NP-Complete Problems Reading Material: Chapter 10 Sections 1, 2, 3, and 4 only.

NP-Complete Problems Problems in Computer Science are classified into –Tractable: There exists a polynomial time algorithm that solves the problem O(n k ) –Intractable: Unlikely for a polynomial time algorithm solution to exist NP-Complete Problems

Decision Problems vs. Optimization Problems Decision Problem: Yes/no answer Optimization Problem: Maximization or minimization of a certain quantity When studying NP-Completeness, it is easier to deal with decision problems than optimization problems

Element Uniqueness Problem Decision Problem: Element Uniqueness –Input: A sequence of integers S –Question: Are there two elements in S that are equal? Optimization Problem: Element Count –Input: A sequence of integers S –Output: An element in S of highest frequency What is the algorithm to solve this problem? How much does it cost?

Coloring a Graph Decision Problem: Coloring –Input: G=(V,E) undirected graph and k , k > 0. –Question: Is G k-colorable? Optimization Problem: Chromatic Number –Input: G=(V,E) undirected graph –Output: The chromatic number of G,  (G) i.e. the minimum number  (G) of colors needed to color a graph in such a way that no two adjacent vertices have the same color.

Clique Definition: A clique of size k in G, for some +ve integer k, is a complete subgraph of G with k vertices. Decision Problem –Input: –Question: Optimization Problem –Input: –Output:

Vertex Cover Definition: A vertex cover of an undirected graph G=(V,E) is a subset C  V such that each edge e=(x,y)  E is incident to at least one vertex in C, i.e. x  C or y  C. Decision Problem –Input: –Question: Optimization Problem –Input: –Output:

Independent Set Definition: Given an undirected graph G=(V,E) a subset S  V is called an independent set if for each pair of vertices x, y  S, (x,y)  E Decision Problem –Input: –Question: Optimization Problem –Input: –Output:

From Decision To Optimization For a given problem, assume we were able to find a solution to the decision problem in polynomial time. Can we find a solution to the optimization problem in polynomial time also?

Deterministic Algorithms Definition: Let A be an algorithm to solve problem . A is called deterministic if, when presented with an instance of the problem , it has only one choice in each step throughout its execution. –If we run A again and again, is there a possibility that the output may change? What type of algorithms did we have so far?

The Class P Definition: The class of decision problems P consists of those whose yes/no solution can be obtained using a deterministic algorithm that runs in polynomial time of steps, i.e. O(n k ), where k is a non-negative integer and n is the input size.

Examples Sorting: Given n integers, are they sorted in non-decreasing order? Set Disjointness: Given two sets of integers, are they disjoint? Shortest path: 2-coloring: –Theorem: A graph G is 2-colorable if and only if G is bipartite

Closure Under Complementation C C CA class C of problems is closed under complementation if for any problem   C the complement of  is also in C. Theorem: The class P is closed under complementation

Non-Deterministic Algorithms A non-deterministic algorithm A on input x consists of two phases: –Guessing: An arbitrary “string of characters y” is generated in polynomial time. It may Correspond to a solution Not correspond to a solution Not be in proper format of a solution Differ from one run to another –Verification: A deterministic algorithm verifies The generated “string of characters y” is in proper format Whether y is a solution in polynomial time

Non-Deterministic Algorithms (Cont.) Definition: Let A be a nondeterministic algorithm for a problem . We say that A accepts an instance I of  if and only if on input I, there exists a guess that leads to a yes answer. –Does it mean that if an algorithm A on a given input I leads to an answer of no for a certain guess, that it does not accept it? What is the running time of a non-deterministic algorithm?

The Class NP Definition: The class of decision problems NP consists of those decision problems for which there exists a nondeterministic algorithm that runs in polynomial time

Example Show that the coloring problem belongs to the class of NP problems

P and NP Problems What is the difference between P problems and NP Problems? We can decide/solve problems in P using deterministic algorithms that run in polynomial time We can check or verify the solution of NP problems in polynomial time using a deterministic algorithm What is the set relationship between the classes P and NP?

NP-Complete Problems Definition: Let  and  ’ be two decision problems. We say that  ’ reduces to  in polynomial time, denoted by  ’  poly , if there exists a deterministic algorithm A that behaves as follows: When A is presented with an instance I’ of problem  ’, it transforms it into an instance I of problem  in polynomial time such that the answer to I’ is yes if and only if the answer to I is yes.

NP-Hard and NP-Complete Definition: A decision problem  is said to be NP-hard if  ’  NP,  ’  poly . Definition: A decision problem  is said to be NP-complete if –  NP –  ’  NP,  ’  poly . What is the difference between an NP-complete problem and an NP-hard problem?

Conjunctive Normal Forms Definition: A clause is the disjunction of literals, where a literal is a boolean variable or its negation –E.g., x 1  x 2  x 3  x 4 Definition: A boolean formula f is said to be in conjunctive normal form (CNF) if it is the conjunction of clauses. –E.g., (x 1  x 2 )  (x 1  x 5 )  (x 2  x 3  x 4  x 6 ) Definition: A boolean formula f is said to be satisfiable if there is a truth assignment to its variables that makes it true.

The Satisfiability Problem Input: A CNF boolean formula f. Question: Is f satisfiable? Theorem: Satisfiability is NP-Complete –Satisfiability is the first problem to be proven as NP-Complete –The proof includes reducing every problem in NP to Satisfiability in polynomial time.