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# Module35 - Hardness Results for Problems Example Problems P...

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1 Hardness Results for Problems Example Problems P: Class of “easy to solve” problems Absolute hardness results Relative hardness results Reduction technique

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2 Example: Clique Problem Input: Undirected graph G = (V,E), integer k Y/N Question: Does G contain a clique of size ≥ k? k=4 k=5
3 Example: Vertex Cover * Input: Undirected graph G = (V,E), integer k Y/N Question: Does G contain a vertex cover of size ≤ k? Vertex cover: A set of vertices C such that for every edge (u,v) in E, either u is in C or v is in C (or both are in C) k=3 k=2

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4 Example: Satisfiability * Input: Set of variables X and set of clauses C over X Y/N Question: Is there a satisfying truth assignment T for the variables in X such that all clauses in C are true?
5 Hardness Results for Problems Example Problems P: Class of “easy to solve” problems Absolute hardness results Relative hardness results Reduction technique

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6 Fundamental Setting When faced with a new problem Π, we alternate between the following two goals 1. Find a “good” algorithm for solving Π Use algorithm design techniques 2. Prove a “hardness result” for problem Π No “good” algorithm exists for problem Π
7 Complexity Class P P is the set of problems that can be solved using a polynomial-time algorithm Sometimes we focus only on decision problems The task of a decision problem is to answer a yes/no question If a problem belongs to P, it is considered to be “efficiently solvable” If a problem is not in P, it is generally considered to be NOT “efficiently solvable” Looking back at previous slide, our goals are to: 1. Prove that Π belongs to P 2. Prove that Π does not belong to P

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Module35 - Hardness Results for Problems Example Problems P...

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