Packing Under Convex Quadratic Constraints

In this paper, we propose a branch-and-bound algorithm for finding a global optimal solution for a nonconvex quadratic program with convex quadratic constraints (NQPCQC). We first reformulate NQPCQC by adding some nonconvex quadratic constraints induced by eigenvectors of negative eigenvalues associated with the nonconvex quadratic objective function to Shor's semidefinite relaxation. Under the assumption of having a bounded feasible domain, these nonconvex quadratic constraints can be further relaxed into linear ones to form a special semidefinite programming relaxation. Then an efficient branch-and-bound algorithm branching along the eigendirections of negative eigenvalues is designed. The theoretic convergence property and the worst-case complexity of the proposed algorithm are proved. Numerical experiments are conducted on several types of quadratic programs to show the efficiency of the proposed method.

This book is intended to be used as a textbook for graduate students studying theoretical computer science. It can also be used as a reference book for researchers in the area of design and analysis of approximation algorithms. Design and Analysis of Approximation Algorithms is a graduate course in theoretical computer science taught widely in the universities, both in the United States and abroad. There are, however, very few textbooks available for this course. Among those available in the market, most books follow a problem-oriented format; that is, they collected many important combinatorial optimization problems and their approximation algorithms, and organized them based on the types, or applications, of problems, such as geometric-type problems, algebraic-type problems, etc. Such arrangement of materials is perhaps convenient for a researcher to look for the problems and algorithms related to his/her work, but is difficult for a student to capture the ideas underlying the various algorithms. In the new book proposed here, we follow a more structured, technique-oriented presentation. We organize approximation algorithms into different chapters, based on the design techniques for the algorithms, so that the reader can study approximation algorithms of the same nature together. It helps the reader to better understand the design and analysis techniques for approximation algorithms, and also helps the teacher to present the ideas and techniques of approximation algorithms in a more unified way.

When the follower's optimality conditions are both necessary and sufficient, the nonlinear bilevel program can be solved as a global optimization problem. The complementary slackness condition is usually the complicating constraint in such problems. We show how this constraint can be replaced by an equivalent system of convex and separable quadratic constraints. In this paper, we propose different methods for finding the global minimum of a concave function subject to quadratic separable constraints. The first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex, the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints, we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.

Suppose we are given a set of objects that cover a region and a duration associated with each object. Viewing the objects as jobs, can we schedule their beginning times to maximize the length of time that the original region remains covered? We call this problem the SENSOR COVER PROBLEM. It arises in the context of covering a region with sensors. For example, suppose you wish to monitor activity along a fence (interval) by sensors placed at various fixed locations. Each sensor has a range (also an interval) and limited battery life. The problem is then to schedule when to turn on the sensors so that the fence is fully monitored for as long as possible.This one-dimensional problem involves intervals on the real line. Associating a duration to each yields a set of rectangles in space and time, each specified by a pair of fixed horizontal endpoints and a height. The objective is to assign a bottom position to each rectangle (by moving them up or down) so as to maximize the height at which the spanning interval is fully covered. We call this one-dimensional problem RESTRICTED STRIP COVERING. If we replace the covering constraint by a packing constraint (rectangles may not overlap, and the goal is to minimize the highest point covered), then the problem becomes identical to DYNAMIC STORAGE ALLOCATION, a well-studied scheduling problem, which is in turn a restricted case of the well known problem STRIP PACKING.We present a collection of algorithms for RESTRICTED STRIP COVERING. We show that the problem is NP-hard and present an O(log log log n)-approximation algorithm. We also present better approximation or exact algorithms for some special cases, including when all intervals have equal width. For the general SENSOR COVER PROBLEM, we distinguish between cases in which elements have uniform or variable durations. The results depend on the structure of the region to be covered: We give a polynomial-time, exact algorithm for the uniform-duration case of RESTRICTED STRIP COVERING but prove that the uniform-duration case for higher-dimensional regions is NP-hard. We give some more specific results for two-dimensional regions. Finally, we consider regions that are arbitrary sets, and we present an O(log n)-approximation algorithm for the most general case.

A solution procedure for linear programs with one convex quadratic constraint is suggested. The method finds an optimal solution in finitely many iterations.

In recent years, the data from Internet of Things (IoT) devices is growing at a rapid pace, and the data collection issues have attracted more and more attention. Distinct from existing solutions which usually adopted traditional wireless technologies achieving low-bandwidth data connections towards IoT devices, this paper adopts the 60 GHz communication technology which is with increasing maturity, providing high-bandwidth wireless transmission for data-intensive IoT devices, such as high-definition (HD) cameras. However, 60 GHz links are subject to line-of-sight short communication range, therefore, we propose to use unmanned aerial vehicles (UAVs) to fly over data-intensive IoT devices and achieve short-range high-throughput 60 GHz transmission in this paper. Moreover, for a set of HD cameras deployed in the linear scenario, multiple UAV flights are assigned to collect data by 60 GHz communication. To this end, we investigate the UAV flight scheduling (UFS) problem which aims to minimize the number of UAV flights while satisfying the data requirements of all ground cameras (GCs) with limitations of UAV’s energy and data storage. We prove that the UFS problem is NP-hard and design efficient algorithms with constant theoretical approximation ratios. Specifically, we first study a special case of the UFS problem where all the cameras are on the same direction of the UAV ground station, and propose two algorithms NF_SUFS and FF_SUFS, whose approximation ratios are both proved to be 2 by theoretical analysis. Then, we extend the algorithms to a more general case with the cameras on both directions of the ground station along the road, and put forward the FF_UFS algorithm that achieves an approximation ratio of 3. Finally, we conduct experiments to validate the effectiveness and efficiency of our algorithms.

