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hw3soln - CME 305 Discrete Mathematics and Algorithms...

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CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi ([email protected]) HW#3 – Due 03/04/11 1. Recall the minimum vertex cover problem: given a graph G ( V, E ) find a subset S * V with minimum cardinality such that every edge in E has at least one endpoint in S * . (a) Consider the following greedy algorithm. Find the highest degree vertex, add it to the vertex cover S and remove it along with all incident edges. Repeat until all no edges remain. Prove that this algorithm has an unbounded approximation factor i.e. for any c there exists a graph G such that | S | ≥ c OPT. (b) Prove that when the graph G is bipartite we can find the minimum vertex cover in polynomial time. State your algorithm. Solution: (a) Consider a bipartite graph with partition ( A, B ). Let | A | = n and partition B into n disjoint sets { B i } n 1 with | B i | = b n/i c . Then | B | = n + b n/ 2 c + b n/ 3 c + · · · + 1 = O ( n log n ). For each vertex a A place an edge to a vertex b B k such that each vertex in B k has degree at least k . Repeat this for k = 1 , 2 , . . . , n and note that this construction is possible since each | B k | ≤ n/k . Clearly, OPT for this graph is just | A | = n . But the algorithm will (depending on how degree ties are broken) remove all vertices in B . The approximation ratio is ALG/OPT = O ( n log( n )) /n = O (log( n )) Since the ratio depends on n it is unbounded. (b) Let ( A, B ) be the bipartite partitioning of the graph and M be the maximum matching. The following definition is useful. A path P is called an alternating path e 1 , e 2 , . . . , e k if and only if e i
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