CS 1510 Approximiation Algorithms Problems
1. Consider the vertex cover problem, that is, given a graph Gfind a minimal cardinality collection Sof
vertices with the property that every edge in Gis incident to a vertex in S.
Consider the following algorithm:
(a) Pick an arbitrary edge e= (v, w) from G
(b) Add vand wto S
(c) Remove vand wand all incident edges from G
(d) Go to step a.
Show that the performance ratio of this algorithm is at most 2.
HINT: First consider why the size of any matching (a collection of edges such that no pair are incident
on a common vertex) is a lower bound on the size of the vertex cover.
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2. Give an example instance of the vertex cover problem where the above algorithm does not give an
optimal solution.
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3. In this problem you have nboxes b1,...,bnthat you wish to load onto ktrucks. You know the integral
weight in kilograms wiof each box bi. You goal is to minimize the weight on more heavily loaded
truck.
Consider the obvious greedy algorithm that considers the boxes in an arbitrary order, and always places
the box under consideration into the least heavily loaded truck. Show that this algorithm guarantees
that the weight on the more heavily loaded truck is at most 2 times optimal. That is this algorithm
has performance ratio of at most 2.
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4. In this problem you have nboxes b1,...,bnthat you wish to load onto two trucks. You know the
integral weight in kilograms wiof each box bi. You goal is to minimize the weight on more heavily
loaded truck.
Consider the obvious greedy algorithm that considers the boxes in an arbitrary order, and always places
the box under consideration into the least heavily loaded truck. Show that this algorithms guarantees
that the weight on the more heavily loaded truck is at most 3
2times optimal. That is this algorithm
has performance ratio 3
2.
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5. Give an example of an instance to the above problem where the load on the more heavily loaded truck
is fifty percent larger than the load on the more heavily loaded truck in the optimal solution.
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6. In this problem you have nboxes b1,...,bnthat you wish to load onto two trucks. You know the
integral weight in kilograms wiof each box bi. You goal is to minimize the weight on more heavily
loaded truck.
Consider generalizations of the obvious greedy algorithm that considers the boxes in an arbitrary order,
and always places the box under consideration into one of the trucks based on some rule that relies
only on the load of the two trucks at that time. Show no such algorithm can have performance ratio
strictly better than 3
2.
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