NP-Completeness

1
NP-Completeness

• CS 583 Fall 2006
• Analysis of Algorithms

2
Outline

• HW2/3 Status
• Overview
• Showing a problem to be NPC
• Polynomial time
• Abstract problems
• Encodings
• Verification in Polynomial Time
• NP-completeness
• Self-test
• 34.1-2, 34.2-3
• Final exam details

3
HW 2/3 Status

• It is very likely that HW2 grades have been lost.
• HW2 solutions are posted on the web.
• Therefore, HW3 becomes mandatory to avoid
  losing 10 points.
• The deadline is extended to Saturday, December 9
  (midnight).
• The online submission needs to be e-mailed to the
  instructor.
• The answers will posted on 12/10.
• If HW2 grades are recovered before 12/16, they
  will be taken into account.

4
Overview

• Three classes of problems
• Class P consists of those problems that are
  solvable in polynomial time.
• An algorithm runs in O(nk) for some constant k,
  where n is the size of the input.
• Class NP consists of problems that are
  verifiable in polynomial time.
• It can be verified in polynomial time that a
  given solution is feasible.
• A problem is in the class NPC if it is in NP, and
  is as hard as any other problem in NP.
• If any NP-complete problem can be solved in
  polynomial time, then every problem in NP has a
  polynomial time algorithm.
• Most computer scientists believe NPC problems are
  intractable, i.e. P ? NP.

5
Showing a Problem to be NPC

• Decision problems versus optimization problems.
• Many problems of interest are optimization
  problems
• Each feasible solution has an associated value,
  and we wish to find the one with the best value.
• For example, the graph coloring problem use the
  least number of colors, so that no two adjacent
  vertices are colored the same.
• NP-completeness applies to decision problems
• The answer to a problem is yes or no.
• For example, the graph coloring decision problem
  can a graph be colored with k colors?
• An optimization problem can be cast to a decision
  problem.
• The decision problem is easier as once an
  optimization problem is solved, its related
  decision problem is solved as well.

6
Reductions

• We show that one decision problem is no easier or
  harder than another one.
• Suppose we have a decision problem A that we wish
  to solve in polynomial time.
• Assume we have a decision problem B that we can
  solve in polynomial time.
• Develop a procedure that converts an instance
  (input) ? of A into instance ? of B as follows.
• The transformation takes polynomial time.
• The answers are the same the answer for ? is
  yes if and only if the answer for ? is also
  yes.

7
Reductions (cont.)

• Such procedure is called a reduction algorithm as
  it allows solving A in polynomial time
• Given an instance ? of A, use a polynomial-time
  reduction algorithm to transform it to an
  instance ? of B.
• Run B in polynomial time on input ?.
• Use the answer for ? as the answer for ?.
• It can be shown that if it is proven that A is
  intractable, then B cannot be solved in
  polynomial time as well.
• Proof by contradiction.

8
A First NPC Problem

• The technique of reduction relies on having a
  problem known to be NP-complete.
• Hence we need to start with some NPC problem.
• The first such problem we use is the
  circuit-satisfiability problem
• Given a boolean combinational circuit composed of
  AND, OR, and NOT gates,
• We wish to know if theres any set of boolean
  inputs to this circuit that causes its output to
  be 1.

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Polynomial time

• The polynomial time problems are considered
  tractable for the following reasons
• A problem that runs in ?(n100) is practically
  intractable. However,
• In practice there are very few such problems.
• When a problem running in ?(n100) is solved, it
  is likely that much better algorithms are
  discovered.
• A problem that can be solved in polynomial time
  in one model, can be solved in P time in another.
• Parallel algorithms.
• Class P has closure properties.
• An algorithm composed of P-time algorithms is
  also polynomial.

10
Abstract Problems

• An abstract problem Q is a binary relation on a
  set I of problem instances, and a set S of
  solutions
• Q I ? S, e.g., (i1, s1) ? Q, where i1 ? I, s1 ?
  S.
• Decision problems have a yes/no solution
• S 0, 1
• Example graph coloring decision problem COLOR
• If i (G, k) is an instance of the problem
  COLOR, then
• COLOR(i) 1 (yes) if graph G can be colored with
  k colors
• COLOR(i) 0 (no) otherwise
• Recall that more practical optimization problems
  can be recast as decision problems.

11
Concrete Problems

• An encoding of a set S of abstract objects is a
  mapping e from S to the set of binary strings.
• Example natural numbers (set N0,1,2,...) can
  be represented as binary numbers
• e(3) 11, e(4) 100, ... , e(17) 10001
• An algorithm that solves some abstract decision
  problem takes an encoding of a problem instance
  as input.
• A problem whose instance set is a set of binary
  strings is called a concrete problem.
• An algorithm solves a concrete problem in O(T(n))
  time, where n i
• A problem is polynomial-time solvable if T(n)
  nk for some constant k.
• The complexity class P is a set of concrete
  decision problems that are polynomial-time
  solvable.

