Improved
Approximation
Algorithms
for
Maximum
Cut
and
Satisfiability
Problems
Using
Semidefinite
Programming
MIC13EL
X.
GOEMANS
Massachusetts
Institute
of
Technology,
Cambridge,
Massachusetts
AND
DAVID
P.
WILLIAMSON
IBM
T. J. Watson Research
Center,
Yorktown
Heights,
New
York
Abstract.
We present randomized
approximation
algorithms
for the maximum cut (MAX
CUT)
and maximum
2-satisfiability
(MAX
2SAT) problems
that
always deliver
solutions
of expected
value
at
least
.87856 times
the
optimal
value.
These
algorithms
use a simple
and
elegant
technique
that
randomly
rounds
the
solution
to
a nonlinear
programming
relaxation.
This
relaxation
can be interpreted
both as a semidefinite
program and as an eigenvalue minimization
problem.
The best previously known approximation
algorithms
for
these problems
had perfc~r-
mance guarantees of ~ for MAX
CUT
and ~ for MAX
2SAT. Slight extensions of our analysis
lead to a .79607-approximation
algorithm
for the maximum directed cut problem (MAX
DICUT)
and a .758-approximation
algorithm
for MAX
SAT, where the best previously known approxim a-
tion algorithms had performance guarantees of ~ and ~, respectively. Our algorithm
gives the first
substantial progress in approximating
MAX
CUT in nearly twenty years, and represents the first
use of :semidefinite programming
in the design of approximation
algorithms.
Categories
and
Subject
Descriptors:
F2.2 [Analysis
of Algorithms
and
Problem
Complexity]:
Nonumerical
Algorithms
and Problems—computations
on discrete structures;
G2.2
[Discrete Math-
A preliminary
version has appeared in Proceedings of the 26th AnnualACM
Symposium
on Theory
of Computing
(Montreal,
Que., Canada). ACM,
New York,
1994, pp. 422–431.
The research of M. X. Goemans was supported in part by National
Science Foundation
(NSF)
contract CCR 93-02476 and DARPA
contract NOO014-92-J-1799.
The
research of
D.
P. Williamson
was supported
by
an NSF Postdoctoral
Fellowship.
This
research was conducted while the author was visiting MIT.
Authors’
addresses: M. X. Goemans, Department
of Mathematics,
Room 2-382, Massachusetts
Institute
of Technology, Cambridge, MA 02139, e-mail: goemans@math.mit. edu; D. P. Williamson,
IBM
T, J. Watson Research Center, Room 33-219, P.O. Box 218, Yorktown
Heights, NY
1059I8,
e-mail: dpw@watson.ibm. corn.
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01995
ACM
0004-5411/95/1100-1115
$03.50
Journalof theAssociationfor ComputinsMachinery,Vol. 42,No.6,November1995,pp.1115-1145.