P vs NP problem

Transcript

1. Problems those we can solve, and those we can not ... yet david.ng@oursky.com
2. Satisfiability Problems Existence Search Decision
3. Example Given a list... [20, 1, 5323, 2319, 2387, 0, 27, 23] Question: Is 20 in the list ? Question: What is the biggest number in the list? Question: What should be the next number?
4. Some problems are easy...
5. P P : all decision problems , some of them: 1. You can verify the solution easily. 2. You can (or some people can) solve it in a polynomial-time A turing machine
6. Determinism DFA NFA
7. Example 3-SUM Given a set S of n integers, are there elements a, b, c in S ? a + b + c = 0 You can do so in O(n2) time
8. Example -25 -10 -7 -3 2 4 8 10 (a+b+c==-25) -25 -10 -7 -3 2 4 8 10 (a+b+c==-22) . . . -25 -10 -7 -3 2 4 8 10 (a+b+c==-7) -25 -10 -7 -3 2 4 8 10 (a+b+c==-7) -25 -10 -7 -3 2 4 8 10 (a+b+c==-3) -25 -10 -7 -3 2 4 8 10 (a+b+c==2) -25 -10 -7 -3 2 4 8 10 (a+b+c==0) Yet, to solve 3SUM, we used a subquadratic-time algorithm
9. Example Minimum spanning tree Given a connected, undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together. Famous DFS/ BFS / Dijkstra's algorithm Work within O(|V|2)
10. Some problems are hard...
11. How hard? 1. You can verify the solution easily. 2. But you (nor neither others in the world) can not solve it in a polynomial-time Definition NP stands for “nondeterministic polynomial time,” a term going back to the roots of complexity theory. Intuitively, it means that a solution to any search problem can be found and verified in polynomial time by a special (and quite unrealistic) sort of algorithm, called a nondeterministic algorithm. Such an algorithm has the power of guessing correctly at every step.
12. Example Euler's path Find a path that one exactly walks across each bridge once
13. Example Clique finding the maximum clique (a clique with the largest number of vertices) finding a maximum weight clique in a weighted graph listing all maximal cliques (cliques that cannot be enlarged) solving the decision problem of testing whether a graph contains a clique larger than a given size. I suspect it's hard... but how to show its HARDNESS?
14. Approach to solve... Anyway, when we try to solve a problem.. 1. You know how to solve it... (DONE) 2. You don't know how to solve the current problem... Reduce it to another similar problem! If you know how to solve it
15. Reduction Reducing A -> B Try to solve B
16. Reduction Reduction between search problems
17. Reduction Independent set Vertex cover Find maximum independent sets such that v 1 , v 2 , v n do not have an edge connecting them Find min. vertex number to cover the graph G VertexCover(G,k) G' = G k' = |V| - k return IndependentSet(G',k')
18. NP-complete THEOREM If any NP-complete problem is solvable in polynomial time, then P = NP. If any NP-complete problem is not solvable in polynomial time, then P != NP. THEOREM 2 (Cook (1971) and Levin (1973)) Satisfiability is NP-complete. A search problem is NP-complete if all other search problems reduce to it
19. NP-complete Space NP f all search problems, assuming P != NP
20. Basic NP-complete Problems 3-Satisfiability (3SAT) Vertex Cover Instance: An undirected graph G and an integer K Question: Is there a vertex cover of size K or less for G, i. e., a subset V' of V with the size of V' less than K such that every edge has at least one endpoint in V'. C1 = (x1 V x2 V x3) C2 = (x2 V x3 V x4) C3 = (x1 V x2 V x4) C4 = (x1 V x3 V x4) Question: Find values for x1 to x4 that C1 && C2 && C3 && C4 -> true
21. Deal with NP Intelligent exhaustive search Backtracking
22. Deal with NP Heuristics Relies on ingenuity, intuition, good understanding of the application, experiments and often insights from physics or biology to attack a problem Approximation algorithms So, when faced with an NP-complete optimization problem, a reasonable goal is to look for an approximation algorithm A to obtain a nearest solution for the problem. Fixed-Parameter Tractable algorithm (FPT) Parameterized the problem, which makes it to be solvable in polynomial complexity time limit. Reduce the scale of the NP-hard problem.
23. NP vs P Some people are trying to prove P == NP One of the Millennium Prize Problems
24. Implications We are making good use of P every day E.g. Is Key K the correct key for decryption function f(x,k) ? Some NP question maybe... E.g. Find ALL correct key for decryption function f(x,k) .
25. Implications Knowing NP Design specific algorithm to tackle it. To solve in other ways Appoximation Fixed-Parameter Tractable algorithm (FPT)
26. Implications If P = NP ? Many of the problems we know to be in NP or NP-complete are problems that we actually want to solve. By reduction ... NP P SOLVED polynomial time polynomial time
27. Know more General definition and examples http://compgeom.cs.uiuc.edu/~jeffe/teaching/algorithms/notes/21-nphard.pdf http://www.cs.berkeley.edu/~vazirani/algorithms/chap8.pdf Battleship example http://www.mountainvistasoft.com/docs/BattleshipsAsDecidabilityProblem.pdf List of NPC problems http://en.wikipedia.org/wiki/List_of_NP-complete_problems