On sorting by 3-bounded transpositions

Heath and Vergara [Sorting by short block moves, Algorithmica 28 (2000) 323-352] proved the equivalence between sorting by 3-bounded transpositions and sorting by correcting skips and correcting hops. This paper explores various algorithmic as well as combinatorial aspects of correcting skips/hops, with the aim of understanding 3-bounded transpositions better.We show that to sort any permutation via correcting hops and skips, ⌊n/2⌋ correcting skips suffice. We also present a tighter analysis of the approximation algorithm of Heath and Vergara, and a possible simplification. Along the way, we study the class H_n of those permutations of S_n which can be sorted using correcting hops alone, and characterize large subsets of this class. We obtain a combinatorial characterization of the set G_n⊆S_n of all correcting-hop-free permutations, and describe a linear-time algorithm to optimally sort such permutations. We also show how to efficiently sort a permutation with a minimum number of correcting moves.     

A primal-dual method for approximating tree cover with two weights

The tree cover (TC) problem is to compute a minimum weight connected edge set, given a connected and edge-weighted graph G, such that its vertex set forms a vertex cover for G. Unlike related problems of vertex cover or edge dominating set, weighted TC is not yet known to be approximable in polynomial time as well as the unweighted version is. Moreover, the best approximation algorithm known so far for weighted TC is far from practical in its efficiency. In this paper we consider a restricted version of weighted TC, as a first step towards better approximation of general TC, where only two edge weights differing by at least a factor of 2 are available. It will be shown that a factor 2 approximation can be attained efficiently (in the complexity of max flow) in this case by a primal-dual method. Even under the limited weights as such, the primal-dual arguments used will be seen to be quite involved, having a nontrivial style of dual assignments as an essential part, unlike the case of uniform weights.     

Threshold-based preprocessing for approximating the weighted dense k-subgraph problem

Based on an application in forestry, we study the dense k  -subgraph problem: Given a parameter k∈N$k\in \mathbb{N}$ and an undirected weighted graph G, the task is to find a subgraph of G with k vertices such that the sum of the weights of the induced edges is maximized. The problem is well-known to be NP-hard and difficult to approximate if the underlying graph does not satisfy the triangle inequality.     

An approximation algorithm for sorting by reversals and transpositions

Genome rearrangement algorithms are powerful tools to analyze gene orders in molecular evolution. Analysis of genomes evolving by reversals and transpositions leads to a combinatorial problem of sorting by reversals and transpositions, the problem of finding a shortest sequence of reversals and transpositions that sorts one genome into the other. In this paper we present a 2k-approximation algorithm for sorting by reversals and transpositions for unsigned permutations where k is the approximation ratio of the algorithm used for cycle decomposition. For the best known value of k our approximation ratio becomes 2.8386+δ for any δ>0. We also derive a lower bound on reversal and transposition distance of an unsigned permutation.     

R-bounded approximating sequences and applications to semigroups

It is shown that on certain Banach spaces, including C[0,1] and L₁[0,1], there is no strongly continuous semigroup (T_t)_0<t<1 consisting of weakly compact operators such that (T_t)_0<t<1 is an R-bounded family. More general results concerning approximating sequences are included and some variants of R-boundedness are also discussed.     

Approximating reversal distance for strings with bounded number of duplicates

For a string A=a₁…a_n, a reversalρ(i,j), 1?i?j?n, transforms the string A into a string A^′=a₁…a_i-1a_ja_j-1…a_ia_j+1…a_n, that is, the reversal ρ(i,j) reverses the order of symbols in the substring a_i…a_j of A. In the case of signed strings, where each symbol is given a sign + or -, the reversal operation also flips the sign of each symbol in the reversed substring. Given two strings, A and B, signed or unsigned, sorting by reversals (SBR) is the problem of finding the minimum number of reversals that transform the string A into the string B.Traditionally, the problem was studied for permutations, that is, for strings in which every symbol appears exactly once. We consider a generalization of the problem, k-SBR, and allow each symbol to appear at most k times in each string, for some k?1. The main result of the paper is an O(k²)-approximation algorithm running in time O(n). For instances with , this is the best known approximation algorithm for k-SBR and, moreover, it is faster than the previous best approximation algorithm.     

The complexity of approximating a nonlinear program

We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadratic programming) there is no polynomial time approximation unless NP is contained in quasi-polynomial time.Our results rely on recent advances in the theory of interactive proof systems. They exemplify an interesting interplay of discrete and continuous mathematics—using a combinatorial argument to get a hardness result for a continuous optimization problem.     

Short proofs for cut-and-paste sorting of permutations

We consider the problem of determining the maximum number of moves required to sort a permutation of [n] using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of [n] can be transformed to the identity in at most ⌊2n/3⌋ such moves and that some permutations require at least ⌊n/2⌋ moves.     

