NP-Complete Chapter 8

Transcription

1 NP-omplete hapter 8 10/3/2005 Kumar S Polnomial lgorithms Problems encountered so far are polnomial time algorithms The worst-case running time of most of these algorithms is On k time, for some constant k. ll problems cannot be solved in polnomial time There are problems that cannot be solved at all Unsolvable There are problems that can be solved but not in On k time for some constant time. Problems that are solvable in polnomial time b polnomial-time algorithms are said to be tractable or eas or efficient. Problems that require superpolnomial time are said to be intractable or hard. 10/3/2005 Kumar S5311 2

2 P and NP lass P problems are solvable in polnomial time. lass NP problems are verifiable in polnomial time. For eample, given a problem, we can verif the solution in polnomial time n problem in P is also in NP P NP It is NOT KNOWN whether P is a proper subset of NP. 10/3/2005 Kumar S NP What are NP-complete Problems? problem is said to be NP-omplete if it is as hard as an problem in NP. No polnomial time algorithm has et been discovered for an NP-omplete problem. However, it has not been proven that NO polnomial time algorithm can eist for an NP- omplete problem. This problem was first posed b ook in The issue of, P=NP or P NP is an open research problem 10/3/2005 Kumar S5311 4

3 amples of NP-omplete problems Shortest vs. longest simple paths Finding the shortest paths from a single source in a directed graph G =V, can be completed in OV time. ven with negative edge weights. However, finding the longest simple path between two vertices is NP-complete. It is NP-omplete even if each of edge weights is equal to one. n uler tour of a connected directed graph G=V,, can be completed in O time. However, the Hamiltonian cle is NP-omplete. The traveling salesman problem is a variation of the Hamiltonian ccle. 10/3/2005 Kumar S Polnomial Time Reductions ecision Problems: Problems whose answer is es or no!! Most problems can be converted to decision problems. Language recognition problem is a decision problem. Suppose L U is the set of all inputs for which the answer to the problem is es U is the input space and L is the language that returns a true or es S = L is called the language corresponding to the problem Turing machines. The terms language and problem are used interchangeabl. Given a problem, with an input language X, Now the decision problem can be defined as the problem to recognie whether or not X belongs to L. 10/3/2005 Kumar S5311 6

4 Reductions ontd. efinition: Let L1 and L2 be two languages from input spaces U1 and U2. We sa that L1 is polnomiall reducible to L2 if there eists a polnomial-time algorithm that converts each input u1 U1 to another input u2 U2 such that u1 L1 iff u2 L2. The algorithm is polnomial in the sie of the input u1. If we have an algorithm for L2 then we can compose the two algorithms to produce an algorithm for L1. If L1 is polnomiall reducible to L2 and there is a polnomial-time algorithm for L2, then there is a polnomial time algorithm for L1. Reducibilit is not smmetric L1 is polnomiall reducible to L2 does not impl L2 is polnomiall reducible to L1 10/3/2005 Kumar S The Satisfiabilit ST Problem S oolean epression in onjunctive Normal Form NF Product N of Sums ORs For eample S = The ST problem is to determine whether a given oolean epression is Satisfiable without necessaril finding a satisfing assignment We can guess a truth assignment and check that it satisfies the epression in polnomial time. ST is NP hard Turing machine and all of its operations on a given input can be described b a oolean pression. The epression will be satisfiable iff the Turing machine will terminate at an accepting state for the given input. 10/3/2005 Kumar S5311 8

5 10/3/2005 Kumar S lique Problem 10/3/2005 Kumar S lique Problem

6 10/3/2005 Kumar S lique Problem 10/3/2005 Kumar S lique Problem

7 10/3/2005 Kumar S lique Problem 10/3/2005 Kumar S Verte over problem verte cover of G = V, is a set of vertices such that ever edge in is incident to at least one of the vertices in the verte cover.

8 Verte over problem verte cover of G = V, is a set of vertices such that ever edge in is incident to at least one of the vertices in the verte cover. G G complement of G 10/3/2005 Kumar S Verte over problem verte cover of G = V, is a set of vertices such that ever edge in is incident to at least one of the vertices in the verte cover. G has a maimum clique of sie V -k 10/3/2005 Kumar S

9 Verte over problem verte cover of G = V, is a set of vertices such that ever edge in is incident to at least one of the vertices in the verte cover. complement of G G has a minimum verte cover of sie k if and onl if G has a maimum clique of sie V -k G 10/3/2005 Kumar S Verte over problem verte cover of G = V, is a set of vertices such that ever edge in is incident to at least one of the vertices in the verte cover. G G complement of G 10/3/2005 Kumar S

10 ominating Set G = V, is an undirected graph. dominating set of G is a set of vertices a subset of V such that ever verte of G is either in or is adjacent to at least one verte from. G has a verte cover of sie m 10/3/2005 Kumar S ominating Set G = V, is an undirected graph. dominating set of G is a set of vertices a subset of V such that ever verte of G is either in or is adjacent to at least one verte from. 10/3/2005 Kumar S

11 ominating Set G = V, is an undirected graph. dominating set of G is a set of vertices a subset of V such that ever verte of G is either in or is adjacent to at least one verte from. G has a verte cover of sie m 10/3/2005 Kumar S ominating Set G = V, is an undirected graph. dominating set of G is a set of vertices a subset of V such that ever verte of G is either in or is adjacent to at least one verte from. G has a dominating set of sie m if and onl if G has a verte cover of sie m 10/3/2005 Kumar S

12 ominating Set G = V, is an undirected graph. dominating set of G is a set of vertices a subset of V such that ever verte of G is either in or is adjacent to at least one verte from. G has a dominating set of sie m if and onl if G has a verte cover of sie m 10/3/2005 Kumar S ealing with NP omplete problems Proving that a given problem is NP-omplete does not make the problem go awa!! Udi Manber n NP-omplete problem cannot be solved precisel in polnomial time We make compromises in terms of optimalit, robustness, efficienc, or completeness of the solution. pproimation algorithms do not lead to optimal solutions Probabilistic algorithms ranch and bound acktracking 10/3/2005 Kumar S