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.