Approximation Algorithms for the Multiple Knapsack Problem with Assignment Restrictions

Motivated by a real world application, we study the multiple knapsack problem with assignment restrictions (MKAR). We are given a set of items, each with a positive real weight, and a set of knapsacks, each with a positive real capacity. In addition, for each item a set of knapsacks that can hold that item is specified. In a feasible assignment of items to knapsacks, each item is assigned to at most one knapsack, assignment restrictions are satisfied, and knapsack capacities are not exceeded. We consider the objectives of maximizing assigned weight and minimizing utilized capacity.We focus on obtaining approximate solutions in polynomial computational time. We show that simple greedy approaches yield 1/3-approximation algorithms for the objective of maximizing assigned weight. We give two different 1/2-approximation algorithms: the first one solves single knapsack problems successively and the second one is based on rounding the LP relaxation solution. For the bicriteria problem of minimizing utilized capacity subject to a minimum requirement on assigned weight, we give an (1/3,2)-approximation algorithm.     

Approximation Algorithms for Bounded Facility Location Problems

The bounded k-median problem is to select in an undirected graph G = (V,E) a set S of k vertices such that the distance from any vertex v V to S is at most a given bound d and the average distance from vertices V\S to S is minimized. We present randomized algorithms for several versions of this problem and we prove some inapproximability results. We also study the bounded version of the uncapacitated facility location problem and present extensions of known deterministic algorithms for the unbounded version.     

Approximation Algorithms for Certain Network Improvement Problems

We study budget constrained network upgrading problems. Such problems aim at finding optimal strategies for improving a network under some cost measure subject to certain budget constraints. Given an edge weighted graph G = (V, E), in the edge based upgrading model, it is assumed that each edge e of the given network also has an associated function ce (t) that specifies the cost of upgrading the edge by an amount t. A reduction strategy specifies for each edge e the amount by which the length (e) is to be reduced. In the node based upgrading model, a node v can be upgraded at an expense of c(v). Such an upgrade reduces the delay of each edge incident on v. For a given budget B, the goal is to find an improvement strategy such that the total cost of reduction is at most the given budget B and the cost of a subgraph (e.g. minimum spanning tree) under the modified edge lengths is the best over all possible strategies which obey the budget constraint.After providing a brief overview of the models and definitions of the various problems considered, we present several new results on the complexity and approximability of network improvement problems.     

Approximation and Exact Algorithms for Constructing Minimum Ultrametric Trees from Distance Matrices

An edge-weighted tree is called ultrametric if the distances from the root to all the leaves in the tree are equal. For an n by n distance matrix M, the minimum ultrametric tree for M is an ultrametric tree T = (V, E, w) with leaf set {1,..., n} such that d_T(i, j) M[i, j] for all i, j and is minimum, where d_T(i, j) is the distance between i and j on T. Constructing minimum ultrametric trees from distance matrices is an important problem in computational biology. In this paper, we examine its computational complexity and approximability. When the distances satisfy the triangle inequality, we show that the minimum ultrametric tree problem can be approximated in polynomial time with error ratio 1.5(1 + log n), where n is the number of species. We also develop an efficient branch-and-bound algorithm for constructing the minimum ultrametric tree for both metric and non-metric inputs. The experimental results show that it can find an optimal solution for 25 species within reasonable time, while, to the best of our knowledge, there is no report of algorithms solving the problem even for 12 species.     

Approximation Algorithms for Quadratic Programming

We consider the problem of approximating the global minimum of a general quadratic program (QP) with n variables subject to m ellipsoidal constraints. For m=1, we rigorously show that an -minimizer, where error (0, 1), can be obtained in polynomial time, meaning that the number of arithmetic operations is a polynomial in n, m, and log(1/). For m 2, we present a polynomial-time (1- )-approximation algorithm as well as a semidefinite programming relaxation for this problem. In addition, we present approximation algorithms for solving QP under the box constraints and the assignment polytope constraints.     

