Optimal Linear Arrangement Problem Research Articles

Molecular solution to the optimal linear arrangement problem based on DNA computation

Due to massive parallelism, enormous memory storage and very low energy consumption, biomolecular operations have been suggested to solve various NP-hard problems that are beyond the capability of the fastest known digital computer. The optimal linear arrangement (OLA) problem is a well-known NP-hard combinatorial optimization problem. Based on a DNA computational model, this paper describes a novel algorithm for the OLA problem, which is executed in O(n 3log2 n) DNA operations on tubes of (nK + n + m + L + 1)-bits DNA strands, where $$K = \left\lceil {\log_2 n} \right\rceil$$ and $$L = \left\lceil {\log _2 \left({nm} \right)} \right\rceil + 1$$ . With the advance in molecular biology techniques, this algorithm may be of practical utility.

Chromosome Reconstruction from Physical Maps Using a Cluster of Workstations

Ordering clones from a genomic library into physical maps of whole chromosomes presents a central computational problem in genetics. Chromosome reconstruction via clone ordering is shown to be isomorphic to the classical NP-complete Optimal Linear Arrangement problem. Parallel algorithms for simulated annealing and microcanonical annealig based on Markov chain decomposition are proposed and applied to the problem of chromosome reconstruction via clone ordering. These algorithms are implemented on a cluster of UNIX workstations using the Parallel Virtual Machine (PVM) system. PVM is a software system that permits a heterogeneous collection of networked computers to be viewed by a user‘s program as a single monolithic parallel computing resource. The parallel algorithms are implemented and tested on clonal data derived from Chromosome IV of the fungus Asperigillus nidulans Perturbation methods and problem-specific annealing heuristics for the parallel simulated annealing and parallel microcanonical annealing algorithms are proposed and described. Convergence, speedup and scalability characteristics of the various parallel algorithms are analyzed and discussed.

The use of dynamic programming in genetic algorithms for permutation problems

To deal with computationally hard problems, approximate algorithms are used to provide reasonably good solutions in practical time. Genetic algorithms are an example of the meta-heuristics which were recently introduced and which have been successfully applied to a variety of problems. We propose to use dynamic programming in the process of obtaining new generation solutions in the genetic algorithm, and call it a genetic DP algorithm. To evaluate the effectiveness of this approach, we choose three representative combinatorial optimization problems, the single machine scheduling problem, the optimal linear arrangement problem and the traveling salesman problem, all of which ask to compute optimum permutations of n objects and are known to be NP-hard. Computational results for randomly generated problem instances exhibit encouraging features of genetic DP algorithms.

Parallel Computing of Physical Maps— A Comparative Study in SIMD and MIMD Parallelism

Ordering clones from a genomic library into physical maps of whole chromosomes presents a central computational problem in genetics. Chromosome reconstruction via clone ordering is usually isomorphic to the NP-complete Optimal Linear Arrangement problem. Parallel SIMD and MIMD algorithms for simulated annealing based on Markov chain distribution are proposed and applied to the problem of chromosome reconstruction via clone ordering. Perturbation methods and problem-specific annealing heuristics are proposed and described. The SIMD algorithms are implemented on a 2048 processor MasPar MP-2 system which is an SIMD 2-D toroidal mesh architecture whereas the MIMD algorithms are implemented on an 8 processor Intel iPSC/860 which is an MIMD hypercube architecture. A comparative analysis of the various SIMD and MIMD algorithms is presented in which the convergence, speedup, and scalability characteristics of the various algorithms are analyzed and discussed. On a fine-grained, massively parallel SIMD architecture with a low synchronization overhead such as the MasPar MP-2, a parallel simulated annealing algorithm based on multiple periodically interacting searches performs the best. For a coarse-grained MIMD architecture with high synchronization overhead such as the Intel iPSC/860, a parallel simulated annealing algorithm based on multiple independent searches yields the best results. In either case, distribution of clonal data across multiple processors is shown to exacerbate the tendency of the parallel simulated annealing algorithm to get trapped in a local optimum.

Branch distance optimization of structured programs

One of the functions of code generation (or peephole optimization of generated code) is to find a linear arrangement of code segments such that the sum of branch distances between code segments is minimized. The problem is a generalization of the optimal linear arrangement problem which is known to be NP-hard. However, it has important applications in generating object code for pipeline and cache machines. This paper presents a branch-and-bound method which incorporates a heuristic for deriving a near-optimal initial solution.

Linear arrangement problems on recursively partitioned graphs

We analyze the complexity of the restrictions of linear arrangement problems that are obtained if the legal permutations of the nodes are restricted to those that can be obtained by orderings of a binary tree structuring the nodes of the graph, the so-called p-tree. These versions of the linear arrangement problems occur in several places in current circuit layout systems. There the p-tree is the result of a recursive partitioning process of the graph. We show that the MINCUT LINEAR ARRANGEMENT problem and the OPTIMAL LINEAR ARRANGEMENT problem can be solved in polynomial time, if the p-tree is balanced. All other versions of the linear arrangement problems we analyzed are NP-complete.

Some simplified NP-complete graph problems

It is widely believed that showing a problem to be NP-complete is tantamount to proving its computational intractability. In this paper we show that a number of NP-complete problems remain NP-complete even when their domains are substantially restricted. First we show the completeness of Simple Max Cut (Max Cut with edge weights restricted to value 1), and, as a corollary, the completeness of the Optimal Linear Arrangement problem. We then show that even if the domains of the Node Cover and Directed Hamiltonian Path problems are restricted to planar graphs, the two problems remain NP-complete, and that these and other graph problems remain NP-complete even when their domains are restricted to graphs with low node degrees. For Graph 3-Colorability, Node Cover, and Undirected Hamiltonian Circuit, we determine essentially the lowest possible upper bounds on node degree for which the problems remain NP-complete.