Solving operations research problems with approximation algorithms has been an important topic since approximation algorithm can provide near-optimal solutions to NP-hard problems while achieving computational efficiency. In this thesis, we consider two different problems in the field of optimal control and scheduling theory respectively and develop efficient approximation algorithms for those problems with performance guarantee. Chapter 2 presents approximation algorithms for solving the optimal execution problem for demand-response contract in electricity markets. Demand side participation is essential for achieving real-time energy balance in today's electricity grid. Demand-response contracts, where an electric utility company buys options from consumers to reduce their load in the future, are an important tool to increase demand-side participation. In this chapter, we consider the operational problem of optimally exercising the available contracts over the planning horizon such that the total cost to satisfy the demand is minimized. In particular, we consider the objective of minimizing the sum of the expected ℓ_β-norm of the load deviations from given thresholds and the contract execution costs over the planning horizon. For β=∞, this reduces to minimizing the expected peak load. The peak load provides a good proxy to the total cost of the utility as spikes in electricity prices are observed only in peak load periods. We present a data driven near-optimal algorithm for the contract execution problem. Our algorithm is a sample average approximation (SAA) based dynamic program over a multi-period planning horizon. We provide a sample complexity bound on the number of demand samples required to compute a (1+ε)-approximate policy for any ε>0. Our SAA algorithm is quite general and we show that it can be adapted to quite general demand models including Markovian demands and objective functions. For the special case where the demand in each period is i.i.d., we show that a static solution is optimal for the dynamic problem. We also conduct a numerical study to compare the performance of our SAA based DP algorithm. Our numerical experiments show that we can achieve a (1+ε)-approximation in significantly smaller numbers of samples than what is implied by the theoretical bounds. Moreover, the structure of the approximate policy also shows that it can be well approximated by a simple affine function of the state. In Chapter 3, we study the NP-hard coflow scheduling problem and develop a polynomial-time approximation algorithm for the problem with constant approximation ratio. Communications in datacenter jobs (such as the shuffle operations in MapReduce applications) often involve many parallel flows, which may be processed simultaneously. This highly parallel structure presents new scheduling challenges in optimizing job-level performance objectives in data centers. Chowdhury and Stoica [13] introduced the coflow abstraction to capture these communication patterns, and recently Chowdhury et al. [15] developed effective heuristics to schedule coflows. In this chapter, we consider the problem of efficiently scheduling coflows so as to minimize the total weighted completion time, which has been shown to be strongly NP-hard [15]. Our main result is the first polynomial-time deterministic approximation algorithm for this problem, with an approximation ratio of $64/3$, and a randomized version of the algorithm, with a ratio of 8+16sqrt{2}/3. Our results use techniques from both combinatorial scheduling and matching theory, and rely on a clever grouping of coflows. In Chapter 4, we carry out a comprehensive experimental analysis on a Facebook trace and extensive simulated instances to evaluate the practical performance of several algorithms for coflow scheduling, including our approximation algorithms developed in Chapter 3. Our experiments suggest that simple algorithms provide effective approximations of the optimal, and that the performance of the approximation algorithm of Chapter 3 is relatively robust, near optimal, and always among the best compared with the other algorithms, in both the offline and online settings.