Simple Approximation Algorithms for MAXNAESP and Hypergraph 2-colorability

Hypergraph 2-colorability, also known as set splitting, is a widely studied problem in graph theory. In this paper we study the maximization version of the same. We recast the problem as a special type of satisfiability problem and give approximation algorithms for it. Our results are valid for hypergraph 2-colorability, set splitting and MAX-CUT (which is a special case of hypergraph 2-colorability) because the reductions are approximation preserving. Here we study the MAXNAESP problem, the optimal solution to which is a truth assignment of the literals that maximizes the number of clauses satisfied. As a main result of the paper, we show that any locally optimal solution (a solution is locally optimal if its value cannot be increased by complementing assignments to literals and pairs of literals) is guaranteed a performance ratio of . This is an improvement over the ratio of attributed to another local improvement heuristic for MAX-CUT (C. Papadimitriou, Computational Complexity, Addison Wesley, 1994). In fact we provide a bound of for this problem, where k 3 is the minimum number of literals in a clause. Such locally optimal algorithms appear to subsume typical greedy algorithms that have been suggested for problems in the general domain of satisfiability. It should be noted that the NAESP problem where each clause has exactly two literals, is equivalent to MAX-CUT. However, obtaining good approximation ratios using semi-definite programming techniques (M. Goemans and D.P. Williamson, in Proceedings of the 26th Annual ACM Symposium on Theory of Computing, 1994a, pp. 422–431) appears difficult. Also, the randomized rounding algorithm as well as the simple randomized algorithm both (M. Goemans and D.P. Williamson, SIAM J. Disc. Math, vol. 7, pp. 656–666, 1994b) yield a bound of for the MAXNAESP problem. In contrast to this, the algorithm proposed in this paper obtains a bound of for this problem.     

An Improved Randomized Approximation Algorithm for Max TSP

We present an O(n³)-time randomized approximation algorithm for the maximum traveling salesman problem whose expected approximation ratio is asymptotically , where n is the number of vertices in the input (undirected) graph. This improves the previous best.Part of work done while visiting City University of Hong Kong.     

Approximation and Exact Algorithms for RNA Secondary Structure Prediction and Recognition of Stochastic Context-free Languages

For a basic version (i.e., maximizing the number of base-pairs) of the RNA secondary structure prediction problem and the construction of a parse tree for a stochastic context-free language, O(n³) time algorithms were known. For both problems, this paper shows slightly improved O(n³(log log n)^1/2/(log n)^1/2) time exact algorithms, which are obtained by combining Valiant's algorithm for context-free recognition with fast funny matrix multiplication. Moreover, this paper shows an O(n^2.776 + (1/)^O(1)) time approximation algorithm for the former problem and an O(n^2.976 log n + (1/)^O(1)) time approximation algorithm for the latter problem, each of which has a guaranteed approximation ratio 1 – for any positive constant , where the absolute value of the logarithm of the probability is considered as an objective value in the latter problem. The former algorithm is obtained from a non-trivial modification of the well-known O(n³) time dynamic programming algorithm, and the latter algorithm is obtained by combining Valiant's algorithm with approximate funny matrix multiplication. Several related results are shown too.     

Generalized Steiner Problems and Other Variants

In this paper, we examine combinatorial optimization problems by considering the case where the set N (the ground set of elements) is expressed as a union of a finite number of m nonempty distinct subsets N ₁,...,N _m. The term we use is the generalized Steiner problems coined after the Generalized Traveling Salesman Problem. We have collected a short list of classical combinatorial optimization problems and we have recast each of these problems in this broader framework in an attempt to identify a linkage between these “generalized” problems. In the literature one finds generalized problems such as the Generalized Minimum Spanning Tree (GMST), Generalized Traveling Salesman Problem (GTSP) and Subset Bin-packing (SBP). Casting these problems into the new problem setting has important implications in terms of the time effort required to compute an optimal solution or a “good” solution to a problem. We examine questions like “is the GTSP “harder” than the TSP?” for a number of paradigmatic problems starting with “easy” problems such as the Minimal Spanning Tree, Assignment Problem, Chinese Postman, Two-machine Flow Shop, and followed by “hard” problems such as the Bin-packing, and the TSP.     

