Approximation Algorithms for Combinatiorial Optimization: International Workshop APPROX'98 Aalborg, Denmark, July 18–19, 1998 Proceedings

202 Pages·1998·2.84 MB·English

by  MagnÚs M. Halldórsson (auth.)

Lecture Notes in Computer Science 1444
Edited by G. Goos, J. Hartmanis and J. van Leeuwen
Klaus Jansen Jos6 Rolim (Eds.)
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International Workshop APPROX'98
Aalborg, Denmark, July 18-19, 1998
Proceedings
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Series Editors
Gerhard Goos, Karlsruhe University, Germany
Juris Hartmanis, Cornell University, NY, USA
Jan van Leeuwen, Utrecht University, The Netherlands
Volume Editors
Klaus Jansen
IDSIA Lugano
Corso Elvezia 36, CH-6900 Lugano, Switzerland
E-mail: [email protected]
Jos6 Rolim
University of Geneva, Computer Science Center
23, Rue Gtntral Dufour, CH-1211 Geneva 4, Switzerland
E-mail: jose.rolim @cui.unige.ch
Cataloging-in-Publication data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufuahme
Approximation algorithms for combinatorial optimization :
proceedings / International ICALP '98 Workshop, APPROX '98,
Aalborg, Denmark, July 18 - 19, 1998. Klaus Jansen ; Jos~ Rolim
(ed.). - Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong
Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 1998
(Lecture notes in computer science ; Vol. 1444)
ISBN 3-540-64736-8
CR Subject Classification (1991): F.2.2, G.1.2, G.1.6, G.3, 1.3.5
ISSN 0302-9743
ISBN 3-540-64736-8 Springer-Verlag Berlin Heidelberg New York
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Preface
The Workshop on Approximation Algorithms for Combinatorial Optimization
smelborP APPROX'98 focuses on algorithmic and complexity aspects arising
in the development of efficient approximate solutions to computationally difficult
problems. It aims, in particular, at fostering cooperation among algorithmic and
complexity researchers in the field. The workshop, to be held at the University
of Aalborg, Denmark, on July 18 - 19, 1998, co-locates with ICALP'98. We
would like to thank the organizer of ICALP'98, Kim Larsen, for this opportunity.
A previous event in Europe on approximate solutions of hard combinatorial
problems consisting in a school followed by a workshop was held in Udine (Italy)
in 1996.
Topics of interest for APPROX'98 are: design and analysis of approxima-
tion algorithms, inapproximability results, on-line problems, randomization tech-
niques, average-case analysis, approximation classes, scheduling problems, rout-
ing and flow problems, coloring and partitioning, cuts and connectivity, packing
and covering, geometric problems, network design, and various applications. The
number of submitted papers to APPROX'98 was 37. Only 14 papers were se-
lected. This volume contains the selected papers plus papers by invited speakers.
All papers published in the workshop proceedings were selected by the program
committee on the basis of referee reports. Each paper was reviewed by at least
three referees who judged the papers for originality, quality, and consistency with
the topics of the conference.
We would like to thank all authors who responded to the call for papers and
our invited speakers: Magnds M. Halld6rsson (Reykjavik), David B. Shmoys
(Cornell), and Vijay V. Vazirani (Georgia Tech). Furthermore, we thank the
members of the program committee:
- Ed Coffman (Murray Hill),
Pierluigi Crescenzi (Florence),
- - Ulrich Faigle (Enschede),
Michel X. Goemans (Louvain and Cambridge),
- Peter Gritzmann (Mfinchen),
- Magnfis M. Halld6rsson (Reykjavik),
- Johan Hs (Stockholm),
- Klaus Jansen (Saarbr/icken and Lugano, chair),
-
Claire Kenyon (Orsay),
- - Andrzej Lingas (Lund),
- George Lueker (Irvine),
- Ernst W. Mayr (Miinchen),
- Jose D.P. Rolim (Geneva, chair),
Andreas Schulz (Berlin),
- David B. Shmoys (Cornell),
- Jan van Leeuwen (Utrecht).
