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Approximation Algorithms Lecture for CS 302

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What is a NP problem? Given an instance of the problem, V, and a ‘certificate’, C, we can verify V is in the language in polynomial time All problems in P are NP problems – Why?

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VERTEX-COVER Given a graph, G, return the smallest set of vertices such that all edges have an end point in the set

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HAMILTONIAN PATH Given a graph, G, find a path that visits every vertex exactly once Alt version: Find the path with the minimum weight

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What is NP-Complete? A problem is NP-Complete if: – It is in NP – Every other NP problem has a polynomial time reduction to this problem NP-Complete problems: – 3-SAT – VERTEX-COVER – CLIQUE – HAMILTONIAN-PATH (HAMPATH)

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Applications

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SAT is used to verify circuit design NEED MORE EXAMPLES. WANT PICTURES

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Dilemma NP problems need solutions in real-life We only know exponential algorithms What do we do?

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Accuracy NP problems are often optimization problems It’s hard to find the EXACT answer Maybe we just want to know our answer is close to the exact answer?

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Approximation Algorithms Can be created for optimization problems The exact answer for an instance is OPT The approximate answer will never be far from OPT We CANNOT approximate decision problems

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k-approximation S is the approx. answer, OPT is optimal Maximization – k*OPT ≤ S ≤ OPT Minimization – OPT ≤ S ≤ k*OPT

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Approximate Vertex-Cover Let S be the cover Pick an edge (u,v) in the graph Add it’s end-points u and v to S Remove any edge that neighbors u or v Repeat until there are no edges left

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Approximate Vertex-Cover OPT must cover every edge so either u or v must be in the cover => OPT > ½|S| => 2*OPT ≥ |S| We have a 2-approximation

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Traveling Salesperson Problem Given a graph, find a minimum weight hamiltonian path There is a 2-approximation based on MINIMUM SPANNING TREES

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Minimum Spanning Tree Given a graph, G, a Spanning Tree of G is a subgraph with no cycles that connects every vertex together A MST is a Spanning Tree with minimal weight

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Finding a MST Finding a MST can be done in polynomial time using PRIM’S ALGORITHM or KRUSKAL’S ALGORITHM Both are greedy algorithms Details can be found on Wikipedia

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MST vs HAMPATH Any HAMPATH becomes a Spanning Tree by removing an edge cost(MST) ≤ cost(min-HAMPATH)

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Approximate TSP Given a complete graph G Compute G’s MST, M The tour is the pre-order traversal of M This is a 2-approximation

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Approximating 3-SAT f is a Boolean formula in 3-CNF form if: What’s the optimization version of 3-SAT? Satisfy as many clauses as you can

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Approximating 3-SAT Algorithm: – For each variable x i assign True with probability ½, False with probability ½ – This satisfies 7/8ths of the clauses in expectation

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Approximating 3-SAT x1x1 x2x2 x3x3 TTTT TTTF TTFT TTFF TFTT TFTF TFFT FFFF The only way we don’t satisfy the clause is if we select the last assignment. This happens only 1/8 th of the time.

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