NP-Completeness : Concepts Why Studying NP-Completeness ? - PDF document

NP-Completeness : Concepts • Why Studying NP-Completeness ? • • • ♣ Pursuing your Ph.D. ♣ Keeping your job Before studying NP-completeness : “I can’t find an efficient algorithm, I guess I’m just too dumb.” 1

After studying NP-completeness : “I can’t find an efficient algorithm, because no such algorithm is possible !” “I can’t find an efficient algorithm, but neither can all these famous people.” 2

• Measure to Time Complexity • • • l : measure to the time complexity of an algorithm The discussion of NP-completeness considers l the input size, i.e., the total length of all inputs to the algorithm. Two assumptions : (1) all inputs are integers (a rational number can be represented by a pair of integers); (2) each integer has a binary representation. Ex. Sorting a 1 , a 1 , …, a n . n � � � � ) log a 1 . l = ( � + i 2 i = 1 3

Ex. Consider the following procedure. input ( n ); s ← ← 0; ← ← for i ← ← 1 to n do ← ← s ← ← s + i ; ← ← output ( s ). l = log 2 n + 1. The procedure takes O ( n ) = O (2 l ) time. � an exponential-time algorithm ! 4

• Polynomial-Time Algorithms vs. • • • Exponential-Time Algorithms Suppose that your computer takes 1 second to perform 10 6 operations. The following is the time requirement for your computer to perform f ( n ) operations, where f ( n ) = n , n 2 , n 3 , n 5 , 2 n , 3 n and n = 10, 20, 30, 40, 50, 60. 5

The following shows the largest value of n such that f ( n ) operations can be performed in 1 hour on a faster computer. 6

An algorithm is referred to as a polynomial-time algorithm , if its time complexity can be bounded above by a polynomial function of input size. An algorithm is referred to as an exponential-time algorithm , if its time complexity cannot be thus bounded (even if the function is not normally regarded as an exponential one, like n log n ). Usually, a problem is referred to as tractable if it can be solved with a polynomial-time algorithm, and intractable otherwise. The two tables above give us a reason why polynomial-time algorithms are much more desirable than exponential-time algorithms. They also motive us to study the theory of NP-completeness. 7

• Maximal vs. Maximum • • • Ex. maximal cliques : {1, 2, 3}, {2, 3, 4, 5}, {4, 6} maximum cliques : {2, 3, 4, 5} 8

• Decision Problems vs. Optimization • • • Problems A decision problem asks a solution of “ yes ” or “ no ”. An optimization problem asks a solution of an optimal value (a maximum or a minimum). Ex. The maximum clique problem can be expressed as a decision problem as follows. Instance : An undirected graph G = ( V , E ) and a positive integer k ≤ ≤ | V |. ≤ ≤ Question : Does G contain a clique of size ≥ ≥ k ? ≥ ≥ It can be also expressed as an optimization problem as follows. Instance : An undirected graph G = ( V , E ). Question : What is the size of a maximum clique of G ? 9

Ex. The traveling salesman problem can be expressed as a decision problem as follows. Instance : A set C of m cities, distances d i , j > 0 for all pairs of cities i , j ∈ ∈ C , and a ∈ ∈ positive integer k . Question : Is there a tour of length ≤ ≤ k that starts ≤ ≤ at any city, visits each of the other m − − 1 − − cities exactly once, and returns to the initial city ? It can be also expressed as an optimization problem as follows. Instance : A set C of m cities and distances d i , j > 0 for all pairs of cities i , j ∈ ∈ C . ∈ ∈ Question : What is the length of a shortest tour that starts at any city, visits each of the other m − − 1 cities exactly once, − − and returns to the initial city ? 10

Ex. The problem of sorting a 1 , a 1 , …, a n can be expressed as a decision problem as follows. Instance : Given a 1 , a 2 , …, a n and a positive integer k . Question : Is there a permutation of a 1 , a 2 , …, a n , denoted by a ’ 1 , a ’ 2 , …, a ’ n , such that | a ’ 2 − − a ’ 1 | + | a ’ 3 − − a ’ 2 | + … + | a ’ n − − a ’ n − − 1 | ≤ ≤ k ? − − − − − − ≤ ≤ − − An optimization problem is “harder” than its corresponding decision problem. Since the NP-completeness concerns whether or not a problem can be solved in polynomial time, the discussion of NP-completeness considers only decision problems. (If a decision problem is not polynomial-time solvable, then its corresponding optimization problem is not polynomial-time solvable either.) 11

