ICS 353: Design and Analysis of Algorithms King Fahd University of Petroleum & Minerals Information & Computer Science Department ICS 353: Design and Analysis of Algorithms NP-Complete Problems

Reading Assignment M. Alsuwaiyel, Introduction to Algorithms: Design Techniques and Analysis, World Scientific Publishing Co., Inc. 1999. Chapter 10 Sections 1 – 4

Problems in Computer Science are classified into NP-Complete Problems Problems in Computer Science are classified into Tractable: There exists a polynomial time algorithm that solves the problem O(nk) 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 Output: An element in S of highest frequency What is the algorithm to solve this problem? How much does it cost?

Decision Problem: Coloring 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 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 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 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(nk), where k is a non-negative integer and n is the input size.

Sorting: Given n integers, are they sorted in non-decreasing order? 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 A 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., x1  x2  x3  x4 Definition: A boolean formula f is said to be in conjunctive normal form (CNF) if it is the conjunction of clauses. E.g., (x1  x2)  (x1  x5)  (x2  x3  x4  x6) 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.

Transitivity of poly Theorem: Let , ’, and ’’ be three decision problems such that  poly ’ and ’poly ’’. Then poly ’’. Proof: Corollary: If , ’ NP such that ’ poly  and ’  NP-complete, then   NP-complete How can we prove that   NP-hard? How can we prove that   NP-complete?

Proving NP-Completeness SAT 3-CNF-SAT Clique Hamiltonian Cycle Vertex-Cover Traveling Salesman Subset-Sum

Example NP-Complete Problems 3-CNF-SAT Input: Boolean formula f in CNF, such that each clause consists of exactly three literals. Question: Is f satisfiable. Hamiltonian Cycle Input: G = (V,E), undirected graph. Does G have a cycle that visits each vertex exactly once (Hamiltonian Cycle)? Traveling Salesman Input: A set of n cities with their intercity distances and an integer k. Question: Does there exist a tour of length less than or equal to k? A tour is a cycle that visits each vertex exactly once.

Example 1 Show that the traveling salesman problem is NP-complete, assuming that the Hamiltonian cycle problem is NP-complete.

Example 2 Prove that the Problem Clique is NP-Complete. Proof: Clique  NP Clique  NP-Hard SAT poly Clique Given an instance of satisfiability f = C1 C2  …  Cm with m clauses and n boolean variables x1, x2, …, xn we construct a graph G = (V, E), where V is the set of all occurrences of the 2n literals x1, x2, …, xn, and E = {(xi, xj) | xi  x’j and xi and xj belong to two different clauses}. Lemma 10.1 f is satisfiable if and only if G has a clique of size m.

Example 3 Prove that the problem Vertex Cover is NP-Complete Proof: Given an instance of satisfiability f = C1 C2  …  Cm with m clauses and n boolean variables x1, x2, …, xn we construct a graph G = (V, E) as follows: xi  f, two vertices, xi and x’i are added to V with the edge (xi, x’i) added to E.  Cj containing nj literals, add the nj vertices to V and add to E all the edges that make of the nj vertices a clique. w  Cj, connect (w, xw) where xw has been constructed in 1. These are called connection edges. Let k = n + Sigma_(j=1)^(m) (nj – 1)

Example 4 Prove that the problem Independent Set is NP-Complete Proof: Vertex Cover poly Independent Set S is an independent set iff V – S is a vertex cover

Example NP-Complete Problems (Cont.) Subset Sum 3-Coloring 3D-Matching Hamiltonian Path Partition Knapsack Bin Packing Set Cover Multiprocessor Scheduling Longest Path …