Improved Approximation Algorithms for Maximum Graph Partitioning Problems

We consider the design of approximation algorithms for a number of maximum graph partitioning problems, among others MAX-k-CUT, MAX-k-DENSE-SUBGRAPH, and MAX-k-DIRECTED-UNCUT. We present a new version of the semidefnite relaxation scheme along with a better analysis, extending work of Halperin and Zwick (2002). This leads to an improvement over known approximation factors for such problems. The key to the improvement is the following new technique: It was already observed by Han et al. (2002) that a parameter-driven choice of the random hyperplane can lead to better approximation factors than obtained by Goemans and Williamson (1995). But it remained difficult to find a “good” set of parameters. In this paper, we analyze random hyperplanes depending on several new parameters. We prove that a sub-optimal choice of these parameters can be obtained by the solution of a linear program which leads to the desired improvement of the approximation factors. In this fashion a more systematic analysis of the semidefinite relaxation scheme is obtained.     

Multi-phase Algorithms for Throughput Maximization for Real-Time Scheduling

We consider the problem of off-line throughput maximization for job scheduling on one or more machines, where each job has a release time, a deadline and a profit. Most of the versions of the problem discussed here were already treated by Bar-Noy et al. (Proc. 31st ACM STOC, 1999, pp. 622–631; http://www.eng.tau.ac.il/amotz/). Our main contribution is to provide algorithms that do not use linear programming, are simple and much faster than the corresponding ones proposed in Bar-Noy et al. (ibid., 1999), while either having the same quality of approximation or improving it. More precisely, compared to the results of in Bar-Noy et al. (ibid., 1999), our pseudo-polynomial algorithm for multiple unrelated machines and all of our strongly-polynomial algorithms have better performance ratios, all of our algorithms run much faster, are combinatorial in nature and avoid linear programming. Finally, we show that algorithms with better performance ratios than 2 are possible if the stretch factors of the jobs are bounded; a straightforward consequence of this result is an improvement of the ratio of an optimal solution of the integer programming formulation of the JISP2 problem (see Spieksma, Journal of Scheduling, vol. 2, pp. 215–227, 1999) to its linear programming relaxation.     

Approximation algorithms and hardness results for labeled connectivity problems

Let G=(V,E) be a connected multigraph, whose edges are associated with labels specified by an integer-valued function ℒ:E→ℕ. In addition, each label ℓ∈ℕ has a non-negative cost c(ℓ). The minimum label spanning tree problem (MinLST) asks to find a spanning tree in G that minimizes the overall cost of the labels used by its edges. Equivalently, we aim at finding a minimum cost subset of labels I⊆ℕ such that the edge set {e∈E:ℒ(e)∈I} forms a connected subgraph spanning all vertices. Similarly, in the minimum label s – t path problem (MinLP) the goal is to identify an s–t path minimizing the combined cost of its labels. The main contributions of this paper are improved approximation algorithms and hardness results for MinLST and MinLP.     

New Approximation Algorithms for the Steiner Tree Problems

The Steiner tree problem asks for the shortest tree connecting a given set of terminal points in a metric space. We design new approximation algorithms for the Steiner tree problems using a novel technique of choosing Steiner points in dependence on the possible deviation from the optimal solutions. We achieve the best up to now approximation ratios of 1.644 in arbitrary metric and 1.267 in rectilinear plane, respectively.     

A New Fully Polynomial Time Approximation Scheme for the Knapsack Problem

A fully polynomial time approximation scheme (FPTAS) is presented for the classical 0-1 knapsack problem. The new approach considerably improves the necessary space requirements. The two best previously known approaches need O(n + 1/³) and O(n · 1/) space, respectively. Our new approximation scheme requires only O(n + 1/²) space while also reducing the running time.     