This thesis involves the study of a range of geometric covering and piercing problems, where the unifying thread is approximation using disks. While some of the problems addressed in this work are solved exactly with polynomial time algorithms, many problems are shown to be at least NP-hard. For the latter, approximation algorithms are the best that we can do in polynomial time assuming that P is not equal to NP. One of the best known problems involving unit disks is the Discrete Unit Disk Cover (DUDC) problem, in which the input consists of a set of points P and a set of unit disks in the plane D, and the objective is to compute a subset of the disks of minimum cardinality which covers all of the points. Another perspective on the problem is to consider the centre points (denoted Q) of the disks D as an approximating set of points for P. An optimal solution to DUDC provides a minimal cardinality subset Q*, a subset of Q, so that each point in P is within unit distance of a point in Q*. In order to approximate the general DUDC problem, we also examine several restricted variants. In the Line-Separable Discrete Unit Disk Cover (LSDUDC) problem, P and Q are separated by a line in the plane. We write that l^- is the half-plane defined by l containing P, and l^+ is the half-plane containing Q. LSDUDC may be solved exactly in O(m^2n) time using a greedy algorithm. We augment this result by describing a 2-approximate solution for the Assisted LSDUDC problem, where the union of all disks centred in l^+ covers all points in P, but we consider using disks centred in l^- as well to try to improve the solution. Next, we describe the Within-Strip Discrete Unit Disk Cover (WSDUDC) problem, where P and Q are confined to a strip of the plane of height h. We show that this problem is NP-complete, and we provide a range of approximation algorithms for the problem with trade-offs between the approximation factor and running time. We outline approximation algorithms for the general DUDC problem which make use of the algorithms for LSDUDC and WSDUDC. These results provide the fastest known approximation algorithms for DUDC. As with the WSDUDC results, we present a set of algorithms in which better approximation factors may be had at the expense of greater running time, ranging from a 15-approximate algorithm which runs in O(mn + m log m + n log n) time to a 18-approximate algorithm which runs in O(m^6n+n log n) time. The next problems that we study are Hausdorff Core problems. These problems accept an input polygon P, and we seek a convex polygon Q which is fully contained in P and minimizes the Hausdorff distance between P and Q. Interestingly, we show that this problem may be reduced to that of computing the minimum radius of disk, call it k_opt, so that a convex polygon Q contained in P intersects all disks of radius k_opt centred on the vertices of P. We begin by describing a polynomial time algorithm for the simple case where P has only a single reflex vertex. On general polygons, we provide a parameterized algorithm which performs a parametric search on the possible values of k_opt. The solution to the decision version of the problem, i.e. determining whether there exists a Hausdorff Core for P given k_opt, requires some novel insights. We also describe an FPTAS for the decision version of the Hausdorff Core problem. Finally, we study Generalized Minimum Spanning Tree (GMST) problems, where the input consists of imprecise vertices, and the objective is to select a single point from each imprecise vertex in order to optimize the weight of the MST over the points. In keeping with one of the themes of the thesis, we begin by using disks as the imprecise vertices. We show that the minimization and maximization versions of this problem are NP-hard, and we describe some parameterized and approximation algorithms. Finally, we look at the case where the imprecise vertices consist of just two vertices each, and we show that the minimization version of the problem (which we call 2-GMST) remains NP-hard, even in the plane. We also provide an algorithm to solve the 2-GMST problem exactly if the combinatorial structure of the optimal solution is known. We identify a number of open problems in this thesis that are worthy of further study. Among them: Is the Assisted LSDUDC problem NP-complete? Can the WSDUDC results be used to obtain an improved PTAS for DUDC? Are there classes of polygons for which the determination of the Hausdorff Core is easy? Is there a PTAS for the maximum weight GMST problem on (unit) disks? Is there a combinatorial approximation algorithm for the 2-GMST problem (particularly with an approximation factor under 4)?

The capacitated facility location problem is a well known problem in combinatorial optimization and operations research. In it, we are given a set of clients and a set of possible facility locations. Each client has a certain demand that needs to be satisfied from open facilities, without exceeding their capacity. Whenever we open a facility we incur in a corresponding opening cost. Whenever demand is served, we incur in an assignment cost; depending on the distance the demand travels. The goal is to open a set of facilities that satisfy all demands while minimizing the total opening and assignment costs. In this thesis, we present two novel LP-based approximation algorithms for the capacitated facility location problem. The first algorithm is based on LP-rounding techniques, and is designed for the special case of the capacitated facility location problem where capacities are uniform and assignment costs are given by a tree metric. The second algorithm follows a primal-dual approach, and works for the general case. For both algorithms, we obtain an approximation guarantee that is linear on the size of the problem. To the best of our knowledge, there are no LP-based algorithms known, for the type of instances that we focus on, that achieve a better performance.

No description available.

No description available.

In this work, we study the problem of clearing contamination spreading through a large network where we model the problem as a graph searching game. The problem can be summarized as constructing a search strategy that will leave the graph clear of any contamination at the end of the searching process in as few steps as possible. We show that this problem is NP-hard even on directed acyclic graphs and provide an efficient approximation algorithm. We experimentally observe the performance of our approximation algorithm in relation to the lower bound on several large online networks including Slashdot, Epinions and Twitter. The experiments reveal that in most cases our algorithm performs near optimally. / Graduate

The capacitated facility location problem is a well known problem in combinatorial optimization and operations research. In it, we are given a set of clients and a set of possible facility locations. Each client has a certain demand that needs to be satisfied from open facilities, without exceeding their capacity. Whenever we open a facility we incur in a corresponding opening cost. Whenever demand is served, we incur in an assignment cost; depending on the distance the demand travels. The goal is to open a set of facilities that satisfy all demands while minimizing the total opening and assignment costs. In this thesis, we present two novel LP-based approximation algorithms for the capacitated facility location problem. The first algorithm is based on LP-rounding techniques, and is designed for the special case of the capacitated facility location problem where capacities are uniform and assignment costs are given by a tree metric. The second algorithm follows a primal-dual approach, and works for the general case. For both algorithms, we obtain an approximation guarantee that is linear on the size of the problem. To the best of our knowledge, there are no LP-based algorithms known, for the type of instances that we focus on, that achieve a better performance.

