Approximation Algorithms for Degree-Constrained Minimum-Cost Network-Design Problems

We study network-design problems with two different design objectives: the total cost of the edges and nodes in the network and the maximum degree of any node in the network. A prototypical example is the degree-constrained node-weighted Steiner tree problem: We are given an undirected graph G(V,E) , with a nonnegative integral function d that specifies an upper bound d(v) on the degree of each vertex v ∈ V in the Steiner tree to be constructed, nonnegative costs on the nodes, and a subset of k nodes called terminals. The goal is to construct a Steiner tree T containing all the terminals such that the degree of any node v in T is at most the specified upper bound d(v) and the total cost of the nodes in T is minimum. Our main result is a bicriteria approximation algorithm whose output is approximate in terms of both the degree and cost criteria—the degree of any node v ∈ V in the output Steiner tree is O(d(v) log k) and the cost of the tree is O(log k) times that of a minimum-cost Steiner tree that obeys the degree bound d(v) for each node v . Our result extends to the more general problem of constructing one-connected networks such as generalized Steiner forests. We also consider the special case in which the edge costs obey the triangle inequality and present simple approximation algorithms with better performance guarantees. Received December 21, 1998; revised September 24, 1999.     

Approximation Algorithms for Quickest Spanning Tree Problems

Let $G=(V,E)$ be an undirected multigraph with a special vertex ${\it root} \in V$, and where each edge $e \in E$ is endowed with a length $l(e) \geq 0$ and a capacity $c(e) > 0$. For a path $P$ that connects $u$ and $v$, the {\it transmission time} of $P$ is defined as $t(P)=\mbox{\large$\Sigma$}_{e \in P} l(e) + \max_{e \in P}\!{(1 / c(e))}$. For a spanning tree $T$, let $P_{u,v}^T$ be the unique $u$--$v$ path in $T$. The {\sc quickest radius spanning tree problem} is to find a spanning tree $T$ of $G$ such that $\max _{v \in V} t(P^T_{root,v})$ is minimized. In this paper we present a 2-approximation algorithm for this problem, and show that unless $P =NP$, there is no approximation algorithm with a performance guarantee of $2 - \epsilon$ for any $\epsilon >0$. The {\sc quickest diameter spanning tree problem} is to find a spanning tree $T$ of $G$ such that $\max_{u,v \in V} t(P^T_{u,v})$ is minimized. We present a ${3 \over 2}$-approximation to this problem, and prove that unless $P=NP$ there is no approximation algorithm with a performance guarantee of ${3 \over 2}-\epsilon$ for any $\epsilon >0$.     

Approximation Algorithms for Quickest Spanning Tree Problems

Let $G=(V,E)$ be an undirected multigraph with a special vertex ${\it root} \in V$, and where each edge $e \in E$ is endowed with a length $l(e) \geq 0$ and a capacity $c(e) > 0$. For a path $P$ that connects $u$ and $v$, the {\it transmission time} of $P$ is defined as $t(P)=\mbox{\large$\Sigma$}_{e \in P} l(e) + \max_{e \in P}\!{(1 / c(e))}$. For a spanning tree $T$, let $P_{u,v}^T$ be the unique $u$--$v$ path in $T$. The {\sc quickest radius spanning tree problem} is to find a spanning tree $T$ of $G$ such that $\max _{v \in V} t(P^T_{root,v})$ is minimized. In this paper we present a 2-approximation algorithm for this problem, and show that unless $P =NP$, there is no approximation algorithm with a performance guarantee of $2 - \epsilon$ for any $\epsilon >0$. The {\sc quickest diameter spanning tree problem} is to find a spanning tree $T$ of $G$ such that $\max_{u,v \in V} t(P^T_{u,v})$ is minimized. We present a ${3 \over 2}$-approximation to this problem, and prove that unless $P=NP$ there is no approximation algorithm with a performance guarantee of ${3 \over 2}-\epsilon$ for any $\epsilon >0$.     