• Problem Reduction • • • A problem P 1 reduces to another problem P 2 , denoted by P 1 ∝ ∝ P 2 , if any instance of P 1 can ∝ ∝ be transformed into an instance of P 2 such that the solution for P 1 can be obtained from the solution for P 2 . T ∝ ∝ : the reduction time. ∝ ∝ T : the time required to obtain the solution for P 1 from the solution for P 2 . Since the NP-completeness concerns whether or not a problem can be solved in polynomial time, we consider only the reductions with both T ∝ ∝ and ∝ ∝ T polynomial. ∈ P � � P 1 ∈ ∉ P � � P 2 ∉ � � � � (Thus, P 2 ∈ ∈ P or P 1 ∉ ∉ P.) ∈ ∈ ∈ ∈ ∉ ∉ ∉ ∉ If P 1 ∝ ∝ P 2 and P 2 ∝ ∝ P 3 , then P 1 ∝ ∝ P 3 . ∝ ∝ ∝ ∝ ∝ ∝ 12

• P, NP, and NP-Complete • • • Three classes of decision problems : P, NP, and NP-complete. P : the set of decision problems that can be solved in polynomial time by deterministic algorithms. NP : the set of decision problems that can be solved in polynomial time by non-deterministic algorithms. Any non-deterministic algorithm consists of two phases : guessing and checking . 13

For the maximum clique problem, the guessing phase will return a clique, and the checking phase will decide whether or not the clique size is greater than or equal to k . For the traveling salesman problem, the guessing phase will return a tour, and the checking phase will decide whether or not the tour length is greater than or equal to k . A decision problem has an AFFIRMATIVE answer. ⇔ ⇔ The guessing is SUCCESSFUL. ⇔ ⇔ Notice that non-deterministic algorithms are imaginary. A more detailed description of non- deterministic algorithms and more illustrative examples can be found in Ref. (2). 14

Every decision problem in P is also in NP, i.e., P ⊆ ⊆ NP. ⊆ ⊆ An NP problem is NP-complete if every NP problem can reduce to it in polynomial time. � If any NP-complete problem can be solved in � � � polynomial time, then every NP problem can be solved in polynomial time (i.e., P = NP). (Intuitively, NP-complete problems are the “hardest” problems in NP.) It is one of the most famous open problems in computer science whether P ≠ ≠ NP or P = NP. ≠ ≠ 15

When P ≠ ≠ NP, ≠ ≠ NP NP-Complete P (There exist problems in NP that are neither in P, nor in NP-complete (see Chap. 7 in Ref. (1).) When P = NP, P = NP = NP-Complete Almost all people believe P ≠ ≠ NP. ≠ ≠ 16

A problem is NP-hard if an NP-complete problem can be reduced to it in polynomial time. (Equivalently, a problem is NP-hard if every NP problem can be reduced to it in polynomial time.) � If any NP-hard problem can be solved in � � � polynomial time, then all NP-complete problems can be solved in polynomial time. (Intuitively, NP-hard problems are “harder” than NP-complete problems.) NP NP-hard NP-complete The class of NP-hard problems contains both decision problems and optimization problems. 17

If an NP-hard problem is in NP, then it is an NP-complete problem. (Intuitively, NP-complete problems are an “easier” subclass of NP-hard problems.) The corresponding optimization problems of NP-complete problems are NP-hard. The well-known halting problem (a decision problem), which is to determine whether or not an algorithm will terminate with a given input, is NP-hard, but not NP-complete. 18

• Pseudo-Polynomial Time Algorithms • • • Ex. Given a set S = { a 1 , a 1 , …, a n } of integers and an integer M > 0, the sum - of - subset problem is to determine whether or not there exists a subset of S whose sum is equal to M . This problem can be solved in O ( nM ) time by dynamic programming as follows. Let t ( i , j ) = true , if there exists a subset of { a 1 , a 2 , …, a i } whose sum is equal to j , and false else. Then, t ( i , j ) = t ( i − − 1, j ) + t ( i − − 1, j − − a i ), where i > 1. − − − − − − Initially, t (1, j ) = true , if j = 0 or j = a 1 , and false else. The answer is t ( n , M ). 19

Although the time complexity is exponential with respect to M , the problem is considered polynomial-time solvable, if M is bounded. An algorithm like this is usually referred to as a pseudo-polynomial time algorithm . An NP-complete problem is in the strong sense if and only if there exists no pseudo-polynomial time algorithm for solving it (unless P = NP). Intuitively, NP-complete problems in the strong sense are “harder” NP-complete problems (refer to Ref. (1)). 20