Linear Time Approximation Algorithm Research Articles

Crossing Numbers of Graphs with Rotation Systems

We show that computing the crossing number and the odd crossing number of a graph with a given rotation system is NP-complete. As a consequence we can show that many of the well-known crossing number notions are NP-complete even if restricted to cubic graphs (with or without rotation system). In particular, we can show that Tutte’s independent odd crossing number is NP-complete, and we obtain a new and simpler proof of Hliněný’s result that computing the crossing number of a cubic graph is NP-complete. We also consider the special case of multigraphs with rotation systems on a fixed number k of vertices. For k=1 we give an O(mlog m) algorithm, where m is the number of edges, and for loopless multigraphs on 2 vertices we present a linear time 2-approximation algorithm. In both cases there are interesting connections to edit-distance problems on (cyclic) strings. For larger k we show how to approximate the crossing number to within a factor of ${k+4\choose4}/5$ in time O(m k log m) on a graph with m edges.

New Approximation Algorithms for Minimum Cycle Bases of Graphs

We consider the problem of computing an approximate minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices; the extension to directed graphs is also discussed. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over $\mathbb{F}_{2}$ generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Cycle bases of low weight are useful in a number of contexts, e.g. the analysis of electrical networks, structural engineering, chemistry, and surface reconstruction. Although in most such applications any cycle basis can be used, a low weight cycle basis often translates to better performance and/or numerical stability. Despite the fact that the problem can be solved exactly in polynomial time, we design approximation algorithms since the performance of the exact algorithms may be too expensive for some practical applications. We present two new algorithms to compute an approximate minimum cycle basis. For any integer k≥1, we give (2k−1)-approximation algorithms with expected running time O(kmn 1+2/k +mn (1+1/k)(ω−1)) and deterministic running time O(n 3+2/k ), respectively. Here ω is the best exponent of matrix multiplication. It is presently known that ω<2.376. Both algorithms are o(m ω ) for dense graphs. This is the first time that any algorithm which computes sparse cycle bases with a guarantee drops below the Θ(m ω ) bound. We also present a 2-approximation algorithm with expected running time $O(m^{\omega}\sqrt{n\log n})$ , a linear time 2-approximation algorithm for planar graphs and an O(n 3) time 2.42-approximation algorithm for the complete Euclidean graph in the plane.

Linear time 3-approximation for the MAST problem

Given a set of leaf-labeled trees with identical leaf sets, the well-known Maximum Agreement SubTree (MAST) problem consists in finding a subtree homeomorphically included in all input trees and with the largest number of leaves. MAST and its variant called Maximum Compatible Tree (MCT) are of particular interest in computational biology. This article presents a linear-time approximation algorithm to solve the complement version of MAST, namely identifying the smallest set of leaves to remove from input trees to obtain isomorphic trees. We also present an O ( n 2 + kn ) algorithm to solve the complement version of MCT. For both problems, we thus achieve significantly lower running times than previously known algorithms. Fast running times are especially important in phylogenetics where large collections of trees are routinely produced by resampling procedures, such as the nonparametric bootstrap or Bayesian MCMC methods.

L(h, 1, 1)-labeling of outerplanar graphs

An L(h, 1, 1)-labeling of a graph is an assignment of labels from the set of integers {0, . . . , λ} to the nodes of the graph such that adjacent nodes are assigned integers of at least distance h ≥ 1 apart and all nodes of distance three or less must be assigned different labels. The aim of the L(h, 1, 1)-labeling problem is to minimize λ, denoted by λ h, 1, 1 and called span of the L(h, 1, 1)-labeling. As outerplanar graphs have bounded treewidth, the L(1, 1, 1)-labeling problem on outerplanar graphs can be exactly solved in O(n 3), but the multiplicative factor depends on the maximum degree Δ and is too big to be of practical use. In this paper we give a linear time approximation algorithm for computing the more general L(h, 1, 1)-labeling for outerplanar graphs that is within additive constants of the optimum values.

On k-connectivity problems with sharpened triangle inequality

The k-connectivity problem is to find a minimum-cost k-edge- or k-vertex-connected spanning subgraph of an edge-weighted, undirected graph G for any given G and k. Here, we consider its NP-hard subproblems with respect to the parameter β, with 1 2 < β < 1 , where G = ( V , E ) is a complete graph with a cost function c satisfying the sharpened triangle inequality c ( { u , v } ) ⩽ β ⋅ ( c ( { u , w } ) + c ( { w , v } ) ) for all u , v , w ∈ V . First, we give a simple linear-time approximation algorithm for these optimization problems with approximation ratio β 1 − β for any 1 2 ⩽ β < 1 , which improves the known approximation ratios for 1 2 < β < 2 3 . The analysis of the algorithm above is based on a rough combinatorial argumentation. As the main result of this paper, for k = 3 , we sophisticate the combinatorial consideration in order to design a ( 1 + 5 ( 2 β − 1 ) 9 ( 1 − β ) + O ( 1 | V | ) ) -approximation algorithm for the 3-connectivity problem on graphs satisfying the sharpened triangle inequality for 1 2 ⩽ β ⩽ 2 3 . As part of the proof, we show that for each spanning 3-edge-connected subgraph H, there exists a spanning 3-regular 2-vertex-connected subgraph H ′ of at most the same cost, and H can be transformed into H ′ efficiently.