In this work, we are interested in the convex hull of the region determined by two quadratic polynomial constraints. Our main result is that if this region is nonempty, its convex hull is either IR or the feasible set of another pair of quadratic constraints which are, in fact, positive linear combinations of the original ones. We propose an algorithm to find these positive combinations efficiently and convert them into linear matrix inequalities (LMI). This result can be utilized to find the optimal value of a linear objective function over a set determined by two general quadratic inequality constraints. As a result, we show that such problems can be solved in polynomial time. To the best of our knowledge, this was not known until now. When the literature is investigated it can be seen that one of the most suitable approaches that can be utilized to obtain the solution of this problem are relaxation methods. However, using relaxations one can efficiently compute bounds on the global optimal solution which may not turn out to be exact. We show that our approach always finds the exact solution more efficiently (asymptotically) when compared with the available relaxation. Moreover, by an example, we also show that our method may perform way better these approaches for some choice of quadratic constraints. Lastly, we show how the results developed can be applied to a certain class of control problems.

The quantum approximate optimization algorithm (QAOA) is a method of approximately solving combinatorial optimization problems. While QAOA is developed to solve a broad class of combinatorial optimization problems, it is not clear which classes of problems are best suited for it. One factor in demonstrating quantum advantage is the relationship between a problem instance and the circuit depth required to implement the QAOA method. As errors in noisy intermediate-scale quantum (NISQ) devices increase exponentially with circuit depth, identifying lower bounds on circuit depth can provide insights into when quantum advantage could be feasible. Here, we identify how the structure of problem instances can be used to identify lower bounds for circuit depth for each iteration of QAOA and examine the relationship between problem structure and the circuit depth for a variety of combinatorial optimization problems including MaxCut and MaxIndSet. Specifically, we show how to derive a graph, G, that describes a general combinatorial optimization problem and show that the depth of circuit is at least the chromatic index of G. By looking at the scaling of circuit depth, we argue that MaxCut, MaxIndSet, and some instances of vertex covering and Boolean satisfiability problems are suitable for QAOA approaches while knapsack and traveling salesperson problems are not.

In recent years approximation algorithms based on primal-dual methods have been successfully applied to a broad class of discrete optimization problems. In this paper, we propose a generic primal-dual framework to design and analyze approximation algorithms for integer programming problems of the covering type that uses valid inequalities in its design. The worst-case bound of the proposed algorithm is related to a fundamental relationship (called strength) between the set of valid inequalities and the set of minimal solutions to the covering problems. In this way, we can construct an approximation algorithm simply by constructing the required valid inequalities. We apply the proposed algorithm to several problems, such as covering problems related to totally balanced matrices, cyclic scheduling, vertex cover, general set covering, intersections of polymatroids, and several network design problems attaining (in most cases) the best worst-case bound known in the literature.

In the k-Connectivity Augmentation Problem we are given a k-edge-connected graph and a set of additional edges called links. Our goal is to find a set of links of minimum size whose addition to the graph makes it (k + 1)-edge-connected. There is an approximation preserving reduction from the mentioned problem to the case k = 1 (a.k.a. the Tree Augmentation Problem or TAP) or k = 2 (a.k.a. the Cactus Augmentation Problem or CacAP). While several better-than-2 approximation algorithms are known for TAP, for CacAP only recently this barrier was breached (hence for k-Connectivity Augmentation in general). As a first step towards better approximation algorithms for CacAP, we consider the special case where the input cactus consists of a single cycle, the Cycle Augmentation Problem (CycAP). This apparently simple special case retains part of the hardness of the general case. In particular, we are able to show that it is APX-hard. In this paper we present a combinatorial left (frac {3}{2}+varepsilon right )-approximation for CycAP, for any constant ε > 0. We also present an LP formulation with a matching integrality gap: this might be useful to address the general case of the problem.