A New Approximation Algorithm for the Capacitated Vehicle Routing Problem on a Tree

This paper presents a new approximation algorithm for a vehicle routing problem on a tree-shaped network with a single depot. Customers are located on vertices of the tree, and each customer has a positive demand. Demands of customers are served by a fleet of identical vehicles with limited capacity. It is assumed that the demand of a customer is splittable, i.e., it can be served by more than one vehicle. The problem we are concerned with in this paper asks to find a set of tours of the vehicles with minimum total lengths. Each tour begins at the depot, visits a subset of the customers and returns to the depot without violating the capacity constraint. We propose a 1.35078-approximation algorithm for the problem (exactly, ), which is an improvement over the existing 1.5-approximation.     

Performance Analysis and Improvement for Some Linear On-Line Bin-Packing Algorithms

The study on one-dimensional bin packing problem may bring about many important applications such as multiprocessor scheduling, resource allocating, real-world planning and packing. Harmonic algorithms (including H _K, RH, etc.) for bin packing have been famous for their linear-time and on-line properties for a long time. This paper profoundly analyzes the average-case performance of harmonic algorithms for pieces of i.i.d. sizes, provides the average-case performance ratio of H _K under (0,d] (d 1) uniform distribution and the average-case performance ratio of RH under (0,1] uniform distribution. We also finished the discussion of the worst-case performance ratio of RH. Moreover, we propose a new improved version of RH that obtains better worst- and average-case performance ratios.     

The Enhanced Double Digest Problem for DNA Physical Mapping

The double digest problem is a common NP-hard approach to constructing physical maps of DNA sequences. This paper presents a new approach called the enhanced double digest problem. Although this new problem is also NP-hard, it can be solved in linear time in certain theoretically interesting cases.     

An Approximation Scheme for Bin Packing with Conflicts

In this paper we consider the following bin packing problem with conflicts. Given a set of items V = {1,..., n} with sizes s₁,..., s (0,1) and a conflict graph G = (V, E), we consider the problem to find a packing for the items into bins of size one such that adjacent items (j, j) E are assigned to different bins. The goal is to find an assignment with a minimum number of bins. This problem is a natural generalization of the classical bin packing problem.We propose an asymptotic approximation scheme for the bin packing problem with conflicts restricted to d-inductive graphs with constant d. This graph class contains trees, grid graphs, planar graphs and graphs with constant treewidth. The algorithm finds an assignment for the items such that the generated number of bins is within a factor of (1 + ) of optimal provided that the optimum number is sufficiently large. The running time of the algorithm is polynomial both in n and in .     

Facility Location with Dynamic Distance Functions

Facility location problems have always been studied with theassumption that the edge lengths in the network are static anddo not change over time. The underlying network could be used to model a city street networkfor emergency facility location/hospitals, or an electronic network for locating information centers. In any case, it is clear that due to trafficcongestion the traversal time on links changes with time. Very often, we have estimates as to how the edge lengths change over time, and our objective is to choose a set of locations (vertices) ascenters, such that at every time instant each vertex has a center close to it (clearly, the center close to a vertex may change over time). We also provide approximation algorithms as well as hardness results forthe K-center problem under this model. This is the first comprehensive study regarding approximation algorithmsfor facility location for good time-invariant solutions.     

Pipage Rounding: A New Method of Constructing Algorithms with Proven Performance Guarantee

The paper presents a general method of designing constant-factor approximation algorithms for some discrete optimization problems with assignment-type constraints. The core of the method is a simple deterministic procedure of rounding of linear relaxations (referred to as pipage rounding). With the help of the method we design approximation algorithms with better performance guarantees for some well-known problems including MAXIMUM COVERAGE, MAX CUT with given sizes of parts and some of their generalizations.