40144651
On the Approximation of Shortest Common Supersequences and Longest Common Subsequences
The problems of finding shortest common supersequences (SCS) and longest common subsequences (LCS) are two well-known {\bf NP}-hard problems that have applications in many areas including computational molecular biology, data compression, robot motion planning and scheduling, text editing, etc. A lot of fruitless effort has been spent in searching for good approximation algorithms for these problems. In this paper, we show that these problems are inherently hard to approximate in the worst case. In particular, we prove that (i) SCS does not have a polynomial time linear approximation algorithm, unless {\bf P} = {\bf NP}; (ii) There exists a constant $\delta > 0$ such that, if SCS has a polynomial time approximation algorithm with ratio $\log^{\delta} n$, where $n$ is the number of input sequences, then {\bf NP} is contained in {\bf DTIME}$(2^{\polylog n})$; (iii) There exists a constant $\delta > 0$ such that, if LCS has a polynomial time approximation algorithm with performance ratio $n^{\delta}$, then {\bf P} = {\bf NP}. The proofs utilize the recent results of Arora et al. [Proc. 23rd IEEE Symposium on Foundations of Computer Science, 1992, pp. 14-23] on the complexity of approximation problems. 
In the second part of the paper, we introduce a new method for analyzing the average-case performance of algorithms for sequences, based on Kolmogorov complexity. Despite the above nonapproximability results, we show that near optimal solutions for both SCS and LCS can be found on the average. More precisely, consider a fixed alphabet $\Sigma$ and suppose that the input sequences are generated randomly according to the uniform probability distribution and are of the same length $n$. Moreover, assume that the number of input sequences is polynomial in $n$. Then, there are simple greedy algorithms which approximate SCS and LCS with expected additive errors $O(n^{0.707})$ and $O(n^{\frac{1}{2}+\epsilon})$ for any $\epsilon > 0$, respectively. 
Incidentally, our analyses also provide tight upper and lower bounds on the expected LCS and SCS lengths for a set of random sequences, solving a generalization of another well-known open question on the expected LCS length for two random sequences [K. Alexander, The rate of convergence of the mean length of the longest common subsequence, 1992, manuscript],[V. Chvatal and D. Sankoff, J. Appl. Probab., 12 (1975), pp. 306-315], [D. Sankoff and J. Kruskall, eds., Time Warps, String Edits, and Macromolecules: The Theory and Practice of Sequence Comparison, Addison-Wesley, Reading, MA, 1983].
871253
Kolmogorov Complexity Arguments in Combinatorics
Abstract The utility of a Kolmogorov complexity method in combinatorial theory is demonstrated by several examples.
553776
Linear approximation of shortest superstrings
We consider the following problem: given a collection of strings s1, ••• ,sm, find the shortest string s such that each s; appears as a substring (a consecutive block) of s. Although this problem is known to be NP-hard, a simple greedy procedure appears to do quite well and is routinely used in DNA sequencing and data compression practice, namely: repeatedly merge the pair of (distinct) strings with maximum overlap until only one string remains. Let n denote the A. Blum was supported by a National Science Foundation (NSF) Graduate Fellowship. Part of this work was done while A. Blum was visiting AT & T Bell Labs. T. Jiang was supported in part by a grant from SERB, McMaster University and NSERC Operating Grant OGP0046613. M. Li was supported in part by the NSERC Operating Grants OGP0036747 and OGP0046506. J. Tromp was supported in part by NSERC Grant OGP0036747 while the author was visiting at the University of Waterloo. Authors' present addresses: A. Blum, School of Computer Science, Carnegie-Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213. E-mail: avrim@theory.cs.cmu.edu; T. Jiang, Department of Computer Science, McMaster University, Hamilton, Ont., Canada L8S 4Kl. E-mail: jiang<Q•maccs. mcmaster.ca; M. Li, Department of Computer Science, University of Waterloo, Waterloo, Ont., Canada N3L 30 I. E-mail: mli@watmath.waterloo.edu; J. Tromp, CWI, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands. E-mail: tromp@cwi.nl; M. Yannakakis, Room 2C-319, AT&T Bell Labs, 600 Mountain Avenue, Murray Hill, NJ 07974. E-mail: mihalis@research.att.com. Permission to copy without fee all or part of this material is granted provided that the copies arc not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. © 1994 ACM 0004-5411/94/0700-0630$03.50 Journal of the Association for Computing Machinery, Vol. 41, No. 4, July 1994, pages 630-647. Linear Approximation of Shortest Superstrings 631 length of the optimal superstring. A common conjecture states that the above greedy procedure produces a superstring of length O(n) (in fact, 2n), yet the only previous nontrivial bound known for any polynomial-time algorithm is a recent O(n log n) result. We show that the greedy algorithm does in fact achieve a constant factor approximation, proving an upper bound of 4n. Furthermore, we present a simple modified version of the greedy algorithm that we show produces a superstring of length at most 3n. We also show the superstring problem to be MAX SNP-hard, which implies that a polynomial-time approximation scheme for this problem is unlikely.
