Algorithm Design

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Chapter 11 Approximation Mgofithms
we seek to minimize maxi Li. This is the definition of the Generalized Load
Balancing Problem.
In addition to containing our initial Load Balancing Problem as a special
case (setting Mj = M for all jobs j), Generalized Load Balancing includes the
Bipartite Perfect Matching Problem as another special case. Indeed, given a
bipartite graph with the same number of nodes on each side, we can view the.
nodes on the left as jobs and the nodes on the right as machines; we define
tj = 1 for all jobs j, and define Mj to be the set of machine nodes i such that
there is an edge (i, j) E E. There is an assignment of maximum load 1 if and
only if there is a perfect matching in the bipartite graph. (Thus, network flow
techniques can be used to find the optimum load in this special case.) The
fact that Generalized Load Balancing includes both these problems as special
cases gives some indication of the challenge in designing an algorithm for it.
i~J Designing and Analyzing the Algorithm
We now develop an approximation algorithm based on linear programming for
the Generalized Load Balancing Problem. The basic plan is the same one we
saw in the previous section: we’ll first formulate the problem as an equivalent
linear program where the variables have to take specific discrete values; we’ll
then relax this to a linear program by dropping this requirement on the values
of the variables; and then we’ll use the resnlting fractional assignment to obtain
an actual assignment that is dose to optimal. We’ll need to be more careful than
in the case of the Vertex Cover Problem in rounding the solution to produce
the actual assignment.
Integer and Linear Programming Formulations First we formulate the Generalized
Load Balancing Problem as a linear program with restrictions on the
variable values. We use variables xq corresponding to each pair (i, j) of machine
i E M and iob j ~ I. Setting xq = 0 will indicate that )oh jis not assigned
to machine i, while setting xq = t~ will indicate that all the load ti of iob j is
assigned to machine i. We can think of x as a single vector with mn coordinates.
We use linear inequalities to encode the requirement that each iob is
assigned to a machine: For each iob j we require that ~_,i xq = t~. The load
of a machine i can then be expressed as L i = ~4 xq. We require that xq = 0
whenever i ~ My. We will use the obiecfive function to encode the goal of
finding an assignment that minimizes the maximum load. To do this, we
will need one more variable, L, that will correspond to the load. We use the
inequalities ~7 xq < L for all machines i. In summary, we have formulated the
following problem.
11.7 Load Balancing Revisited: A More Advanced LP Application
(GL.IP) rain L
~--~xq=ty for all j ~ J
E xq 0 for ally ~ J and i E My. Let (GL.LP) denote the resulting linear program. It
would also be natural to add xq L.
We can use linear programming to obtain such a solution (x, L) in polynomial
time. Our goal will then be to use x to create an assignment. Recall that
the Generalized Load Balancing Problem is NP-hard, and hence we cannot expect
to solve it exactly in polynomial time. Instead, we will find an assignment
with load at most two times the minimum possible. To be able to do this, we
will also need the simple lower bound (11.2), which we used already in the
original Load Balancing Problem.
(11.28) The optimal load is at least L* > maxy ty.
Rounding the Solution When There Are No Cycles The basic idea is to
round the xq values to 0 or ty. However, we cannot use the simple idea of
iust rounding large values up and small values down. The problem is that the
linear programming solution may assign small fractions of a job j to each of
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