NP-Completeness - II

NP-Completeness - II

• 1.
  Design and Analysis of Algorithms SLAYINGTHE NP-HARD DRAGON
• 2.
   Instructor Prof. AmrinderArora amrinder@gwu.edu Please copy TA on emails Please feel free to call as well   Available for study sessions Science and Engineering Hall GWU Algorithms NP-completeness 2 LOGISTICS
• 3.
  Input for SAT Output for SAT Algorithmfor Known NPC Problem – Say SAT Reduction from SAT to X Algorithm for X x R(x) Yes/No Algorithms NP-completeness 3 GOOD NEWS: REDUCIBILITY Problem X is at least as hard as SAT. Problem X is NP-Hard.
• 4.
   Business problemsare not NP-complete Algorithms NP-completeness 4 BAD NEWS: PROVING THAT SOMETHING IS NPC IS NOT THE END OF THE GAME
• 5.
  MECHANISMS TO SOLVEREAL LIFE PROBLEMS – NPC OR OTHERWISE Strategy 1: Look for simplifications that render the problem solvable in polynomial time Strategy 2: Look for improvements in running time which make the exponential “bearable” Strategy 3: Understand performance bounds that are acceptable practically, and use approximation algorithms Strategy 4: Use parallel processing, cloud computing and server farms (Google, Facebook and government agencies do it) Algorithms NP-completeness 5
• 6.
   For generalgraphs, this problem is NPC  What if your graph is a tree (or a forest)?  MIS in Trees is very easily solvable  Recursive Formulation coming up next.. Algorithms NP-completeness 6 MAXIMUM INDEPENDENT SET (MIS)
• 7.
   Consider atree (or subtree) rooted at node v  NMIS(v): maximum independent set for the tree rooted at node v, such that v cannot be in the set.  MIS(v): maximum independent set for the tree rooted at node v.  If v is a leaf node, |NMIS(v)| = 0, |MIS(v)| = 1  Otherwise, MIS(v) = max { {v} union NMIS(u) for all child nodes u of v, MIS(u) for all child nodes u of v }  NMIS(v) = union of MIS(u) for all child nodes u of v  How long does this take? Algorithms NP-completeness 7 MIS IN TREES
• 8.
   Given S⊆ V, Let C(S, j) be the shortest path that starting at 1, visits all nodes in S and ends at j. (1 and j must be in S.)  If |S| = 2, then C(S, k) = d(1,k) for k = 2… n  If |S| > 2, then C(S, k) = the optimal tour from C(1,m) + d(m,k) Algorithms NP-completeness 8 TRAVELING SALESPERSON PROBLEM – A BETTER ALGORITHM USING DP Note to Self: O(n2 2n) is much better than O(n!)
• 9.
   An r-approximationalgorithm is a polynomial time algorithm that  returns a feasible solution in polynomial time  with value at most “r” times worse than optimal  Question: How can we compare our solution to the optimal that we cannot compute?  Answer: lower bound optimal value Algorithms NP-completeness 9 TSP – APPROXIMATION ALGORITHMS
• 10.
   Given ncities with distances between them  Find the shortest tour that visits each city at least once Algorithms NP-completeness 10 EXAMPLE: TSP (1/4) 1 2 3 4 5 d(1,2) 1 2 3 4 5 1 2 3 4 5
• 11.
   Removing oneedge gives a spanning tree  Kruskal’s algorithm runs in time O(n2log n) Algorithms NP-completeness 11 LOWER BOUND (2/4) 1 2 3 4 5 1 2 3 4 5
• 12.
   Run DFSon MST  Each vertex is visited at least once  Each edge traversed at most twice  Value of tour at most twice the MST, if triangle inequalities hold. Algorithms NP-completeness 12 CONSTRUCTING TOUR FROM MST (3/4) 1 23 45
• 13.
  1. Find MSTas lower bound 2. Construct tour by running DFS on MST Algorithms NP-completeness 13 2-APPROXIMATION ALGORITHM (4/4) OPTMST 2*MST
• 14.
   Is theproblem discrete or continuous?  If you have a Yes/No solver for the decision version of IS problem, we can find Maximum Independent Set in O(log n) invocations of Yes/No solver.  So, if problem is discrete, optimization problem is no harder than decision problem (assuming search space if finite)  If the problem is continuous, then read on…  NP-Complete vs. NP-Hard  Given a TSP tour and x: We can verify that the given TSP tour is a valid tour (covers each city exactly once) and its cost is less than x, in polynomial time. The decision version of the problem is in NP.  Given a TSP tour: We do not know of a way to verify that a given solution to TSP is indeed the optimal one. The optimization version is not known to be in NP. It is NP-Hard, but not known to be NP-Complete Algorithms NP-completeness 14 DECISION VERSUS OPTIMIZATION PROBLEM
• 15.
   Even NP-completeproblems deserve solutions  You can use 3 tricks to solve NPC problems  Look for better exponential solutions  Look for approximation scheme  Look for idiosyncrasies in data Algorithms NP-completeness 15 SUMMARY