This thesis involves the study of a range of geometric covering and piercing problems, where the unifying thread is approximation using disks. While some of the problems addressed in this work are solved exactly with polynomial time algorithms, many problems are shown to be at least NP-hard. For the latter, approximation algorithms are the best that we can do in polynomial time assuming that P is not equal to NP. One of the best known problems involving unit disks is the Discrete Unit Disk Cover (DUDC) problem, in which the input consists of a set of points P and a set of unit disks in the plane D, and the objective is to compute a subset of the disks of minimum cardinality which covers all of the points. Another perspective on the problem is to consider the centre points (denoted Q) of the disks D as an approximating set of points for P. An optimal solution to DUDC provides a minimal cardinality subset Q*, a subset of Q, so that each point in P is within unit distance of a point in Q*. In order to approximate the general DUDC problem, we also examine several restricted variants. In the Line-Separable Discrete Unit Disk Cover (LSDUDC) problem, P and Q are separated by a line in the plane. We write that l^- is the half-plane defined by l containing P, and l^+ is the half-plane containing Q. LSDUDC may be solved exactly in O(m^2n) time using a greedy algorithm. We augment this result by describing a 2-approximate solution for the Assisted LSDUDC problem, where the union of all disks centred in l^+ covers all points in P, but we consider using disks centred in l^- as well to try to improve the solution. Next, we describe the Within-Strip Discrete Unit Disk Cover (WSDUDC) problem, where P and Q are confined to a strip of the plane of height h. We show that this problem is NP-complete, and we provide a range of approximation algorithms for the problem with trade-offs between the approximation factor and running time. We outline approximation algorithms for the general DUDC problem which make use of the algorithms for LSDUDC and WSDUDC. These results provide the fastest known approximation algorithms for DUDC. As with the WSDUDC results, we present a set of algorithms in which better approximation factors may be had at the expense of greater running time, ranging from a 15-approximate algorithm which runs in O(mn + m log m + n log n) time to a 18-approximate algorithm which runs in O(m^6n+n log n) time. The next problems that we study are Hausdorff Core problems. These problems accept an input polygon P, and we seek a convex polygon Q which is fully contained in P and minimizes the Hausdorff distance between P and Q. Interestingly, we show that this problem may be reduced to that of computing the minimum radius of disk, call it k_opt, so that a convex polygon Q contained in P intersects all disks of radius k_opt centred on the vertices of P. We begin by describing a polynomial time algorithm for the simple case where P has only a single reflex vertex. On general polygons, we provide a parameterized algorithm which performs a parametric search on the possible values of k_opt. The solution to the decision version of the problem, i.e. determining whether there exists a Hausdorff Core for P given k_opt, requires some novel insights. We also describe an FPTAS for the decision version of the Hausdorff Core problem. Finally, we study Generalized Minimum Spanning Tree (GMST) problems, where the input consists of imprecise vertices, and the objective is to select a single point from each imprecise vertex in order to optimize the weight of the MST over the points. In keeping with one of the themes of the thesis, we begin by using disks as the imprecise vertices. We show that the minimization and maximization versions of this problem are NP-hard, and we describe some parameterized and approximation algorithms. Finally, we look at the case where the imprecise vertices consist of just two vertices each, and we show that the minimization version of the problem (which we call 2-GMST) remains NP-hard, even in the plane. We also provide an algorithm to solve the 2-GMST problem exactly if the combinatorial structure of the optimal solution is known. We identify a number of open problems in this thesis that are worthy of further study. Among them: Is the Assisted LSDUDC problem NP-complete? Can the WSDUDC results be used to obtain an improved PTAS for DUDC? Are there classes of polygons for which the determination of the Hausdorff Core is easy? Is there a PTAS for the maximum weight GMST problem on (unit) disks? Is there a combinatorial approximation algorithm for the 2-GMST problem (particularly with an approximation factor under 4)?

