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Approximation Algorithms

Approximation Algorithms. Motivation and Definitions TSP Vertex Cover Scheduling. Motivation. Consider an NP-hard optimization problem like TSP Input: n cities, pairwise distances d(i,j) Task: find a tour of minimum length

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Approximation Algorithms

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  1. Approximation Algorithms • Motivation and Definitions • TSP • Vertex Cover • Scheduling

  2. Motivation • Consider an NP-hard optimization problem like TSP • Input: n cities, pairwise distances d(i,j) • Task: find a tour of minimum length • Given this is NP-hard, we are unlikely to find an optimal solution in polynomial time. • What are our options? • Give up • Try to build the fastest possible algorithm that returns an optimal solution • Find a polynomial-time algorithm that returns good solutions that are “approximately” optimal. • This is the option we focus on today.

  3. Definitions • For any input instance I, • let OPT(I) denote both the optimal solution and the value of the optimal solution • let A(I) denote both an approximate solution and the value of the approximate solution generated by running polynomial-time algorithm A. • So what do we mean by a good solution that is “approximately” optimal (for a minimization problem)? • Additive approximation • For any input I, A(I) ≤ OPT(I) + c for some constant c. • Multiplicative approximation • For any input I, A(I) ≤ c OPT(I) for some constant c.

  4. Are both approximations possible? • Consider the TSP problem • Suppose c = 3. • Additive: if OPT(I) = 10, then APPROX(I) ≤ 13. If OPT(I2) = 100, then APPROX(I) ≤ 103. • Multiplicative: if OPT(I) = 10, then APPROX(I) ≤ 30. If OPT(I2) = 100, then APPROX(I) ≤ 300. • Argue that at least one of these approximations is impossible unless we can solve TSP optimally in polynomial time. • Hint: think about scaling the integers in your input. • Is the other approximation possible for TSP?

  5. Definitions II • Because we typically cannot achieve additive approximation due to scaling, we try to get multiplicative approximation • For a minimization problem, an algorithm is a c-approximation algorithm if for all inputs I, • A(I) ≤ c OPT(I) • For a maximization problem, an algorithm is a c-approximation algorithm if for all inputs I, • A(I) ≥ 1/c OPT(I) • Approximation goal: for an NP-hard optimization problem, find a polynomial time algorithm that is a c-approximation algorithm for the smallest c possible.

  6. Example Problems • TSP • general TSP • metric TSP • Vertex cover • Scheduling

  7. TSP • We showed earlier there can be no additive approximation algorithm unless P=NP • Show now that for any constant c, there can be no c-approximation algorithm for TSP unless P=NP • Hint: show that if there is a c-approximation algorithm for some constant c, then Hamiltonian circuit can be solved in polynomial time

  8. Metric TSP • In the metric TSP, the city distances must satisfy a triangle inequality. • For any 3 cities i, j, k, it must be the case that d(i,j) ≤ d(i,k) + d(k,j) • Observe how our previous argument violates this triangle inequality • Show how to come up with a c-approximation algorithm for TSP based on a minimum spanning tree • What value of c can you come up with?

  9. Cristofides Improvement • Start with MST T as before • Identify nodes with odd degree • Find a minimum weight matching M on these nodes • Now compute an Euler tour of the graph of T union M (with shortcuts to prevent visiting an edge twice) • This solution is guaranteed to have length at most 3/2.

  10. Vertex Cover • Input: Graph G = (V,E) • Task: Find C subset of V of minimum size such that for each edge (u,v) in E, either u is in C or v is in C • This can be as bad as Θ(lg n) • Hint: make a bipartite graph • Make x nodes in one set all have the same max degree • May y nodes in other set have varying degrees • Greedy picks all the y nodes (with bad tie-breaking), optimal picks all the x nodes

  11. Vertex Cover • Better solution • Find a maximal matching M in G • Matching: set of edges in G that do not share any common vertices • Maximal matching: No edges can be added to the matching to produce a larger matching • Return as C all the nodes in edges in M • What approximation ratio does this guarantee and why?

  12. Scheduling • Input: • Set of m identical machines • Set of n jobs with length pi • Task: Assign the n jobs to the m machines with the goal of minimizing the maximum total length of jobs assigned to any one machine • Greedy strategy • Take the jobs one by one in any order • Assign the current job to the currently least loaded machine • What approximation bound can be derived? • What if we sort the jobs first? Should we do longest or shortest first?

  13. Scheduling • Greedy strategy • Take the jobs one by one in any order • Assign the current job to the currently least loaded machine • What approximation bound can be derived? • Lower bounds on OPT • What bounds can we derive on the best schedule? • How can we relate this algorithm’s schedule to OPT’s?

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