-
IV
and the reviewers Susanne Albers, Abdel-Krim Amoura, Gunnar Andersson,
Christer Berg, Ioannis Caragiannis, Dietmar Cieslik, A. Clementi, Artur Czu-
maj, Elias Dahlhaus, A. Del Lungo, Martin Dyer, Lars Engebretsen, Thomas
Erlebach, Uriel Feige, Stefan Felsner, Rudolf Fleischer, Andras Frank, R. Grossi,
Joachim Gudmundsson, Dagmar Handke, Stephan Hartmann, Dorit .S Hoch-
baum, J.A. Hoogeveen, Sandra Irani, Jesper Jansson, Mark Jerrum, David John-
son, Christos Kaklamanis, Hans KeUerer, Samir Khuller, Ekkehard Koehler, Ste-
fano Leonardi, Joseph .S B. Mitchell, Rolf H. MShring, .S Muthu Muthukrish-
nan, Petra Mutzel, Giuseppe Persiano, Joerg Rambau, Ramamoorthi Ravi, Ingo
Schiermeyer, Martin Skutella, Roberto Solis-Oba, Frederik Stork, Ewald Speck-
enmeyer, C.R. Subramanian, Luca Trevisan, Denis Trystram, John Tsitsiklis,
Marc Uetz, Hans-Christoph Wirth, Gerhard Woeginger, Martin Wolff, Alexan-
der Zelikovsky, and Uri Zwick.
z
We gratefully acknowledge sponsorship from the Max-Planck-Institute for
Computer Science Saarbriicken (AG ,1 Prof. Mehlhorn), ALCOM-IT Algorithms
and Complexity in Information Technology, and Siemens GmbH. We also thank
Luca Gambardella, the research institute IDSIA Lugano, Alfred Hofmann, Anna
Kramer, and Springer-Verlag for supporting our project.
May 8991 Klaus Jansen
Co e s
Invited Talks
Approximations of independent sets in graphs
Magnds M. HalldSrsson
Using linear programming in the design and analysis of 51
approximation algorithms: Two illustrative problems
David B. Shmoys
The steiner tree problem and its generalizations 33
Vijay .V Vazirani
Contributed Talks
Approximation schemes for covering and scheduling on 39
related machines
Yossi Azar and Leah Epstein
One for the price of two: A unified approach for 49
approximating covering problems
Reuven Bar- Yehuda
Approximation of geometric dispersion problems 63
Christoph Baur and Sdndor P. Fekete
Approximating k-outconnected subgraph problems 77
Joseph Cheriyan, Tibor Jorddn and Zeev Nutov
Lower bounds for on-line scheduling with precedence 89
constraints on identical machines
Leah Epstein
Instant recognition of half integrality and 2-approximations 99
Dorit S. Hochbaum
The t - vertex cover problem: Extending the half integrality 111
framework with budget constraints
Dorit .S muabhcoH
vllf
A new fully polynomial approximation scheme for the 321
knapsack problem
Hans Kellerer and Ulrich Pferschy
On the hardness of approximating spanners 531
Guy Kortsarz
Approximating circular arc colouring and bandwidth 741
allocation in all-opticai ring networks
Vijay Kumar
Approximating maximumindependentset in k-clique-free 951
graphs
IngoSchiermeyer
Approximating an interval scheduling problem 961
Frits C.R. Spieksma
Finding dense subgraphs with semidefinite programming 181
Anand Srivastav and Katja Wolf
Best possible approximation algorithm for MAX SAT 391
with cardinality constraint
Maxim I. Sviridenko
Author Index 201
Approximations of Independent Sets in Graphs
Magnfis M. Halld6rsson 2,1
1 Science Institute, University of Iceland, Reykjavik, Iceland. ,ih@almm is
2 Department of Informatics, University of Bergen, Norway.
1 Introduction
The independent set problem is that of finding a maximum size set of mutually
non-adjacent vertices in a graph. The study of independent sets, and their alter
egos, cliques, has had a central place in combinatorial theory.
Independent sets occur whenever we seek sets of items free of pairwise con-
flicts, e.g. when scheduling tasks. Aside from numerous applications (which might
be more pronounced if the problems weren't so intractable), independent sets and
cliques appear frequently in the theory of computing, e.g. in interactive proof
systems 6 or monotone circuit complexity 2. They form the representative
problems for the class of subgraph or packing problems in graphs, are essen-
tial companions of graph colorings, and form the basis of clustering, whether in
terms of nearness or dispersion.
As late as 1990, the literature on independent set approximations was ex-
tremely sparse. In the period since Johnson 13 started the study of algorithms
with good performance ratios in 1974 - and in particular showed that a whole
slew of independent set algorithms had only the trivial performance ratio of n on
general graphs - only one paper had appeared containing positive results ,92
aside from the special case of planar graphs 34, .8 Lower bounds were effec-
tively non-existent, as while it was known that the best possible performance
ratio would not be some fixed constant, there might still be a polynomial-time
approximation scheme lurking somewhere.
Success on proving lower bounds for Independent Set has been dramatic
and received worldwide attention, including the New York Times. Progress on
improved approximation algorithms has been less dramatic, but a notable body
of results has been developed. The purpose of this talk is to bring some of these
results together, consider the lessons learned, and hypothesize about possible
future developments.