12
Formal-Language Framework

• The machinery of formal-language theory is very
  convenient for decision problems.
• An alphabet ? is a finite set of symbols.
• A language L over ? is any set of strings made up
  of symbols from ?.
• For example, ? 0,1, L 10,11,101, ...
  prime numbers
• An empty string is denoted by ?,
• An empty language is denoted by ?.
• The language of all strings over ? is denoted ?.
• For example, ? 0,1, ? ?, 0 , 1, 00, 10,
  11, 101, ... the set of all binary strings.

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Formal-Language Framework (cont.)

• Operations on languages
• Union and intersection follow directly from set
  operations.
• The complement of L is ? L ? - L.
• The concatenation of two languages L1 and L2 is
  the language
• L x1x2 x1 ? L1 and x2 ? L2
• The closure, or Kleene star of language L is
• L ? ? L ? L2 ? L3 ? ...
• Hence, the set of instances for any decision
  problem Q is simply the set ?, where ? 0,1.

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Formal-Language Framework (cont.)

• Q is a language L over ? 0,1, where
• L x ? ? Q(x) 1
• Expressing the relation between decision problems
  and algorithms.
• An algorithm A accepts a string x ? 0,1 if
• given input x, the algorithm outputs A(x) 1.
• An algorithm rejects x if A(x) 0.
• The language accepted by A is
• L x ? 0,1 A(x) 1
• A language L is decided by A if
• every binary string in L is accepted by A, and
• every binary string not in L is rejected by A.

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Formal-Language Framework (cont.)

• A language L is accepted in polynomial time by
  algorithm A if
• it is accepted by A, and
• there is a constant k such that for any x ? L,
  x n it accepts x in O(nk).
• A language L is decided in polynomial time if
• there is a constant k such that for any x ?
  0,1, x n, A correctly decides whether x ?
  L in O(nk).
• Thus, to accept a language, an algorithm only
  needs to consider strings in L, but to decide it,
  it needs to consider every string in 0,1.
• The complexity class P
• P L ? 0,1 there exists an algorithm A
  that decides L in polynomial time

16
Hamiltonian Cycle Problem

• Hamiltonian cycle
• A hamiltonian cycle of a graph G(V,E) is a
  simple cycle that contains all vertices in V.
• A graph that contains a hamiltonian cycle is
  called hamiltonian.
• The Hamiltonian cycle problem defined in a formal
  language
• HAM-CYCLE ltGgt G is a hamiltonian graph
• Brute-force algorithm
• Take each permutation of vertices and check if
  its a hamiltonian graph.
• Running time
• Assume the encoding is its adjacency matrix of
  size n.
• The number of vertices m ?(n1/2).
• There are m! possible permutations of vertices.
• The running time of such algorithm is ?(m!)
  ?(n1/2!) ?(2n1/2) not polynomial!

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Verification Algorithm

• When a certain permutation of vertices is given,
  it can be verified if its a hamiltonian cycle in
  O(n2) time.
• A verification algorithm is a two-argument
  algorithm A
• One argument is an input string x.
• The other argument is a binary string y called a
  certificate.
• A verifies an input x if there exists a
  certificate y such that
• A(x,y) 1
• The language verified by A is
• L x ? 0,1 there exists y ? 0,1 such
  that A(x,y) 1
• A verifies L if for any string x ? L, there is a
  certificate y that A can use to prove that x ? L.
• In the HAM-CYCLE problem, the certificate is the
  list of vertices in some hamiltonian cycle.
• If the graph is hamiltonian, the cycle is
  sufficient to verify the fact.
• If it is not hamiltonian, no such certificate
  exists.

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The Complexity Class NP

• NP is the class of languages that can be verified
  by a polynomial-time algorithm A(x,y)
• L x ? 0,1 there exists a certificate y
  with y O(xc) such that c is constant and
  A(x,y) 1
• We say that algorithm A verifies language L in
  polynomial time.
• If L? P, then L? NP
• If there is a polynomial time algorithm to decide
  L, it can be converted to a verification
  algorithm to ignore the certificate.
• Hence P ? NP.
• It is unknown if P NP
• The class of NP-complete algorithms makes this
  question even more intriguing
• If one NPC algorithm can be solved in polynomial
  time, then every problem in NP has a polynomial
  time solution!

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Final Exam

• Taking the exam
• The exam will be available to take in a
  classroom.
• The exam will be available online on case-by-case
  basis only.
• The exceptions will be granted based on good
  cause.
• The online version will be distributed by e-mail.
• The NEW session will be open during the exam for
  student questions.
• The student evaluations will be taken before the
  exam.
• Exam details
• It is an open everything exam.
• The duration is 2.5 hours, -- from 745 to 1015.
• It consists of 4 main problems (10 points each),
  and 2 bonus questions (2 points each).
• The exam covers the contents of all classes,
  although it will contain more questions from the
  second half of the course (after the midterm
  exam).
• Overall grading
• A 90 points
• B 85 B 80 B- 75
• C 65 D 60