In this doctoral thesis, four questions related to computational geometry are considered. The first is an extremal combinatorics question regarding triangles with vertices taken from a set of n points in convex position. More precisely, two such triangles can exhibit eight distinct configurations and, for each subset of these configurations, we are interested in the asymptotics of how many triangles one can have while avoiding configurations in the subset (as a function of n). For most of these subsets, we answer this question optimally up to a logarithmic factor in the form of several Turán-type theorems. The answers for the remaining few were in turn tied to that of a long-standing open problem which appeared in the literature in the contexts of monotone matrices, tripod packing and 2-comparable sets. The second problem, called Line Segment Covering (LSC), is about covering the cells of an arrangement of line segments with these line segments, where a segment covers the cells it is incident to. Recently, a PTAS, an APX -hardness proof and a FPT algorithm for variants of this problem have been shown. This paper and a new slight generalization of one of its results is included as a chapter. The third problem has been posed in the Sixth Annual Workshop on Geometry and Graphs and concerns the design of road networks to minimize the maximum travel time between two point sets in the plane. Traveling outside the roads costs more time per unit of distance than traveling on the roads and the total length of the roads can not exceed a budget. When the point sets are the opposing sides of a unit square and the budget is at most √2, we were able to come up with a few network designs that cover all possible cases and are provably optimal. Furthermore, when both point sets are the boundary of a unit circle, we managed to disprove the natural conjecture that a concentric circle is an optimal design. Finally, we consider collision-avoiding schedules of unit-velocity axis-aligned trains departing and arriving from points in the integer lattice. We prove a few surprising results on the existence of constant upper bounds on the maximum delay that are independent of the train network. In particular, these upper bounds are shown to always exist in two dimensions and to exist in three dimensions for unit-length trains. We also showed computationally that, for several scenarios, these upper bounds are tight.

The <i>mixed postman problem</i> consists of finding a minimum cost tour of a mixed graph <i>M</i> = (<i>V</i>,<i>E</i>,<i>A</i>) traversing all its edges and arcs at least once. We prove that two well-known linear programming relaxations of this problem are equivalent. The <i>extra cost</i> of a mixed postman tour <i>T</i> is the cost of <i>T</i> minus the cost of the edges and arcs of <i>M</i>. We prove that it is <i>NP</i>-hard to approximate the minimum extra cost of a mixed postman tour. A related problem, known as the <i>windy postman problem</i>, consists of finding a minimum cost tour of an undirected graph <i>G</i>=(<i>V</i>,<i>E</i>) traversing all its edges at least once, where the cost of an edge depends on the direction of traversal. We say that <i>G</i> is <i>windy postman perfect</i> if a certain <i>windy postman polyhedron O</i> (<i>G</i>) is integral. We prove that series-parallel undirected graphs are windy postman perfect, therefore solving a conjecture of Win. Given a mixed graph <i>M</i> = (<i>V</i>,<i>E</i>,<i>A</i>) and a subset <i>R</i> &#8838; <i>E</i> &#8746; <i>A</i>, we say that a mixed postman tour of <i>M</i> is <i>restricted</i> if it traverses the elements of <i>R</i> exactly once. The <i>restricted mixed postman problem</i> consists of finding a minimum cost restricted tour. We prove that this problem is <i>NP</i>-hard even if <i>R</i>=<i>A</i> and we restrict <i>M</i> to be planar, hence solving a conjecture of Veerasamy. We also prove that it is <i>NP</i>-complete to decide whether there exists a restricted tour even if <i>R</i>=<i>E</i> and we restrict <i>M</i> to be planar. The <i>edges postman problem</i> is the special case of the restricted mixed postman problem when <i>R</i>=<i>A</i>. We give a new class of valid inequalities for this problem. We introduce a relaxation of this problem, called the <i>b-join problem</i>, that can be solved in polynomial time. We give an algorithm which is simultaneously a 4/3-approximation algorithm for the edges postman problem, and a 2-approximation algorithm for the extra cost of a tour. The <i>arcs postman problem</i> is the special case of the restricted mixed postman problem when <i>R</i>=<i>E</i>. We introduce a class of necessary conditions for <i>M</i> to have an arcs postman tour, and we give a polynomial-time algorithm to decide whether one of these conditions holds. We give linear programming formulations of this problem for mixed graphs arising from windy postman perfect graphs, and mixed graphs whose arcs form a forest.