A constant approximation algorithm for the densest k-subgraph problem on chordal graphs

The densest k-subgraph (DkS) problem asks for a k-vertex subgraph of a given graph with the maximum number of edges. The DkS problem is NP-hard even for special graph classes including bipartite, planar, comparability and chordal graphs, while no constant approximation algorithm is known for any of these classes. In this paper we present a 3-approximation algorithm for the class of chordal graphs. The analysis of our algorithm is based on a graph theoretic lemma of independent interest.     

Linear structure of bipartite permutation graphs and the longest path problem

The class of bipartite permutation graphs is the intersection of two well known graph classes: bipartite graphs and permutation graphs. A complete bipartite decomposition of a bipartite permutation graph is proposed in this note. The decomposition gives a linear structure of bipartite permutation graphs, and it can be obtained in O(n) time, where n is the number of vertices. As an application of the decomposition, we show an O(n) time and space algorithm for finding a longest path in a bipartite permutation graph.     

A simple algorithm to find Hamiltonian cycles in proper interval graphs

We present an algorithm to find a Hamiltonian cycle in a proper interval graph in O(m+n) time, where m is the number of edges and n is the number of vertices in the graph. The algorithm is simpler and shorter than previous algorithms for the problem.     

A parallel algorithm for generating bicompatible elimination orderings of proper interval graphs

In this paper, we first show how a certain ordering of vertices, called bicompatible elimination ordering (BCO), of a proper interval graph can be used to solve efficiently several problems in proper interval graphs. We, then, propose an NC parallel algorithm (i.e., polylogarithmic-time employing a polynomial number of processors) in SIMD CRCW PRAM (Single Instruction Stream Multiple Data Stream Concurrent Read Concurrent Write Parallel Random Access Machine) model of parallel computation to compute a BCO of a proper interval graph. To the best of our knowledge, this is the first NC parallel algorithm to compute a BCO of a proper interval graph.     

The k-neighbourhood-covering problem on interval graphs

Let G=(V, E) be a simple connected graph and k be a fixed positive integer. A vertex w is said to be a k-neighbourhood-cover (kNC) of an edge (u, v) if d(u, w)≤k and d(v, w)≤k. A set C ? V is called a kNC set if every edge in E is kNC by some vertices of C. The decision problem associated with this problem is NP-complete for general graphs and it remains NP-complete for chordal graphs. In this article, we design an O(n) time algorithm to solve minimum kNC problem on interval graphs by using a data structure called interval tree.     

Improved approximation algorithms for minimum AND-circuits problem via k-set cover

Arpe and Manthey [J. Arpe, B. Manthey, Approximability of minimum AND-circuits, Algorithmica 53 (3) (2009) 337-357] recently studied the minimum AND-circuit problem, which is a circuit minimization problem, and showed some results including approximation algorithms, APX-hardness and fixed parameter tractability of the problem. In this note, we show that algorithms via the k-set cover problem yield improved approximation ratios for the minimum AND-circuit problem with maximum degree three. In particular, we obtain an approximation ratio of 1.199 for the problem with maximum degree three and unbounded multiplicity.     

Approximation algorithm for coloring of dotted interval graphs

Dotted interval graphs were introduced by Aumann et al. [Y. Aumann, M. Lewenstein, O. Melamud, R. Pinter, Z. Yakhini, Dotted interval graphs and high throughput genotyping, in: ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 339-348] as a generalization of interval graphs. The problem of coloring these graphs found application in high-throughput genotyping. Jiang [M. Jiang, Approximating minimum coloring and maximum independent set in dotted interval graphs, Information Processing Letters 98 (2006) 29-33] improves the approximation ratio of Aumann et al. [Y. Aumann, M. Lewenstein, O. Melamud, R. Pinter, Z. Yakhini, Dotted interval graphs and high throughput genotyping, in: ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 339-348]. In this work we improve the approximation ratio of Jiang [M. Jiang, Approximating minimum coloring and maximum independent set in dotted interval graphs, Information Processing Letters 98 (2006) 29-33] and Aumann et al. [Y. Aumann, M. Lewenstein, O. Melamud, R. Pinter, Z. Yakhini, Dotted interval graphs and high throughput genotyping, in: ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 339-348]. In the exposition we develop a generalization of the problem of finding the maximum number of non-attacking queens on a triangle.     

On approximation algorithms for the terminal Steiner tree problem

The terminal Steiner tree problem is a special version of the Steiner tree problem, where a Steiner minimum tree has to be found in which all terminals are leaves. We prove that no polynomial time approximation algorithm for the terminal Steiner tree problem can achieve an approximation ratio less than (1−o(1))lnn unless NP has slightly superpolynomial time algorithms. Moreover, we present a polynomial time approximation algorithm for the metric version of this problem with a performance ratio of 2ρ, where ρ denotes the best known approximation ratio for the Steiner tree problem. This improves the previously best known approximation ratio for the metric terminal Steiner tree problem of ρ+2.     