Approximation Algorithms for Multi-Criteria Traveling Salesman Problems

We analyze approximation algorithms for several variants of the traveling salesman problem with multiple objective functions. First, we consider the symmetric TSP (STSP) with γ-triangle inequality. For this problem, we present a deterministic polynomial-time algorithm that achieves an approximation ratio of and a randomized approximation algorithm that achieves a ratio of . In particular, we obtain a 2+ε approximation for multi-criteria metric STSP. Then we show that multi-criteria cycle cover problems admit fully polynomial-time randomized approximation schemes. Based on these schemes, we present randomized approximation algorithms for STSP with γ-triangle inequality (ratio ), asymmetric TSP (ATSP) with γ-triangle inequality (ratio ), STSP with weights one and two (ratio 4/3) and ATSP with weights one and two (ratio 3/2). A preliminary version of this work has been presented at the 4th Workshop on Approximation and Online Algorithms (WAOA 2006) (Lecture Notes in Computer Science, vol. 4368, pp. 302–315, 2007). B. Manthey is supported by the Postdoc-Program of the German Academic Exchange Service (DAAD). He is on leave from Saarland University and has done part of the work at the Institute for Theoretical Computer Science of the University of Lübeck supported by DFG research grant RE 672/3 and at the Department of Computer Science at Saarland University.     

On-Line and Off-Line Approximation Algorithms for Vector Covering Problems

This paper deals with vector covering problems in d -dimensional space. The input to a vector covering problem consists of a set X of d -dimensional vectors in [0,1] ^d . The goal is to partition X into a maximum number of parts, subject to the constraint that in every part the sum of all vectors is at least one in every coordinate. This problem is known to be NP-complete, and we are mainly interested in its on-line and off-line approximability. For the on-line version, we construct approximation algorithms with worst case guarantee arbitrarily close to 1/(2d) in d≥ 2 dimensions. This result contradicts a statement of Csirik and Frenk in [5] where it is claimed that, for d≥ 2 , no on-line algorithm can have a worst case ratio better than zero. Moreover, we prove that, for d≥ 2 , no on-line algorithm can have a worst case ratio better than 2/(2d+1) . For the off-line version, we derive polynomial time approximation algorithms with worst case guarantee Θ(1/ log d) . For d=2 , we present a very fast and very simple off-line approximation algorithm that has worst case ratio 1/2 . Moreover, we show that a method from the area of compact vector summation can be used to construct off-line approximation algorithms with worst case ratio 1/d for every d≥ 2 . Received November 1996; revised March 1997.     

Approximation Algorithms for the Directed k-Tour and k-Stroll Problems

We consider two natural generalizations of the Asymmetric Traveling Salesman problem: the k-Stroll and the k-Tour problems. The input to the k-Stroll problem is a directed n-vertex graph with nonnegative edge lengths, an integer k, as well as two special vertices s and t. The goal is to find a minimum-length s-t walk, containing at least k distinct vertices (including the endpoints s,t). The k-Tour problem can be viewed as a special case of k-Stroll, where s=t. That is, the walk is required to be a tour, containing some pre-specified vertex s. When k=n, the k-Stroll problem becomes equivalent to Asymmetric Traveling Salesman Path, and k-Tour to Asymmetric Traveling Salesman. Our main result is a polylogarithmic approximation algorithm for the k-Stroll problem. Prior to our work, only bicriteria (O(log² k),3)-approximation algorithms have been known, producing walks whose length is bounded by 3OPT, while the number of vertices visited is Ω(k/log² k). We also show a simple O(log² n/loglogn)-approximation algorithm for the k-Tour problem. The best previously known approximation algorithms achieved min(O(log³ k),O(log² n?logk/loglogn)) approximation in polynomial time, and O(log² k) approximation in quasipolynomial time.     

