Unless P=NP, we cannot obtain a polynomial-time algorithm solving hard combinatorial problems. One practical approach in solving this kind of problem is to relax the condition of always finding the optimal solution for an instance and settle for “good enough” solutions. The kind of algorithms which are guaranteed to obtain a solution with a certain quality are called approximative algorithms. However, not all hard problems are approximable, i.e., we can obtain a polynomial-time algorithm that can guarantee the goodness of the solution for a problem.
In this lecture, we will present the concept of reoptimization. In this approach, given an instance I of some problem Π, an optimal solution OPT for Π in I, and a modified instance I’ resulting from a local perturbation of I, we wish to use OPT in order to solve Π in I’. With this additional information, reoptimization may help to improve the approximability of the problem or the running time of the solution to it. In fact, we can obtain a polynomial-time approximation scheme (PTAS) for a reoptimization variant of a problem given that the unmodified problem is approximable.
Several genes across different genomes are grouped together as a result of functional dependencies. These set of genes that are kept together are called gene cluster. Identifying the set of clustered genes in the literature is widely studied in the fields of Biology and Computer Science. In the field of computation, biological problems as such are modelled as optimizationproblems.
A model introduced by Rahman et.al. is the Approximate Gene Cluster Discovery Problem (AGCDP) where the genes are treated as integers and genomes are treated as a string of integers.
The input of AGCDP is the following.
Set of integer strings which represent the genomes.
Number of genes expected to be in the cluster
The solution of the problem is a set of genes which minimizes a certain score. Details on the score computation is detailed in the paper by Rahman. They also presented an integer linear programming formulation of the problem.
Lecture on Randomized Computation for our Computational Complexity class.
This lecture includes a short introduction to randomized algorithms. It includes discussion on probabilistic Turing Machines (PTMs), TMs that make use of random number in its computation, as well as different complexity classes that it recognize. The complexity classes include the Bounded Error Probability Polynomial (BPP), RP, co-RP, and ZPP. Relationship of the said classes where also presented as well as their relationship to known complexity class such as P and NP.