9397376
Inferring a DNA Sequence from Erroneous Copies
Abstract We suggest a novel approach for efficiently reconstructing an original DNA sequence from erroneous copies.
43310555
Twist-Rotation Transformations of Binary Trees and Arithmetic Expressions
The paper studies the computational complexity and efficient algorithms for the twist?rotation transformations of binary trees, which is equivalent to the transformation of arithmetic expressions over an associative and commutative binary operation. The main results are (1) a full binary tree with n labeled leaves can be transformed into any other in at most 3nlogn+2n twist and rotation operations, (2) deciding the twist?rotation distance between two binary trees is NP-complete, and (3) the twist?rotation transformation can be approximated with ratio 6logn+4 in polynomial time for full binary trees with n uniquely labeled leaves.
14168735
Inductive Reasoning and Kolmogorov Complexity
Abstract Reasoning to obtain the “truth” about reality, from external data, is an important, controversial, and complicated issue in man's effort to understand nature. (Yet, today, we try to make machines do this.) There have been old useful principles, new exciting models, and intricate theories scattered in vastly different areas including philosophy of science, statistics, computer science, and psychology. We focus on inductive reasoning in correspondence with ideas of R. J. Solomonoff. While his proposals result in perfect procedures, they involve the noncomputable notion of Kolmogorov complexity. In this paper we develop the thesis that Solomonoff's method is fundamental in the sense that many other induction principles can be viewed as particular ways to obtain computable approximations to it. We demonstrate this explicitly in the cases of Gold's paradigm for inductive inference, Rissanen's minimum description length (MDL) principle, Fisher's maximum likelihood principle, and Jaynes' maximum entropy principle. We present several new theorems and derivations to this effect. We also delimit what can be learned and what cannot be learned in terms of Kolmogorov complexity, and we describe an experiment in machine learning of handwritten characters. We also give an application of Kolmogorov complexity in Valiant style learning, where we want to learn a concept probably approximately correct in feasible time and examples.
368077
Approximation of Walrasian equilibrium in single-minded auctions
We consider a social optimization model of pricing scheme in single-minded auctions, in cases where Walrasian equilibrium does not exist. We are interested in the maximization of the ratio, R, of happy bidders over all agents, in a feasible allocation-pricing scheme. We show NP-hardness of the optimization problem, establish lower and upper bounds of R, as well as develop greedy algorithms to approximate the optimal value of R.
15025287
An Ontology Based Approach to Construct Behaviors in Web Information Systems
System behaviors specify the major functions of domain specific Web Information Systems (WIS). Traditional techniques can not satisfy various requirements or manage innumerous data while developing WIS behaviors. People appeal to a smart tool for implementing the WIS behaviors. This article makes the following contributions: (1) Proposes the concept of domain ontology and behavior ontology to describe the contents and operations of WIS; (2) Extends traditional ECA model to characterize the triggers, parameters, actions as well as validations of WIS behaviors; (3) Analyses the relationships between domain ontology and behavior ontology with rule sets; (4) Implements a tool named WISE Builder with four algorithms to help users building behavior ontology with domain ontology; (5)Shows the feasibility of this technique in a real application case.
1687904
PGGA: A predictable and grouped genetic algorithm for job scheduling
This paper presents a predictable and grouped genetic algorithm (PGGA) for job scheduling. The novelty of the PGGA is two-fold: (1) a job workload estimation algorithm is designed to estimate a job workload based on its historical execution records and (2) the divisible load theory (DLT) is employed to predict an optimal fitness value by which the PGGA speeds up the convergence process in searching a large scheduling space. Comparison with traditional scheduling methods, such as first-come-first-serve (FCFS) and random scheduling, heuristics, such as a typical genetic algorithm, Min-Min and Max-Min indicates that the PGGA is more effective and efficient in finding optimal scheduling solutions.
32860736
Nonrigid motion recovery for 3D surfaces
We present a spline-based nonrigid motion and point correspondence recovery method for 3D surfaces. This method is based on differential geometry. Shape information is used to recover the point correspondences. In contrast to the majority of shape-based methods, which assume that shape (unit normal, curvature) changes are minimum after motion, our method focuses on the nonrigid relationship between before-motion and after-motion shapes. This nonrigid shape relationship is described by modeling the underlying nonrigid motion; we model it as a spline transformation, which has global control over the entire motion field along with the local deformation integrated within. This provides our method certain advantages over some pure differential geometric methods, which also utilize the nonrigid shape relationship but only work on local areas without a global control. For example, motion regularity is hard to implement in these pure differential geometric methods but is not a problem when the motion field is controlled by a spline transformation. The orthogonal parameterization requirement of the nonrigid shape relationship has to be approximated in these previous methods but is easy to meet in our method. Furthermore, the small deformation constraint introduced by the previous works is relaxed in our method. Experiments on both synthetic and real motions have been conducted. The quantitative and qualitative evaluations of our method are presented. The application of our method to the human tongue motion analysis is also presented in this paper.
41518374
Longest cycles in regular 2-connected claw-free graphs
In this paper, we show that every 2-connected, k-regular claw-free graph on n vertices contains a cycle of length at least min 4k?2, n (k?8), and this result is best possible.