Fixed-Parameter Algorithms for CLOSEST STRING and Related Problems

CLOSEST STRING is one of the core problems in the field of consensus word analysis with particular importance for computational biology. Given k strings of the same length and a nonnegative integer d , find a ``center string'' s such that none of the given strings has the Hamming distance greater than d from s . CLOSEST STRING is NP-complete. In biological applications, however, d is usually very small. We show how to solve CLOSEST STRING in linear time for fixed d —the exponential growth in d is bounded by O(dd) . We extend this result to the closely related problems d -MISMATCH and DISTINGUISHING STRING SELECTION. Moreover, we also show that CLOSEST STRING is solvable in linear time when k is fixed and d is arbitrary. In summary, this means that CLOSEST STRING is fixed-parameter tractable with respect to parameter d and with respect to parameter k . Finally, the practical usefulness of our findings is substantiated by some experimental results.     

Fixed-Parameter Algorithms for CLOSEST STRING and Related Problems

CLOSEST STRING is one of the core problems in the field of consensus word analysis with particular importance for computational biology. Given k strings of the same length and a nonnegative integer d , find a ``center string' s such that none of the given strings has the Hamming distance greater than d from s . CLOSEST STRING is NP-complete. In biological applications, however, d is usually very small. We show how to solve CLOSEST STRING in linear time for fixed d —the exponential growth in d is bounded by O(d ^d ) . We extend this result to the closely related problems d -MISMATCH and DISTINGUISHING STRING SELECTION. Moreover, we also show that CLOSEST STRING is solvable in linear time when k is fixed and d is arbitrary. In summary, this means that CLOSEST STRING is fixed-parameter tractable with respect to parameter d and with respect to parameter k . Finally, the practical usefulness of our findings is substantiated by some experimental results.     

Approximation Algorithms for the Traveling Repairman and Speeding Deliveryman Problems

Constant-factor, polynomial-time approximation algorithms are presented for two variations of the traveling salesman problem with time windows. In the first variation, the traveling repairman problem, the goal is to find a path that visits the maximum possible number of locations during their time windows. In the second variation, the speeding deliveryman problem, the goal is to find a path that uses the minimum possible speedup to visit all locations during their time windows. For both variations, the time windows are of unit length, and the distance metric is based on a weighted, undirected graph. Algorithms with improved approximation ratios are given for the case when the input is defined on a tree rather than a general graph. The algorithms are also extended to handle time windows whose lengths fall in any bounded range.     

一类弱集合覆盖问题的近似算法

在近似算法领域,集合覆盖（Set Cover）是研究的比较早和比较透彻的问题之一．该文提出了一类与集合覆盖很相似的问题：集合击中和弱集合b-覆盖,并且给出了解决它们的近似算法,还证明了它们的不可近似性．     

Exact and Approximation Algorithms for Geometric and Capacitated Set Cover Problems

First, we study geometric variants of the standard set cover motivated by assignment of directional antenna and shipping with deadlines, providing the first known polynomial-time exact solutions. Next, we consider the following general (non-necessarily geometric) capacitated set cover problem. There is given a set of elements with real weights and a family of sets of the elements. One can use a set if it is a subset of one of the sets in the family and the sum of the weights of its elements is at most one. The goal is to cover all the elements with the allowed sets. We show that any polynomial-time algorithm that approximates the uncapacitated version of the set cover problem with ratio r can be converted to an approximation algorithm for the capacitated version with ratio r+1.357. In particular, the composition of these two results yields a polynomial-time approximation algorithm for the problem of covering a set of customers represented by a weighted n-point set with a minimum number of antennas of variable angular range and fixed capacity with ratio?2.357. This substantially improves on the best known approximation ratio for the latter antenna problem equal to?3. Furthermore, we provide a PTAS for the dual problem where the number of sets (e.g., antennas) to use is fixed and the task is to minimize the maximum set load, in case the sets correspond to line intervals or arcs. Finally, we discuss the approximability of the generalization of the antenna problem to include several base stations for antennas, and in particular show its APX-hardness already in the uncapacitated case.     

