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# Outline for MATH 4315 at UH

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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 30 views.

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Date Created: 02/06/15

INFORMAL OUTLINE OF COMPLEXITY GRAPH THEORY MATH 4315 A decision problem is a question stated in a form in which it has a yes 7 or no 7 answer Here are a few typical examples Colorabz39lz39ty Instance G a graph k an integer Question Is chromatic number of G g k ie is it possible to color vertices of a graph with at most k colors so that adjacent vertices have different colors 73rz39mality Instance n an integer Question Is n prime Independence Instance G a graph k an integer Question Is the independence number of G 2 k Eulerz39an Tour Instance G a graph Question Is it possible to list edges of G so that each edge appears on the list exactly once and each of them apart from the rst one is incident with the previus edge on the list Hamiltonian Path Instance G a graph Question Is it possible to list vertices of G so that each appears on the list once and every vertex is adjacent to the previus one on the list E1 Is there a polynomial time algorithm to decide if a nite graph is connected E2 Is there a polynomial time algorithm to decide if a nite graph is Eulerian Ten Instance D a Diophantine equation ie7 a multi variable polynomial equation with integer coef cients Question ls D solvable in integers NP is the class of decision problems for which the answer is 77yes77 and for which there is an answer whose correctness can be veri ed in a polynomial time The latter means that there is an algorithm ie7 a program P which on input consisting of an evidence of correctness will con rm that the answer to the question is 77yes77 after running time t5 0sk for some integer k In this de nition 5 is the size of the input problem7 ie7 a measure of its complexity Size can be de ned in many ways7 and the only essential factor is not to 77in ate77 its de nition too much Size of the problem can be thought of as the minimum amount of information needed to describe this problem A right size for most of decision problems involving graphs is either the number of its vertices or its edges Note7 that as far as the existence of polynomial time algorithms is concerned7 it does not matter which one of the two is selected7 as long as the graph is connected However7 it may not be obvious at rst that for Primalz ty the right size is not 717 but logn lndeed7 an integer n can be de ned basically by logn units of information7 namely the digits of n tn Of means that there is a constant C such that for all suf ciently large 717 l tn l S Cfn Thus tn 0nk for an integer k means that It grows slower than some polynomial Note that the number of digits representing n with respect to any base is Ologn NP stands for the class of nondeterministic polynomial decision problems The class 73 is the class of decision problems which can be solved in a polynomial time From a highly theoretical point of view polynomial time algorithms are considered quick or ef cient Certainly can not be solved in polynomial time computationally intractable Such problems are called exponential It is not known whether classes 73 and N73 are equal7 but it is thought that they are not To prove this would be enough to show that for example Colorabz39lz39ty does not belong to 73 and in particular that there is no polynomial time algorithm for nding the chromatic number of a graph To show that Colorabz39lz39ty belongs to N73 it is enough to observe that if one can guess a right coloration of a graph then the correctness of the coloration can be indeed quickly veri ed In other words a right coloration can be presented as an evidence of correctness of the 77yes77 answer 3 More interestingly if there is a polynomial time algorithm for nding chromatic number of a graph then this would imply that every problem in N73 can be solved in a polynomial time and in particular that 73 N73 Decision problems which have this property an which are in N73 are called NP complete Many problems studied in graph theory including all three examples listed above are NP complete An example of a problem which belongs to 73 and thus certainly is not NP complete is the question whether a graph is Eulerian Pratt proved in 775 that Primalz ty belongs to N73 but it is not know whether there is a polynomial time algorithm for testing whether integer is a prime Pratt7s Theorem answered one of the very few non trivial open problems concerning membership in N73 As far as the tests and the material from this section are concerned you may expect only questions concerning decision problems for which there are polynomial time algorithms as for example E1 and E2

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