A Linear Time Approximation Algorithm for Ruler Folding Problem

A Linear Time Approximation Algorithm for Ruler Folding Problem

Almost optimal solutions for bin coloring problems

In this paper we study two interesting bin coloring problems: Minimum Bin Coloring Problem (MinBC) and Online Maximum Bin Coloring Problem (OMaxBC), motivated from several applications in networking. For the MinBC problem, we present two near linear time approximation algorithms to achieve almost optimal solutions, i.e., no more than OPT+2 and OPT+1 respectively, where OPT is the optimal solution. For the OMaxBC problem, we first introduce a deterministic 2-competitive greedy algorithm, and then give lower bounds for any deterministic and randomized (against adaptive offline adversary) online algorithms. The lower bounds show that our deterministic algorithm achieves the best possible competitive ratio.

Linear time algorithms for approximating the facility terminal cover problem

AbstractIn this paper, we consider an interesting generalization of the weighted vertex cover problem, called the Facility Terminal Cover (FTC) problem. In the FTC problem, each vertex is associated with a positive weight, each edge is associated with a positive demand, and the objective is to determine a subset of vertices and a capacity for each selected vertex so that the demand of each edge is covered by the capacity of one of its two endpoints and the total weighted capacity of all selected vertices is minimized. The FTC problem is motivated by several key network optimization problems, such as the power assignment problem in ad hoc networks, and could be used as a subroutine to solve such problems. No quality‐guaranteed solution is previously known for the FTC problem. In this paper, we present two linear time approximation algorithms for this problem. Our first algorithm achieves deterministically an approximation ratio of 8 by using an interesting rounding technique and a lower‐bounding technique. Based on interesting randomization techniques, our second algorithm further improves the approximation ratio to 2e, where e is the natural logarithmic base. The second algorithm can be easily derandomized in quadratic time. Our algorithms are relatively simple and can be easily implemented for networking applications. Experiments show that the two algorithms behave rather similarly, especially in large‐size graphs, indicating that the solutions yielded by one or both algorithms are much closer to the optimum. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 118–126 2007

A simple linear time approximation algorithm for multi-processor job scheduling on four processors

Multiprocessor job scheduling problem has become increasingly interesting, for both theoretical study and practical applications. Theoretical study of the problem has made significant progress recently, which, however, seems not to imply practical algorithms for the problem, yet. Practical algorithms have been developed only for systems with three processors and the techniques seem difficult to extend to systems with more than three processors. This paper offers new observations and introduces new techniques for the multiprocessor job scheduling problem on systems with four processors. A very simple and practical linear time approximation algorithm of ratio bounded by 1.5 is developed for the multi-processor job scheduling problem P 4|fix|C max, which significantly improves previous results. Our techniques are also useful for multiprocessor job scheduling problems on systems with more than four processors.

Linear-time algorithms for problems on planar graphs with fixed disk dimension

The disk dimension of a planar graph G is the least number k for which G embeds in the plane minus k open disks, with every vertex on the boundary of some disk. Useful properties of graphs with a given disk dimension are derived, leading to an algorithm to obtain an outerplanar subgraph of a graph with disk dimension k by removing at most 2 k − 2 vertices. This reduction is used to obtain linear-time exact and approximation algorithms on graphs with fixed disk dimension. In particular, a linear-time approximation algorithm is presented for the pathwidth problem.

A linear-time approximation algorithm for weighted matchings in graphs

Approximation algorithms have so far mainly been studied for problems that are not known to have polynomial time algorithms for solving them exactly. Here we propose an approximation algorithm for the weighted matching problem in graphs which can be solved in polynomial time. The weighted matching problem is to find a matching in an edge weighted graph that has maximum weight. The first polynomial-time algorithm for this problem was given by Edmonds in 1965. The fastest known algorithm for the weighted matching problem has a running time of O ( nm + n 2 log n ). Many real world problems require graphs of such large size that this running time is too costly. Therefore, there is considerable need for faster approximation algorithms for the weighted matching problem. We present a linear-time approximation algorithm for the weighted matching problem with a performance ratio arbitrarily close to 2/3. This improves the previously best performance ratio of 1/2. Our algorithm is not only of theoretical interest, but because it is easy to implement and the constants involved are quite small it is also useful in practice.

A linear time approximation algorithm for movement minimization in conveyor flow shop processing

We consider the movement minimization problem in a conveyor flow shop processing controlled by one worker for all machines. A machine can only execute tasks if the worker is present. Each machine can serve as a buffer. The worker has to cover a certain distance to move from one machine to the other. The distance between two machines P p and P q is | p− q|. The objective is to minimize the total distance the worker has to cover for the processing of all jobs. We introduce a linear time approximation algorithm for the conveyor flow shop problem with performance 3. Such minimization problems usually appear in conveyor controlled manufacturing systems.

