On the Complexity of Local Search for Weighted Standard Set Problems

In this paper, we study the complexity of computing locally optimal solutions for weighted versions of standard set problems such as SetCover, SetPacking, and many more. For our investigation, we use the framework of PLS, as defined in Johnson et al., [JPY88]. We show that for most of these problems, computing a locally optimal solution is already PLS-complete for a simple neighborhood of size one. For the local search versions of weighted SetPacking and SetCover, we derive tight bounds for a simple neighborhood of size two. To the best of our knowledge, these are one of the very few PLS results about local search for weighted standard set problems

(Almost) tight bounds for randomized and quantum Local Search on hypercubes and grids

The Local Search problem, which finds a local minimum of a black-box function on a given graph, is of both practical and theoretical importance to many areas in computer science and natural sciences. In this paper, we show that for the Boolean hypercube \B^n, the randomized query complexity of Local Search is $\Theta(2^{n/2}n^{1/2})$ and the quantum query complexity is $\Theta(2^{n/3}n^{1/6})$. We also show that for the constant dimensional grid $[N^{1/d}]^d$, the randomized query complexity is $\Theta(N^{1/2})$ for $d \geq 4$ and the quantum query complexity is $\Theta(N^{1/3})$ for $d \geq 6$. New lower bounds for lower dimensional grids are also given. These improve the previous results by Aaronson [STOC'04], and Santha and Szegedy [STOC'04]. Finally we show for $[N^{1/2}]^2$ a new upper bound of $O(N^{1/4}(\log\log N)^{3/2})$ on the quantum query complexity, which implies that Local Search on grids exhibits different properties at low dimensions.Comment: 18 pages, 1 figure. v2: introduction rewritten, references added. v3: a line for grant added. v4: upper bound section rewritte

Travelling salesman paths on Demidenko matrices

In the path version of the Travelling Salesman Problem (Path-TSP), a salesman is looking for the shortest Hamiltonian path through a set of n cities. The salesman has to start his journey at a given city s, visit every city exactly once, and finally end his trip at another given city t. In this paper we show that a special case of the Path-TSP with a Demidenko distance matrix is solvable in polynomial time. Demidenko distance matrices fulfill a particular condition abstracted from the convex Euclidian special case by Demidenko (1979) as an extension of an earlier work of Kalmanson (1975). We identify a number of crucial combinatorial properties of the optimal solution and design a dynamic programming approach with time complexity O(n6)

Approximation algorithms for a graph-cut problem with applications to a clustering problem in bioinformatics

xiii, 71 leaves : ill. ; 29 cm.Clusters in protein interaction networks can potentially help identify functional relationships among proteins. We study the clustering problem by modeling it as graph cut problems. Given an edge weighted graph, the goal is to partition the graph into a prescribed number of subsets obeying some capacity constraints, so as to maximize the total weight of the edges that are within a subset. Identification of a dense subset might shed some light on the biological function of all the proteins in the subset. We study integer programming formulations and exhibit large integrality gaps for various formulations. This is indicative of the difficulty in obtaining constant factor approximation algorithms using the primal-dual schema. We propose three approximation algorithms for the problem. We evaluate the algorithms on the database of interacting proteins and on randomly generated graphs. Our experiments show that the algorithms are fast and have good performance ratio in practice

Large-scale optimization for data placement problem

Large-scale optimization of combinatorial problems is one of the most challenging areas. These problems are characterized by large sets of data (variables and constraints). In this thesis, we study large-scale optimization of the data placement problem with zero storage cost. The goal in the data placement problem is to find the placement of data objects in a set of fixed capacity caches in a network to optimize the latency of access. Data placement problem arises naturally in the design of content distribution networks. We report on an empirical study of the upper bound and the lower bound of this problem for large sized instances. We also study a semi-Lagrangean relaxation of a closely related k-median problem. In this thesis, we study the theory and practice of approximation algorithm for the data placement problem and the k-median problem

Three essays on sequencing and routing problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2005.Includes bibliographical references (leaves 157-162).In this thesis we study different combinatorial optimization problems. These problems arise in many practical settings where there is a need for finding good solutions fast. The first class of problems we study are vehicle routing problems, and the second type of problems are sequencing problems. We study approximation algorithms and local search heuristics for these problems. First, we analyze the Vehicle Routing Problem (VRP) with and without split deliveries. In this problem, we have to route vehicles from the depot to deliver the demand to the customers while minimizing the total traveling cost. We present a lower bound for this problem, improving a previous bound of Haimovich and Rinnooy Kan. This bound is then utilized to improve the worst-case approximation algorithm of the Iterated Tour Partitioning (ITP) heuristic when the capacity of the vehicles is constant. Second, we analyze a particular case of the VRP, when the customers are uniformly distributed i.i.d. points on the unit square of the plane, and have unit demand. We prove that there exists a constant c > 0 such that the ITP heuristic is a 2 - c approximation algorithm with probability arbitrarily close to one as the number of customers goes to infinity. This result improves the approximation factor of the ITP heuristic under the worst-case analysis, which is 2. We also generalize this result and previous ones to the multi-depot case. Third, we study a language to generate Very Large Scale Neighborhoods for sequencing problems. Local search heuristics are among the most popular approaches to solve hard optimization problems.(cont.) Among them, Very Large Scale Neighborhood Search techniques present a good balance between the quality of local optima and the time to search a neighborhood. We develop a language to generate exponentially large neighborhoods for sequencing problems using grammars. We develop efficient generic dynamic programming solvers that determine the optimal neighbor in a neighborhood generated by a grammar for a list of sequencing problems, including the Traveling Salesman Problem and the Linear Ordering Problem. This framework unifies a variety of previous results on exponentially large neighborhoods for the Traveling Salesman Problem.by Agustín Bompadre.Ph.D