Improved approximation algorithms for the Max Edge-Coloring problem

The Max Edge-Coloring problem asks for a proper edge-coloring of an edge-weighted graph minimizing the sum of the weights of the heaviest edges in the color classes. In this paper we present a PTAS for trees and a 1.74-approximation algorithm for bipartite graphs; we also adapt the last algorithm to one for general graphs of the same, asymptotically, approximation ratio.     

Approximating minimum coloring and maximum independent set in dotted interval graphs

Dotted interval graphs are introduced by Aumann et al. as a generalization of interval graphs. We study two optimization problems in dotted interval graphs that find application in high-throughput genotyping. We present improved approximations for minimum coloring and the first approximation for maximum independent set in dotted interval graphs.     

The reverse greedy algorithm for the metric k-median problem

The Reverse Greedy algorithm (RGreedy) for the k-median problem works as follows. It starts by placing facilities on all nodes. At each step, it removes a facility to minimize the total distance to the remaining facilities. It stops when k facilities remain. We prove that, if the distance function is metric, then the approximation ratio of RGreedy is between Ω(logn/loglogn) and O(logn).     

An improved lower bound on approximation algorithms for the Closest Substring problem

The Closest Substring problem (the CSP problem) is a basic NP-hard problem in the study of computational biology. It is known that the problem has polynomial time approximation schemes. In this paper, we prove that unless the Exponential Time Hypothesis fails, the CSP problem has no polynomial time approximation schemes of running time f(1/ε)n^o(1/ε) for any function f. This essentially excludes the possibility that the CSP problem has a practical polynomial time approximation scheme even for moderate values of the error bound ε. As a consequence, it is unlikely that the study of approximation schemes for the CSP problem in the literature would lead to practical approximation algorithms for the problem for small error bound ε.     

Parallel computation on interval graphs: algorithms and experiments

This paper describes efficient coarse‐grained parallel algorithms and implementations for a suite of interval graph problems. Included are algorithms requiring only a constant number of communication rounds for connected components, maximum weighted clique, and breadth‐first‐search and depth‐first‐search trees, as well as communication rounds algorithms for optimization problems such as minimum interval covering, maximum independent set and minimum dominating set, where is the number of processors in the parallel system. This implies that the number of communication rounds is independent of the problem size. Implementations of these algorithms are evaluated on parallel clusters, using both Fast Ethernet and Myrinet interconnection networks, and on a CRAY T3E parallel multicomputer, with extensive experimental results being presented and analyzed. Copyright © 2002 John Wiley & Sons, Ltd.     

A note on improving the performance of approximation algorithms for radiation therapy

The segment minimization problem consists of representing an integer matrix as the sum of the fewest number of integer matrices each of which have the property that the non-zeroes in each row are consecutive. This has direct applications to an effective form of cancer treatment. Using several insights, we extend previous results to obtain constant-factor improvements in the approximation guarantees. We show that these improvements yield better performance by providing an experimental evaluation of all known approximation algorithms using both synthetic and real-world clinical data. Our algorithms are superior for 76% of instances and we argue for their utility alongside the heuristic approaches used in practice.     

A note on “An approximation algorithm for the load-balanced semi-matching problem in weighted bipartite graphs”

We point out an error in the algorithm for the Load Balanced Semi-Matching Problem presented by C.P. Low [C.P. Low, An approximation algorithm for the load-balanced semi-matching problem in weighted bipartite graphs, Information Processing Letters 100 (2006) 154-161]. This problem is equivalent to a parallel machine scheduling problem subject to eligibility constraints, in which each job has a pre-determined set of machines capable of processing the job.     

A note on t-complementing permutations for graphs

In this note we present a necessary and sufficient condition for a permutation to be t-complementing which is a natural generalization of the well-known result concerning self-complementing permutations.     

Fast algorithms for the dominating set problem on permutation graphs

AnO(n log logn) (resp.O(n² log² n)) algorithm is presented to solve the minimum cardinality (resp. weight) dominating set problem on permutation graphs, assuming the input is a permutation. The best-known previous algorithm was given by FÄrber and Keil, where they use dynamic programming to get anO(n² (resp.O(n³)) algorithm. Our improvement is based on the following three factors: (1) an observation on the order among the intermediate terms in the dynamic programming, (2) a new construction formula for the intermediate terms, and (3) efficient data structures for manipulating these terms.This research was supported in part by the National Science Foundation under Grant CCR-8905415 to Northwestern University.