A ternary 4-point approximating subdivision scheme

In the implementation of subdivision scheme, three of the most important issues are smoothness, size of support, and approximation order. Our objective is to introduce an improved ternary 4-point approximating subdivision scheme derived from cubic polynomial interpolation, which has smaller support and higher smoothness, comparing to binary 4-point and 6-point schemes, ternary 3-point and 4-point schemes (see Table 2). The method is easily generalized to ternary (2n + 2)-point approximating subdivision schemes. We choose a ternary scheme because a way to get smaller support is to raise arity. And we use polynomial reproduction to get higher approximation order easily.     

A new approach to multi-criteria sorting based on fuzzy outranking relations: The THESEUS method

In this paper, we propose the THESEUS method, a new approach based on fuzzy outranking relations to multi-criteria sorting problems. Compared with other outranking-based methods, THESEUS is inspired by another view of multi-criteria classification problems. It utilizes a new way of evaluating the assignment of an object to an element of the set of ordered categories that were previously defined. This way is based on comparing every possible assignment with the information from various preference relations that are derived from a fuzzy outranking relation defined on the universe of objects. The appropriate assignment is determined by solving a simple selection problem.The capacity of a reference set for making appropriate assignments is related to a good characterization of the categories. A single reference action characterizing a category may be insufficient to achieve well-determined assignments. In this paper, the reference set capacity to perform appropriate assignments is characterized by some new concepts. This capacity may be increased when more objects are added to the reference set. THESEUS is a method for handling the preference information contained in such larger reference sets.     

Throughput optimization in robotic cells with input and output machine buffers: A comparative study of two key models

We consider the problem of scheduling operations in a robotic cell processing a single part type. Each machine in the cell has a one-unit input buffer and a one-unit output buffer. The machines and buffers are served by one single gripper robot. The domain considered is free-pickup cells with additive inter-machine travel time. The processing constraints specify the cell to be a flow shop. The objective is to find a cyclic sequence of robot moves that minimizes the long-run average time to produce a part or, equivalently, maximizes throughput. Bufferless robotic cells have been studied extensively in the literature. However, the few studies of robotic cells with output buffers at each machine have shown that the throughput can be improved by such a configuration. We show that there is no throughput advantage in providing machine input buffers in addition to output buffers. The equivalence in throughput between the two models has significant practical implications, since the cost of providing additional buffers at each machine is substantial.     

Multicriteria sorting using a valued preference closeness relation

To define a metric or a similarity relation with good properties is a crucial issue in most analogy and rule induction oriented methods for multicriteria sorting. Here, we propose a valued indifference (closeness) relation which is inspired on concordance and discordance measures. The criterion “weights” are obtained from the preferential information embedded in a reference set. This proposal performs very well in several practical problems.     

Inverse sorting problem by minimizing the total weighted number of changes and partial inverse sorting problems

In this paper, we consider two types of inverse sorting problems. The first type is an inverse sorting problem by minimizing the total weighted number of changes with bound constraints. We present an O(n ²) time algorithm to solve the problem. The second type is a partial inverse sorting problem and a variant of the partial inverse sorting problem. We show that both the partial inverse sorting problem and the variant can be solved by a combination of a sorting problem and an inverse sorting problem. Supported by the Hong Kong Universities Grant Council (CERG CITYU 103105) and the National Key Research and Development Program of China (2002CB312004) and the National Natural Science Foundation of China (700221001, 70425004).     

Shelling Coxeter-like complexes and sorting on trees

In their work on ‘Coxeter-like complexes’, Babson and Reiner introduced a simplicial complex ΔT associated to each tree T on n nodes, generalizing chessboard complexes and type A Coxeter complexes. They conjectured that ΔT is (n−b−1)-connected when the tree has b leaves. We provide a shelling for the (n−b)-skeleton of ΔT, thereby proving this conjecture. In the process, we introduce notions of weak order and inversion functions on the labellings of a tree T which imply shellability of ΔT, and we construct such inversion functions for a large enough class of trees to deduce the aforementioned conjecture and also recover the shellability of chessboard complexes Mm,n with n?2m−1. We also prove that the existence or nonexistence of an inversion function for a fixed tree governs which networks with a tree structure admit greedy sorting algorithms by inversion elimination and provide an inversion function for trees where each vertex has capacity at least its degree minus one.     

Approximating Huffman codes in parallel

In this paper we present new results on the approximate parallel construction of Huffman codes. Our algorithm achieves linear work and logarithmic time, provided that the initial set of elements is sorted. This is the first parallel algorithm for that problem with the optimal time and work. Combining our approach with the best known parallel sorting algorithms we can construct an almost optimal Huffman tree with optimal time and work. This also leads to the first parallel algorithm that constructs exact Huffman codes with maximum codeword length H in time O(H) with n/logn processors, if the elements are sorted.     

A new approximation algorithm for the multilevel facility location problem

In this paper we propose a new integer programming formulation for the multilevel facility location problem and a novel 3-approximation algorithm based on LP-rounding. The linear program that we use has a polynomial number of variables and constraints, thus being more efficient than the one commonly used in the approximation algorithms for these types of problems.     

High-multiplicity cyclic job shop scheduling

We consider the High-Multiplicity Cyclic Job Shop Scheduling Problem. There are two objectives of interest: the cycle time and the flow time. We give several approximation algorithms after showing that a very restricted case is APX-hard.