New Techniques in Design and Analysis of Exponential Algorithm

In: Computers and Technology

Submitted By shigopunk
Words 1436
Pages 6
New Techniques in Design and Analysis of Exponential Algorithm Arshi Aftab, Shikha Mahajan and Shruti Jajoo

Abstract: In this paper we first see how NP problems can be categorized and look at some examples. We then analyze the subset sum problem and prove its NP completeness. We look at various algorithms used to solve the problem and the efficiency of each algorithm. Hence we prove that subset sum problem is NP complete.
Keywords: NP complete, NP hard, complexity, subset sum, exponential I. Introduction
In the study of computation complexity of problems, we find out the resources required during computation like time, memory and space required. We check if the problem can be solved in polynomial time by some algorithm. An algorithm is said to be solved in polynomial time if its worst case efficiency belongs to O (p (n)) where p (n) is the polynomial of the problems input size. Problems that can be solved in polynomial time are called tractable and those that cannot be solved in polynomial time are intractable.
The formal definition of NP problem is as follows: A problem is said to be Non-deterministically Polynomial (NP) if we can find a nondeterministic Turing machine that can solve the problem in a polynomial number of nondeterministic moves.
Equivalent definition of NP problem: A problem is said to be NP if
1. its solution comes from a finite set of possibilities, and
2. It takes polynomial time to verify the correctness of a possible solution.
NP-complete problems
NP-complete problems are a subset of class NP. An problem p is NP-complete, if
1. p ∈ NP (you can solve it in polynomial time by a non-deterministic Turing machine) and
2. All other problems in class NP can be reduced to problem p in polynomial time.
NP-hard problems
NP-hard problems are partially alike but more difficult problems than NP complete problems. They don’t themselves…...

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