The current paper is not meant to be the ultimate summary of independent
set approximation algorithms, but an introduction to the performance ratios
known, the strategies that have been applied, and offer glimpses of some of the
results that have been proven.
We prefer to study a range of algorithms, rather than seek only the best
possible performance guarantee. The latter is fine as far as it goes, but is not
the only thing that matters; only so much information is represented by a single
number. Algorithmic strategies vary in their time requirements, temporal access
to data, parallelizability, simplicity and numerous other factors that are far from
irrelevant. Different algorithms may also be incomparable on different classes of
graphs, e.g. depending on the size of the optimal solution. Finally, the proof
techniques are perhaps the most valuable product of the analysis of heuristics.
We look at a slightly random selection of approximation results in the body
of the paper. A complete survey is beyond the scope of this paper but is under
preparation. The primary criteria for selection was simplicity, of the algorithm
and the proof. We state some observations that have not formally appeared
before, give some recent results, and present simpler proofs of other results.
The paper is organized as follows. We define relevant problems and definitions
in the following section. In the body of the paper we present a number of par-
ticular results illustrating particular algorithmic strategies: subgraph removal,
semi-definite programming, partitioning, greedy algorithms and local search. We
give a listing of known performance results and finish with a discussion of open
issues.
2 Problems and definitions
TNEDNEPEDNI SET: Given a graph G = ,V( E), find a maximum cardinality set
I C V such that for each u, v E I, (u, v) (cid:127) E. The independence number of
G, denoted by a(G), is the size of the maximum independent set.
EUQILC PARTITION: Given a graph G -- ,V( E), find a minimum cardinality set
of disjoint cliques from G that contains every vertex.
TES-~t PACKING: Given a collection C of sets of size at most ~ drawn from a
finite set S, find a minimum cardinality collection C ~ such that each element
in S is contained in some set in g .~
These problems may also be weighted, with weights on the vertices (or on
the sets in SET .)GNIKCAP
A set packing instance is a case of an independent set problem. Given a set
system (g, S), form a graph with a vertex for each set in g and edge between
two vertices if the corresponding sets intersect. Observe that if the sets in g are
of size at most ,~t then the graph contains a ~ + 1-claw, which is a subgraph
consisting of a center node adjacent to n + 1 mutually non-adjacent vertices.
The independent set problem in n + 1-claw free graphs slightly generalizes TES-~
,GNIKCAP which in turn slightly generalizes LANOISNEMID-;~ .GNIHCTAM
The performance ratio PA of an independent set algorithm A is given by
pA =- pA(n) ~- max ")G(A
a,aal=,,
Notation
n the number of vertices d(v) the degree of vertex v
m the number of edges N(v) set of neighbors of v
A maximum degree N(v) non-neighbors of v
average degree A(G) the size of solution found by A
f( minimum degree PA performance ratio of A
a independence number clique partition number
maximum claw size
3 Ramsey theory and subgraph removal
The first published algorithm with a non-trivial performance ratio on general
graphs was introduced in 1990. In appreciation of the heritage that the late
master Erd6s left us, we give here a treatment different from Boppana and
Halld6rsson 12 that more closely resembles the original Ramsey theorem of
Erd6s and Szekeres 17.
Ramsey (G) CliqueRemoval (G)
ifG = @then return (0, 0) i+- 1
choose some v E G (Ci,Ii) +- yesmaR (G)
(C1,11) +- Ramsey(N(v)) whileG ~ @do
(C2,I2) --+ Ramsey(N(v)) G+-G-C~
return (larger of IC( U {v}, C2), i+-i+l
larger of (Ix, 21 U {v})) (C~,Ii) ~- yesmaR (g)
od
return ((ma,xj.= 1 Ij), {Cz, C2, ..., Ci})
Fig. 1. Independent set algorithm based on Ramsey theory
Theorem 1. Ramsey finds an independent set I and a clique C such that
11+11/I( cl) - 1 > n. In particular, I/1" ICI _> 1 log2 n.
Proof. The proof is by induction on both II I and ICI. It is easy to verify the
claim when either III or ICI are at most 1. By the induction hypothesis,
n= N(v)l + lN(v), + X _< ((111'1Cl11Cl1)-1)+ ((1121+1C21-1)-1)+1.16,21
Recall that ICl = max(lOll + 1, IC21) and I/I = max(lI11, 12/1 + 1). Thus,
n_ -1) + (III+ICI- 1) -
< C'I IcIICI 1 ICI 1.
The claim now follows from the equality (8+t) = (8+t-l~ + ( s-t-t--1 t )
xs~ kt--l!