Parallel Algorithms for Partitioning Sorted Sets and Related Problems

We consider the following partition problem: Given a set S of n elements that is organized as k sorted subsets of size n/k each and given a parameter h with 1/k ≤ h ≤ n/k , partition S into g = O(n/(hk)) subsets D ₁ , D ₂ , . . . , D _g of size Θ(hk) each, such that, for any two indices i and j with 1 ≤ i < j ≤ g , no element in D _i is bigger than any element in D _j . Note that with various combinations of the values of parameters h and k , several fundamental problems, such as merging, sorting, and finding an approximate median, can be formulated as or be reduced to this partition problem. The partition problem also finds many applications in solving problems of parallel computing and computational geometry. In this paper we present efficient parallel algorithms for solving the partition problem and a number of its applications. Our parallel partition algorithm runs in O( log n) time using processors in the EREW PRAM model. The complexity bounds of our parallel partition algorithm on the respective special cases match those of the optimal EREW PRAM algorithms for merging, sorting, and finding an approximate median. Using our parallel partition algorithm, we are also able to obtain better complexity bounds (even possibly on a weaker parallel model) than the previously best known parallel algorithms for several important problems, including parallel multiselection, parallel multiranking, and parallel sorting of k sorted subsets. Received May 5, 1996; revised July 30, 1998.     

Subquadratic Approximation Algorithms for Clustering Problems in High Dimensional Spaces

One of the central problems in information retrieval, data mining, computational biology, statistical analysis, computer vision, geographic analysis, pattern recognition, distributed protocols is the question of classification of data according to some clustering rule. Often the data is noisy and even approximate classification is of extreme importance. The difficulty of such classification stems from the fact that usually the data has many incomparable attributes, and often results in the question of clustering problems in high dimensional spaces. Since they require measuring distance between every pair of data points, standard algorithms for computing the exact clustering solutions use quadratic or “nearly quadratic” running time; i.e., O(dn ^2?α(d)) time where n is the number of data points, d is the dimension of the space and α(d) approaches 0 as d grows. In this paper, we show (for three fairly natural clustering rules) that computing an approximate solution can be done much more efficiently. More specifically, for agglomerative clustering (used, for example, in the Alta Vista? search engine), for the clustering defined by sparse partitions, and for a clustering based on minimum spanning trees we derive randomized (1 + ∈) approximation algorithms with running times Õ(d ² n ^2?γ) where γ > 0 depends only on the approximation parameter ∈ and is independent of the dimension d.     

Simple Algorithms for the On-Line Multidimensional Dictionary and Related Problems

The on-line multidimensional dictionary problem consists of executing on-line any sequence of the following operations: INSERT(p) , DELETE(p) , and MEM-BER-SHIP(p) , where p is any (ordered) d -tuple (or string with d elements, or points in d -space where the dimensions have been ordered). We introduce a clean structure based on balanced binary search trees, which we call multidimensional balanced binary search trees, to represent the set of d -tuples. We present algorithms for each of the above operations that take O(d + log n) time, where n is the current number of d -tuples in the set, and each INSERT and DELETE operation requires no more than a constant number of rotations. Our structure requires dn words to represent the input, plus O(n) pointers and data indicating the first component where pairs of d -tuples differ. This information, which can be easily updated, enables us to test for MEMBERSHIP efficiently. Other operations that can be performed efficiently in our multidimensional balanced binary search trees are: print in lexicographic order (O(dn) time), find the (lexicographically) smallest or largest d -tuple (O( log n) time), and concatenation (O(d + log n) time). Finding the (lexicographically) k th smallest or largest d -tuple can also be implemented efficiently (O( log n) time), at the expense of adding an integer value at each node. Received June 13, 1997; revised September 3, 1998.     

