6.856 — Randomized Algorithms
David Karger
Handout #11, Mar. 31, 2011 — Homework 7, Due 4/6
1. This problem should be done without collaboration. In class we talked about
how to count the number of satisfying assignments to a DNF, which is equivalent to
estimating the probability of a satisfying assignment if variables get unbiased random
assignments. Suppose that instead each variable gets set to true with probability
p < 1/2. Explain how to estimate the probability of getting a satisfying assignment.
2. Consider a set Sof vectors in d-dimensional space. Suppose that you sample each
of those vectors independently with probability p. Prove that there are at most d/p
vectors of Sin expectation that are not spanned by your sample.
3. A flow in an undirected graph is a set of edge-disjoint paths from a source vertex s
to a sink vertex t. The value of the flow is the number of edge disjoint paths. The
s-tmaximum flow problem aims to a flow of maximum value. This quantity turns out
to be equal to the s-tminimum cut value: the minimum number of edges that must
be removed from the graph in order to disconnect vertex sfrom vertex t. There is an
augmenting path algorithm that, given an s-tflow of value v, finds an s-tflow of value
v+ 1 in O(m) time on an m-edge graph, or else reports that vis the maximum flow.
Consider any undirected graph with medges, s-tmaximum flow v, and minimum cut
c:
(a) Prove for any constant , an s-tcut of value at most (1 + )vcan be found in
˜
O(mv/c2) time.
(b) Prove that for any constant , a flow of value (1 −) can be found in ˜
O(mv/c)
time.
(c) Sketch an algorithm that finds the maximum flow in ˜
O(mv/√c) time, and give a
informal argument as to its correctness.
(d) Use the algorithm of part (c) to improve the running times of the algorithms in
parts (a) and (b)
4. In class we gave an (, δ)-FPRAS for estimating the probability of a graph Gdiscon-
necting when each edge fails with probability p. You will show how to generate a
random disconnected version of Gfrom this distribution.
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