A fully linear-time approximation algorithm for grammar-based compression

A linear-time approximation algorithm for the grammar-based compression is presented. This is an optimization problem to minimize the size of a context-free grammar deriving a given string. For each string of length n, the algorithm guarantees O ( log n g ∗ ) approximation ratio without suffix tree construction.

Radio Labeling with Preassigned Frequencies

A radio labeling of a graph G is an assignment of pairwise distinct, positive integer labels to the vertices of G such that labels of adjacent vertices differ by at least 2. The radio labeling problem (RL) consists in determining a radio labeling that minimizes the maximum label that is used (the so-called span of the labeling). RL is a well-studied problem, mainly motivated by frequency assignment problems in which transmitters are not allowed to operate on the same frequency channel. We consider the special case where some of the transmitters have preassigned operating frequency channels. This leads to the natural variants p-RL(l) and p-RL(*) of RL with l preassigned labels and an arbitrary number of preassigned labels, respectively. We establish a number of combinatorial, algorithmical, and complexity-theoretical results for these variants of radio labeling. In particular, we investigate a simple upper bound on the minimum span, yielding a linear time approximation algorithm with a constant additive error bound for p-RL(*) restricted to graphs with girth $\geq 5$. We consider the complexity of p-RL(l) and p-RL(*) for several cases in which RL is known to be polynomially solvable. On the negative side, we prove that p-RL(*) is NP-hard for cographs and for k-colorable graphs where a k-coloring is given ($k\geq3$). On the positive side, we derive polynomial time algorithms solving p-RL(*)} and p-RL(l) for graphs with bounded maximum degree, and for solving p-RL(l) for k-colorable graphs where a k-coloring is given.

On minimizing the maximum congestion for Weighted Hypergraph Embedding in a Cycle

The problem of Weighted Hypergraph Embedding in a Cycle (WHEC) is to embed the weighted hyperedges of a hypergraph as adjacent paths around a cycle, such that the maximum congestion over any physical link in the cycle is minimized. In this paper, we first show that even when hyperedges contain exactly two vertices, the WHEC problem is NP-complete. Afterwards we formulate the problem as an Integer Linear Program (ILP). Then, a solution with approximation ratio of two is presented by using LP-based rounding algorithm. Finally, to improve the efficiency, we develop a linear-time approximation algorithm to provide an embedding with congestion at most two times the optimum.

A simple approximation algorithm for the weighted matching problem

We present a linear time approximation algorithm with a performance ratio of 1/2 for finding a maximum weight matching in an arbitrary graph. Such a result is already known and is due to Preis [STACS'99, Lecture Notes in Comput. Sci., Vol. 1563, 1999, pp. 259–269]. Our algorithm uses a new approach which is much simpler than the one given by Preis and needs no amortized analysis for its running time.

An approximation algorithm with performance guarantees for the maximum traveling salesman problem on special matrices

In this paper, we investigate the maximum traveling salesman problem (Max-TSP) on quasi-banded matrices. A matrix is quasi-banded with multiplier α if all its elements contained within a band of several diagonals above and below the principal diagonal are non-zero, and any element in the band is at least α times larger than the maximal element outside the band. We investigate the properties of the Max-TSP on the quasi-banded matrices, prove that it is strongly NP-hard and derive a linear-time approximation algorithm with a guaranteed performance.

Scheduling Independent Multiprocessor Tasks

We study the problem of scheduling a set of n independent multiprocessor tasks with prespecified processor allocations on a fixed number of processors. We propose a linear time algorithm that finds a schedule of minimum makespan in the preemptive model, and a linear time approximation algorithm that finds a schedule of makespan within a factor of (1+\eps) of optimal in the non-preemptive model. We extend our results by obtaining a polynomial time approximation scheme for the parallel processors variant of the multiprocessor task model.

APPROXIMATING THE DIAMETER, WIDTH, SMALLEST ENCLOSING CYLINDER, AND MINIMUM-WIDTH ANNULUS

We study (1+ε)-factor approximation algorithms for several well-known optimization problems on a given n-point set: (a) diameter, (b) width, (c) smallest enclosing cylinder, and (d) minimum-width annulus. Among our results are new simple algorithms for (a) and (c) with an improved dependence of the running time on ε, as well as the first linear-time approximation algorithm for (d) in any fixed dimension. All four problems can be solved within a time bound of the form O(n+ε-c) or O(n log (1/ε)+ε-c).

Linear time-approximation algorithms for bin packing

Simchi-Levi (Naval Res. Logist. 41 (1994) 579–585) proved that the famous bin packing algorithms FF and BF have an absolute worst-case ratio of no more than 7 4 , and FFD and BFD have an absolute worst-case ratio of 3 2 , respectively. These algorithms run in time O(n log n) . In this paper, we provide a linear time constant space (number of bins kept during the execution of the algorithm is constant) off-line approximation algorithms with absolute worst-case ratio 3 2 . This result is best possible unless P=NP. Furthermore, we present a linear time constant space on-line algorithm and prove that the absolute worst-case ratio is 7 4 .