Approximation Algorithms for Treewidth

This paper presents algorithms whose input is an undirected graph, and whose output is a tree decomposition of width that approximates the optimal, the treewidth of that graph. The algorithms differ in their computation time and their approximation guarantees. The first algorithm works in polynomial-time and finds a factor-O(log OPT) approximation, where OPT is the treewidth of the graph. This is the first polynomial-time algorithm that approximates the optimal by a factor that does not depend on n, the number of nodes in the input graph. As a result, we get an algorithm for finding pathwidth within a factor of O(log OPT⋅log n) from the optimal. We also present algorithms that approximate the treewidth of a graph by constant factors of 3.66, 4, and 4.5, respectively and take time that is exponential in the treewidth. These are more efficient than previously known algorithms by an exponential factor, and are of practical interest. Finding triangulations of minimum treewidth for graphs is central to many problems in computer science. Real-world problems in artificial intelligence, VLSI design and databases are efficiently solvable if we have an efficient approximation algorithm for them. Many of those applications rely on weighted graphs. We extend our results to weighted graphs and weighted treewidth, showing similar approximation results for this more general notion. We report on experimental results confirming the effectiveness of our algorithms for large graphs associated with real-world problems.     

Fast Multiresolution Algorithms and Their Related Variational Problems for Image Denoising

When considering fast multiresolution techniques for image denoising problems, there are three important aspects. The first one is the choice of the specific multiresolution, the second one the choice of a proper filter function and the third one the choice of the thresholding parameter. Starting from the classical one, namely, linear wavelet algorithms with Donoho and Johnstone’s Soft-thresholding with the universal shrinkage parameter, the first aim of this paper is to improve it in the three mentioned directions. Thus, a new nonlinear approach is proposed and analyzed. On the other hand, the linear approach of Donoho and Johnstone is related with a well known variational problem. Our second aim is to find a related variational problem, more adapted to the denoising problem, for the new approach. We would like to mention that the analysis of theoretical properties in a nonlinear setting are usually notoriously more difficult. Finally, a comparison with other approaches, including linear and nonlinear multiresolution schemes, SVD-based schemes and filters with a non-multiresolution nature, is presented.     

Exact and Approximation Algorithms for Clustering

In this paper we present an n^ O(k ^1-1/d ) -time algorithm for solving the k -center problem in \reals ^d , under L _∈ fty - and L ₂ -metrics. The algorithm extends to other metrics, and to the discrete k -center problem. We also describe a simple (1+ɛ) -approximation algorithm for the k -center problem, with running time O(nlog k) + (k/ɛ)^ O(k ^1-1/d ) . Finally, we present an n^ O(k ^1-1/d ) -time algorithm for solving the L -capacitated k -center problem, provided that L=Ω(n/k ^1-1/d ) or L=O(1) . Received July 25, 2000; revised April 6, 2001.     

Improved Approximation Algorithms for the Min-max Tree Cover and Bounded Tree Cover Problems

In this paper we provide improved approximation algorithms for the Min-Max Tree Cover and Bounded Tree Cover problems. Given a graph G=(V,E) with weights w:E→?⁺, a set T ₁,T ₂,…,T _k of subtrees of G is called a tree cover of G if $V=\bigcup_{i=1}^{k} V(T_{i})$ . In the Min-Max k-tree Cover problem we are given graph G and a positive integer k and the goal is to find a tree cover with k trees, such that the weight of the largest tree in the cover is minimized. We present a 3-approximation algorithm for this improving the two different approximation algorithms presented in Arkin et al. (J. Algorithms 59:1–18, 2006) and Even et al. (Oper. Res. Lett. 32(4):309–315, 2004) with ratio 4. The problem is known to have an APX-hardness lower bound of $\frac{3}{2}$ (Xu and Wen in Oper. Res. Lett. 38:169–173, 2010). In the Bounded Tree Cover problem we are given graph G and a bound λ and the goal is to find a tree cover with minimum number of trees such that each tree has weight at most λ. We present a 2.5-approximation algorithm for this, improving the 3-approximation bound in Arkin et al. (J. Algorithms 59:1–18, 2006).