In the previous lecture, we discussed NP-completeness and gave some strong results pertaining to the complexity of SAT, viz. that it is complete for NP under a reduction computable in logarithmic space and polylogarithmic time, and that it is complete for NQLIN (the set of NP problems solvable in quasilinear time) under quasi-linear time mapping reductions. The conclusion that all naturally occ...

Unknown relation between P and NP [3] complexity classes remains to be one of significant non solved problems in complexity theory. P complexity class consists of problems solvable by Deterministic Turing Machine (DTM) in polynomially bounded time, while NP complexity class consists of problem solvable by Non Deterministic Turing Machine (NDTM) in polynomially bounded time. This means that DTM ...

A strategy for using Genetic Algorithms (GAs) to solve NP-complete problems is presented. The key aspect of the approach taken is to exploit the observation that, although all NP-complete problems are equally difficult in a general computational sense, some have much better GA representations than others, leading to much more successful use of GAs on some NP-complete problems than on others. Si...

In this paper, we consider two new variants of the unit covering problem in color-spanning set model: Given a set of n points in d-dimensional plane colored with m colors, the MinCSBC problem is to select m points of different colors minimizing the minimum number of unit balls needed to cover them. Similarly, the MaxCSBC problem is to choose one point of each color to maximize the minimum numbe...

We consider a scenario where stops are to be placed along an already existing public transportation network in order to improve its attractiveness for the customers. The core problem is a geometric set covering problem which is NP-hard in general. However, if the corresponding covering matrix has the consecutive ones property, it is solvable in polynomial time. In this paper, we present data re...

In this note we address the question of whether any any quantum computational model that allows adaptive measurements can be simulated by a model that allows postselected measurements. We argue in the favor of this question and prove that adaptive measurements can be simulated by postselection. We also discuss some potentially stunning consequences of this result such as the ability to solve #P...

Greedy algorithms are often the first algorithm that one considers for various optimization problems, and ,in particular, covering problems. The idea is very simple: try to build a solution incrementally by augmenting a partial solution. In each iteration, select the “best” augmentation according to a simple criterion. The term greedy is used because the most common criterion is to select an au...

In David Hilbert’s lecture in 1900, he presented 15 difficult problems that guided much of the research of mathematicians in the last century. Today, only 12 of these 15 problems have been solved. In honor of this great mathematician and to celebrate mathematics of the new millennium, the Clay Mathematics Institute of Cambridge, Massachusetts is offering a one million dollar prize for the solut...