PRIMALDUAL RNC APPROXIMATION ALGORITHMS FOR SET
COVER AND COVERING INTEGER PROGRAMS
*
SRIDHAR RAJAGOPALAN
†
AND
VIJAY V. VAZIRANI
‡
SIAM J. C
OMPUT
.
c
1998 Society for Industrial and Applied Mathematics
Vol. 28, No. 2, pp. 525–540
Abstract.
We build on the classical greedy sequential set cover algorithm, in the spirit of the
primaldual schema, to obtain simple parallel approximation algorithms for the set cover problem
and its generalizations. Our algorithms use randomization, and our randomized voting lemmas may
be of independent interest. Fast parallel approximation algorithms were known before for set cover,
though not for the generalizations considered in this paper.
Key words.
algorithms, set cover, primaldual, parallel, approximation, voting lemmas
PII.
S0097539793260763
1. Introduction.
Given a universe
U
, containing
n
elements, and a collection,
S
·
=
{
S
i
:
S
i
⊆ U}
, of subsets of the universe, the
set cover
problem asks for the
smallest subcollection
C ⊆ S
that covers all the
n
elements in
U
(i.e.,
S
S
∈
C
S
=
U
).
In a more general setting, one can associate a cost,
c
S
, with each set
S
∈ S
and ask
for the minimum cost subcollection which covers all of the elements.
1
We will use
m
to denote
S
.
Set multicover
and
multiset multicover
are successive natural generalizations of
the set cover problem.
In both problems, each element
e
has an integer coverage
requirement
r
e
, which specifies how many times
e
has to be covered. In the case of
multiset multicover, element
e
occurs in a set
S
with arbitrary multiplicity, denoted
m
(
S, e
). Setting
r
e
= 1 and choosing
m
(
S, e
) from
{
0
,
1
}
to denote whether
S
contains
e
gives back the set cover problem.
The most general problems we address here are
covering integer programs
. These
are integer programs that have the following form:
min
c
·
x
,
s.t.
M
x
≥
r
,
x
∈
Z
+
;
the vectors
c
and
r
and the matrix
M
are all nonnegative rational numbers.
Because of its generality, wide applicability, and clean combinatorial structure, the
set cover problem occupies a central place in the theory of algorithms and approxima
tions. Set cover was one of the problems shown to be NPcomplete in Karp’s seminal
paper [Ka72]. Soon after this, the natural greedy algorithm, which repeatedly adds the
set that contains the largest number of uncovered elements to the cover, was shown to
be an
H
n
factor approximation algorithm for this problem (
H
n
= 1+1
/
2+
. . .
+1
/n
)
by Johnson [Jo74] and Lovasz [Lo75]. This result was extended to the minimum cost
case by Chvatal [Ch79].
Lovasz establishes a slightly stronger statement, namely,
that the ratio of the greedy solution to the optimum fractional solution is at most
*
Received by the editors November 26, 1993; accepted for publication (in revised form) October
21, 1996; published electronically July 28, 1998.
http://www.siam.org/journals/sicomp/282/26076.html
†
DIMACS, Princeton University, Princeton, NJ 08544. Current address: IBM Almaden Research
Center, San Jose, CA 95120 ([email protected]). This research was done while the author
was a graduate student at the University of California, Berkeley, supported by NSF PYI Award CCR
8896202 and NSF grant IRI 9120074. Part of this work was done while the author was visiting IIT,
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Algorithms, Linear Programming Relaxation, RNC, multiset multicover, VIJAY VAZIRANI

Click to edit the document details