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NP-Completeness Introduction Decision Problems Polynomial-Time Reductions Introduction .............................................. 2 Decision Problems .......................................... 3 Reductions ............................................... 4 Problem Reduction Sequence ................................... 5 3SAT ................................................... 6 CLIQUE ................................................. 7 Reducing 3SAT to CLIQUE .................................... 8 VERTEX-COVER .......................................... 9 Reducing CLIQUE to VERTEX-COVER .......................... 10 SUBSET-SUM ........................................... 11 Reducing 3SAT to SUBSET-SUM .............................. 12 Complexity Class P ........................................ 13 Verification Algorithms ...................................... 14 Complexity Class NP ....................................... 15 Polynomial-Time Reducibility.................................. 16 Complexity Class NPC ...................................... 17 1 Introduction NP-Complete problems are of interest because they are problems we would like to solve, but it is unknown whether they have efficient algorithms. This includes: traveling salesman, knapsack, analysis of boolean expressions (always/never/sometimes true). We will start with examples of reductions , then to the definition of NP and NP-complete. CS 3343 Analysis of Algorithms NP-Completeness – 2 Decision Problems A problem specifies an input-output relationship, e.g., sorting a sequence of numbers, finding the minimum spanning tree. An instance is a single, specific input of a problem, a specific sequence of numbers, a specific weighted graph. In a decision problem, the output is either yes or no. Is the median greater than 50? Does the minimum spanning tree have a weight less than 50? CS 3343 Analysis of Algorithms NP-Completeness – 3 2