Consideration is given to the problem of finding a shortest path between two points in 3-D space with a restricted class of polyhedral obstacles (vertical buildings with a fixed number k of distinct heights). For the case when all the obstacles have equal heights, a shortest-path algorithm is presented with complexity O(n/sup 2/), i.e. the same complexity as for the 2-D case (n is the total number of corners in all the obstacles). For the general case (k distinct heights), an algorithm is presented for finding a shortest path in time O(n/sup 6k-1/). Also presented is an O(n/sup 2/) approximation algorithm that finds paths that are, at most, 8% longer than the shortest path for the case of k distinct heights when certain minimum separation requirements are satisfied, and a description is given of how the approximation algorithm can be extended to the general case (arbitrary separations). >

We consider approximation algorithms for the metric labeling problem. This problem was introduced in a recent paper by Kleinberg and Tardos [20], and captures many classification problems that arise in computer vision and related fields. They gave an O(log k log log k) approximation for the general case where k is the number of labels and a 2-approximation for the uniform metric case. More recently, Gupta and Tardos [15] gave a 4-approximation for the truncated linear metric, a natural non-uniform metric motivated by practical applications to image restoration and visual correspondence. In this paper we introduce a natural integer programming formulation and show that the integrality gap of its linear relaxation either matches or improves the ratios known for several cases of the metric labeling problem studied until now, providing a unified approach to solving them. In particular, we show that the integrality gap of our LP is bounded by O(log k log log k) for general metric and 2 for the uniform metric thus matching the ratios in [20]. We also develop an algorithm based on our LP that achieves a ratio of 2 + √2 s 3.414 for the truncated linear metric improving the ratio given in [15]. Our algorithm uses the fact that the integrality gap of our LP is 1 on a linear metric. We believe that our formulation has the potential to provide improved approximation algorithms for the general case and other useful special cases. Finally, our formulation admits general non-metric distance functions. This leads to a non-trivial approximation guarantee for a non-metric case that arises in practice [21], namely the truncated quadratic distance function. We note here that there are non-metric distance functions for which no bounded approximation ratio is achievable.

In the general search problem we want to identify a specific element using a set of allowed tests. The general goal is to minimize the number of tests performed, although different measures are used to capture this goal. In this work we introduce a novel greedy approach that achieves the best known approximation ratios simultaneously for many different variations of this identification problem. In addition to this flexibility, our algorithm admits much shorter and simpler analyses than previous greedy strategies. As a second contribution, we investigate the potential of greedy algorithms for the more restricted problem of identifying elements of partially ordered sets by comparison with other elements. We prove that the latter problem is as hard to approximate as the general identification problem. As a positive result, we show that a natural greedy strategy achieves an approximation ratio of 2 for tree-like posets, improving upon the previously best known 14-approximation for this problem.

A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin—i.e., for each bin, a family of subsets of items that fit in to that bin. The goal is to pack items into bins to maximize the aggregate value. This class of problems includes the maximum generalized assignment problem (GAP)1 and a distributed caching problem (DCP) described in this paper. Given a β-approximation algorithm for finding the highest value packing of a single bin, we give A polynomial-time LP-rounding based ((1 − 1/e)β)-approximation algorithm. A simple polynomial-time local search (β/(β + 1) − ε)-approximation algorithm, for any ε > 0. Therefore, for all examples of SAP that admit an approximation scheme for the single-bin problem, we obtain an LP-based algorithm with (1 − 1/e − ε)-approximation and a local search algorithm with (½ - ε)-approximation guarantee. Furthermore, for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LP-based algorithm analysis can be strengthened to give a guarantee of 1 − 1/e. The best previously known approximation algorithm for GAP is a ½-approximation by Shmoys and Tardos and Chekuri and Khanna. Our LP algorithm is based on rounding a new linear programming relaxation, with a provably better integrality gap. To complement these results, we show that SAP and DCP cannot be approximated within a factor better than 1 − 1/e unless NP ⫅ DTIME(nO(log log n)), even if there exists a polynomial-time exact algorithm for the single-bin problem. We extend the (1 − 1/e)-approximation algorithm to a constant-factor approximation algorithms for a nonseparable assignment problem with applications in maximizing revenue for budget-constrained combinatorial auctions and the AdWords assignment problem. We generalize the local search algorithm to yield a ½ - ε approximation algorithm for the maximum k-median problem with hard capacities.

An information dissemination campaign is often multifaceted, involving several facets or pieces of information disseminating from different sources. The question then arises, how should we assign such pieces to eligible sources so as to achieve the best viral dissemination results? Past research has studied the problem of Influence Maximization (IM), which is to select a set of k promoters that maximizes the expected reach of a message over a network. However, in this classical IM problem, each promoter spreads out the same unitary piece of information. In this paper, we propose the Optimal Influential Pieces Assignment (OIPA) problem, which is to assign k distinct pieces of an information campaign OIPA to k promoters, so as to achieve the highest viral adoption in a network. We express adoption by users with a logistic model, and show that approximating OIPA within any constant factor is NP-hard. Even so, we propose a branch-and-bound framework for problem with an (1-1/e) approximation ratio. We further optimize this framework with a pruning-intensive progressive upper-bound estimation approach, yielding a (1-1/e-\varepsilon) approximation ratio and significantly lower time complexity, as it relies on the power-law properties of real-world social networks to run efficiently. Our extensive experiments on several real-world datasets show that the proposed approaches consistently outperform intuitive baselines, adopted from state-of-the-art IM algorithms. Furthermore, the progressive approach demonstrates superior efficiency with an up to 24-fold speedup over the plain branch-and-bound approach.