# hw6 - CS 573 Graduate Algorithms Fall 2011 HW 6(will not be...

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CS 573: Graduate Algorithms, Fall 2011 HW 6 (will not be graded) This homework is on approximation algorithms. Solve as many as you can as practice for the ﬁnal exam. Discuss the problems on the newsgroup and we will aid the process. 1. The input to the Bin Packing problem consists of n items where item i has a given size s i [0 , 1]. The goal is to pack these items into the fewest possible number of bins each of which is of size 1. You have seen that this problem is NP-Hard. Typical heuristics consider the items in some (adaptive) order and pack the the current item into some existing bin or open a new bin. A greedy heuristic is one which opens a new bin only if the current item does not ﬁt into an existing bin. Show that any greedy policy uses at most 2 OPT bins. Can you prove a better approximation using a speciﬁc policy? Consider the First-Fit-Decreasing policy that considers the items in non-increasing size order and places the item in the ﬁrst bin that it ﬁts into. Show an approximation ratio strictly better than 2 for this policy. A well-known result is that this policy uses at most 11 9 OPT + 4 bins. 2. The maximum independent set problem is NP-Hard and morever it is known that unless P = NP one cannot obtain a 1 /n 1 - ± -approximation for any ﬁxed ± > 0. However, reasonable results can be obtained in various special cases. Consider a simple greedy algorithm that picks a vertex of minimum degree, removes it and its neighbors from the graph and continues as long as the remaining graph is non-empty. Show that this heuristic always outputs a solution of size Ω( n/ (1 + Δ)) where Δ is the average degree of the graph. First ﬁgure out the case when Δ is the maximum degree. Show that the greedy algorithm gives a constant factor

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hw6 - CS 573 Graduate Algorithms Fall 2011 HW 6(will not be...

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