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 NPHard. 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 FirstFitDecreasing
policy that considers the items in nonincreasing 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
wellknown result is that this policy uses at most
11
9
OPT
+ 4 bins.
2. The maximum independent set problem is NPHard 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 nonempty. 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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 Fall '08
 Chekuri,C
 Algorithms, WI, Bin packing problem, RandomVertexCover, VertexCover, RandomWeightedVertexCover

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