Algorithm Design

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Chapter 7 Network Flow
51.
The machines run at different speeds, but jobs are identical in their
processing need.s. More formally, each machine i has a parameter e~, and
each job requ~es e~ time ff assigned to machine i.
There are n jobs. Jobs have identical processing needs but different
levels of urgency. For each job j, we are given a cost function q(t) that
is the cost of completing job ] at time t. We assume that the costs are
nonnegative, and monotone in t.
A schedule consists of an assignment of jobs to machines, and on
each machine the schedule gives the order in which the jobs are done.
The job assigned to machine i as the first job will complete at time
the second job at time 2eg and so on. For a schedule S, let ts(J) denoke
the completion time of job j in this schedule. The cost of the schedule is
cost(S) = ~j q(ts(~)).
Give a polynomial-time algorithm to find a schedule of minimum cost.
Some friends of yours have grown tired of the game "SLx Degrees of Kevin
Bacon" (after a11, they ask, isn’t it just breadth-first search?) and decide to
invent a game with a little more punch, algorithmically speaking. Here’s
how it works.
You start with a set X of n actresses and a set Y of n actors, and two
players P0 and P1. Player P0 names an actress xl ~ X, player P1 names an
actor Yl who has appeared in a movie with xl, player P0 names an actress x2
who has appeared in a movie with yl, and so on. Thus, P0 and P1 collectively
generate a sequence x~, y~, xz, Yz .... such that each actor/actress in the
sequence has costarred with the actress/actor immediately preceding. A
player P~ (~ = 0, 1) loses when it is P~’s turn to move, and he/she cannot
name a member of his/her set who hasn’t been named before.
Suppose you are given a specific pair of such sets X antl Y, with
complete information on who has appeared in a movie with whom. A strategy
for Pg, in our setting, is an algorithm that takes a current sequence
x~, y~, x2, Yz .... and generates a legal next move for P~ (assuming it’s P~’s
turn to move). Give a polynomial-time algorithm that decides which of
the two players can force a win, in a particular instance of this game.
Notes and Further Reading .----
Network flow emerged as a cohesive subject through the work of Ford and
Fulkerson (1962). It is now a field of research in itself, and one can easily
Notes and Fun_her Reading
devote an entire course to the topic; see, for example, the survey by Goldberg,
Tardos, and Tarian (1990) and the book by Ahuja, Magnanti, and Orlin (1993).
Schrijver (2002) provides an interesting historical account of the early
work by Ford and Fulkerson on the flow problem. Lending further support
to those of us who always felt that the Minimum-Cut Problem had a slightly
destructive overtone, this survey cites a recently declassified U.S. Air Force
report to show that in the original motivating application for minimum cuts,
the network was a map of rail lines in the Soviet Union, and the goal Was to
disrupt transportation through it.
As we mention in the text, the formulations of the Bipartite MatChing
and Disjoint Paths Problems predate the Maximum-Flow Problem by several
decades; it was through the development of network flows that these were all
placed on a common methodological footing. The rich structure of matchings
in bipartite graphs has many independent discoverers; P. Hall (1935) and
K6nig (1916) are perhaps the most frequently cited. The problem of finding
edge-disjoint paths from a source to a sink is equivalent to the Maximum-
Flow Problem with all capacities equal to 1; this special case was solved (in
essentially equivalent form) by Menger (1927).
The Preflow-Push Maximum-Flow Algorithm is due to Goldberg (1986),
and its efficient implementation is due to Goldberg and Tarjan (1986). Highperformance
code for this and other network flow algorithms can be found at
a Web site maintained by Andrew Goldberg.
The algorithm for image segmentation using minimum cuts is due to
Greig, Porteous, and Seheult (!989), and the use of minimum cuts has become
an active theme in computer vision research (see, e.g., Veksler (1999)
and Kolmogorov and Zabih (2004) for overviews); we will discuss some further
extensions of this approach in Chapter 12. Wayne (2001) presents further
results on baseball elimination and credits Alan Hoffman with initially popularizing
this example in the 1960s. Many further applications of network flows
and cuts are discussed in the book by Ahuja, Magnanti, and Orlin (1993).
The problem of finding a minimum-cost perfect matching is a special case
of the Minimum-Cost Flow Problem, which is beyond the scope of our coverage
here. There are a number of equivalent ways to state the Minimum-Cost Flow
Problem; in one formulation, we are given a flow network with both capacities
c e and costs Ce on the edges; the cost of a flow [ is equal to the sum of the edge
costs weighted by the amount of flow they carry, ~e Cef(e), and the goal is
to produce a maximum flow of minimum total cost. The Minimum-Cost Flow
Problem can be solved in polynomial time, and it too has many applications;
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