.5 From deadlines to time-windows . 30 2.5 A bicriteria approximation for the TIME-WINDOWS-TSP . 31 2.5.1 The small margin case . 32 2.5.2 The large margin case . 34 2.5.3 The general case . 36 2.6 Extensions . 38 vii viii CONTENTS 2.6.1 Max-Value Tree and Max-Value Cycle . 38 2.6.2 Multiple-Path ORIENTEERING . 38 2.6.3 ORIENTEERING on special metrics . 39 2.7 Hardness of approximation . 40 2.8 Concluding remarks . 41 3 Planning under Uncertainty 43 3.1 Introduction . 43 3.1.1 Markov decision processes . 44 3.1.2 Strategies and policies . 44 3.1.3 Objectives . 45 3.1.4 Stochastic optimization . 47 3.1.5 Our Results . 48 3.2 Maximum Discounted-Reward Path . 48 3.3 Stochastic TSP . 51 3.3.1 The complexity of computing the optimal strategy . 52 3.3.2 The adaptivity gap . 53 3.3.3 Approximation via static policies . 56 3.3.4 Approximation via dynamic strategies . 58 3.4 Concluding remarks . 59 4 Correlation Clustering 61 4.1 Introduction . 61 4.1.1 Related and subsequent work . 63 4.2 Notation and definitions . 64 4.3 NP-completeness . 65 4.4 A constant factor approximation for minimizing disagreements . 66 4.5 A PTAS for maximizing agreements . 73 4.5.1 A PTAS doubly-exponential in 1/ . 74 4.5.2 A singly-exponential PTAS . 75 4.6 Random noise . 81 4.7 Extensions . 82 4.8 Concluding remarks . 85 5 Min-Ratio Partitioning 87 5.1 Introduction . 87 5.1.1 SPARSEST-CUT and low-distortion metric embeddings . 88 5.1.2 The Unique Games Conjecture . 89 5.2 Our results . 90 CONTENTS ix 5.3 Related and subsequent work . 92 5.4 An improved approximation for SPARSEST-CUT . 94 5.4.1 A high-level overview . 94 5.4.2 Notation and definitions . 95 5.4.3 Creating decomposition suites . 97 5.4.4 Obtaining decomposition bundles: weighting and watching . 101 5.4.5 Embedding via decomposition bundles . 103 5.5 Hardness of approximating SPARSEST-CUT and MULTICUT . 108 5.5.1 Preliminaries . 108 5.5.2 Hardness of bicriteria approximation for MULTICUT . 109 5.5.3 Hardness of approximating SPARSEST-CUT . 114 5.5.4 Hardness of approximating MIN-2CNF DELETION . 116 ≡ 5.5.5 Limitations of our approach . 117 5.6 Concluding remarks . 117 Bibliography 118 x CONTENTS List of Figures 1.1 Approximation factors and references for a few path-planning problems. Prior re- sults are accompanied by references. 6 2.1 Segment partition of a path in graph G . 19 2.2 Algorithm P2P— 3-approximation for s t ORIENTEERING . 23 − 2.3 A bad example for Orienteering. The distance between each consecutive pair of points is 1. 24 2.4 An example illustrating the construction of the rectangles (x; y; z). 27 R 2.5 Algorithm DTSP—3 log n-approximation for DEADLINES-TSP . 30 2.6 Division of set V into segments Sj. Note that segments Si, Si+3 and Si+6 are disjoint along the time axis. 32 2.7 Algorithm —Bicriteria approximation for the small margin case . 33 A 2.8 Algorithm —Approximation for the large margin case . 35 B 2.9 Algorithm —Bicriteria approximation for the general case . 37 C 3.1 A (6:75 + δ)-approximation for DISCOUNTED-REWARD-TSP . 50 3.2 Exact iterative PSPACE algorithm ITER for STOCHASTIC-TSP . 53 4.1 Construction of a triangle packing for Lemma 4.5 . 68 5.1 Projection and Bucketing . 98 5.2 The distribution of projected edge lengths in the proof of Lemma 5.9. If y falls into a light-shaded interval, the pair (x; y) is split. 99 xi xii LIST OF FIGURES Chapter 1 Introduction We study two classes of optimization problems on graphs, namely, planning and partitioning, and present improved approximations and hardness of approximation results for them. Broadly, the goal in these problems is to find structure in a set of objects, satisfying some constraints, such that the value of the structure is maximized (or its cost is minimized). In planning problems, the objects are tasks to be performed, and the structure is an ordering of these tasks satisfying scheduling constraints; The goal is to perform a large subset of the tasks in the most efficient manner. In partitioning problems, the structure is a grouping of the objects into classes based on their similarity or closeness. Planning and partitioning are fundamental combinatorial problems and capture a wide-variety of natural optimization problems; Examples arise in transportation problems, supply chain man- agement, document classification in machine learning, and reconstructing phylogenies in computa- tional genomics. Unfortunately most of these problems are NP-hard. Therefore, a natural approach to solving them is to look for approximate solutions that can be computed in polynomial time. In this thesis, we study the approximability of several partitioning and planning problems. We provide the first approximation algorithms for some of the problems, and improve upon previously known approximations for others. For some of these problems, we provide hardness of approxima- tion results showing that these problems are NP-hard to approximate better than a certain factor. We begin with some basic definitions. Approximation algorithms and hardness of approximation As mentioned earlier, most of.
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