Table Of ContentLecture Notes in Computer Science 1084
Edited by G. Goos, J. Hartmanis and J. van Leeuwen
Advisory Board: W. Brauer D. Gries J. Stoer
William .H Cunningham
.S Thomas McCormick
Maurice Queyranne ).sdE(
regetnI gnimmargorP
dna lairotanibmoC
noitazimitpO
5th International IPCO Conference
Vancouver, British Columbia, Canada
June 3-5, 1996
Proceedings
r e g ~ n i r p S
Series Editors
Gerhard Goos, Karlsruhe University, Germany
Juris Hartmanis, Cornell University, ,YN USA
Jan van Leeuwen, Utrecht University, The Netherlands
Volume Editors
William H. Cunningham
University of Waterloo, Department of Combinatorics and Optimization
Waterloo, Ontario, N2L 3Gl, Canada
S.Thomas McCormick
Maurice Queyranne
University of British Columbia
Faculty of Commerce and BusinessAdministration
Vancouver, British Columbia, V6T lZ2 Canada
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Integer programming and combinatorial optimization :
proceedings / 5th International IPCO Conference, Vancouver,
British Columbia, Canada, June 1996. William H. Cunningham
... (ed.). - Berlin ; Heidelberg ; New York ; Barcelona ;
Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ;
Singapore ; Tokyo (cid:12)9 Springer, 1996
(Lecture notes in computer science ; Vol. )4801
ISBN 3-540-61310-2
NE: Cunningham, William H. Hrsg.; International IPCO Conference
<5, 1996, Vancouver, British Columbia>; GT
CR Subject Classification (1991): G.!.6, G.2.1-2, E2.2
1991 Mathematics Subject Classification: 90Cxx, 65Kxx, 05-06, 90-06
ISBN 3-540-61310-2 Springer-Verlag Berlin Heidelberg New York
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Preface
This volume contains the papers selected for presentation at IPCO V, the Fifth
Integer Programming and Combinatorial Optimization Conference, Vancouver,
June 3-5, 1996. The IPCO series of conferences highlights recent developments in
theory, computation, and applications of integer programming and combinatorial
optimization.
These conferences are sponsored by the Mathematical Programming Society,
and are held in the years in which no International Symposium on Mathemat-
ical Programming takes place. Earlier IPCO conferences were held in Waterloo
(Ontario) in May 1990; Pittsburgh (Pennsylvania) in May 1992; Erice (Sicily) in
April 1993; and Copenhagen (Denmark) in May 1995. See the article ;'History
and Scope of IPCO", by Karen Aardal, Ravi Kannan, and William R. Pulley-
blank, published in 1994 in OPTIMA 43 (the newsletter of The Mathematical
Programming Society) for further details.
The proceedings of the first three IPCO conferences were published by or-
ganizing institutions. Since then they have been published by Springer in the
series, Lecture Notes in Computer Science. The proceedings of IPCO IV, Integer
Programming and Combinatorial Optimization, edited by Egon Balas and Jens
Clausen, were published in 1995 as Volume 920 of this series.
The 36 papers presented at IPCO V were selected from a total of 99 extended
abstracts. The overall quality of these submissions was extremely high. As a
result, many excellent papers could not be chosen.
The papers included in this volume in most cases represent expanded versions
of the submitted abstracts, within a stated 15-page limit. They have not been
refereed. We expect more detailed versions of most of these works to appear
eventually in scientific journals.
The Program Committee thanks all the authors of submitted abstracts and
papers for their support of the IPCO conferences.
March 1996
William It. Cunningham
S. Thomas McCormick
Maurice Queyranne
IPCO V Program Committee
William J. Cook, Rice University
G&ard Cornu~jols, Carnegie Mellon University
William H. Cunningham, University of Waterloo (chair)
Jan Karel Lenstra, Eindhoven University of Technology
Lgszld Lovgsz, Yale University
Thomas L. Magnanti, Massachusetts Institute of Technology
Maurice Queyranne, University of British Columbia
Giovanni Rinaldi, Istituto de Ana!isi dei Sistemi ed Informatica, Rome
Table of Contents
Session 1: Integer Programming Theory
Colourful Linear Programming ............................................. 1
L Bdrdny and S. Onn
Test Sets and Inequalities for Integer Programs ............................ 16
R.R. Thomas and R. Weismantel
An Optimal, Stable Continued Fraction Algorithm
for Arbitrary Dimension ................................................... 31
C. Rb'ssner and C.P. Schnorr
Session 2: Integer Programming Models
Algorithms and Extended Formulations for One and Two Facility
Network Design ........................................................... 44
S. Chopra, L Gilboa and S.T. Sastry
Integer Multicommodity Flow Problems ................................... 58
.C Barnhart, C.A. Itane and P.H. Vance
A Heuristic Algorithm for the Set Covering Problem ....................... 72
A. Caprara, M. Fischetti and P. Toth
Session 3: Network Flow Algorithms
An ~-Relaxation Method for Generalized Separable Convex Cost
Network Flow Problems ................................................... 85
P, Tseng and D.P. Bertsekas
Finding Real-Valued Single-Source Shortest Paths
in o(n 3) Expected Time ................................................... 94
S.G. Kolliopoulos and .C Stezn
A Network-Flow Technique for Finding Low-Weight Bounded-Degree
Spanning qYees .......................................................... 105
S.P. Fekete, S. Khuller, M. Klemmstein, B. Raghavachari and
N. Young
IIIV
Session 4: Approximation Algorithms
Approximating k-Set Cover and Complementary Graph Coloring ......... 811
M.M. Itallddrsson
On Minimum 3-Cuts and Approximating k-Cuts Using Cut Trees ......... 231
S. Kapoor
PrimM-Dual Approximation Algorithms for Feedback Problems
in Planar Graphs ........................................................ 741
M.X. Goemans and D.P. Williamson
Session 5: Semi-Definite Methods
Cone-LP's and Semidefinite Programs: Geometry
and a Simplex-Type Method ............................................. 261
.G Pataki
Quadratic Knapsack Relaxations Using Cutting Planes
and Semidefinite Programming ........................................... 571
.C Hehnberg, F. Rendl and R. Weismantel
A Semidefinite Bound for Mixing Rates of Markov Chains ................ 091
N. Kahale
Session 6: Matrix Models
The Quadratic Assignment Problem with a Monotone Anti-Monge
and a Symmetric Toeplitz Matrix: Easy and Hard Cases .................. 204
R.E. Burkard, E. ~ela, .G Rote and G.J. Woeginger
On Optimizing Multiplications of Sparse Matrices ........................ 219
E. Cohen
Continuous Relaxations for Constrained Maximum-Entropy Sampling ..... 234
K.M. Anstreicher, M. ~apmaF .Y eeL and .Y Williams
XI
Session 7: Set Systems and Submodularity
A Submodular Optimization Problem with Side Constraints .............. 249
D. Itartvigsen
Convexity and Steinitz's Exchange Property .............................. 260
K. Murota
On Ideal Clutters, Metrics and Multiflows ................................ 275
B. Novick and A. "6beS
Session 8: Scheduling I
A Supermodular Relaxation for Scheduling with Release Dates ............ 288
M.X. Goemans
Scheduling to Minimize Total Weighted Completion Time:
Performance Guarantees of LP-Based Heuristics and Lower Bounds ....... 301
A.S. Schulz
Implementation of a Linear Time Algorithm for
Certain Generalized Traveling Salesman Problems ........................ 316
N. Simonelti and E. Balas
Session 9: Probabilistic Methods
On Dependent Randomized Rounding Algorithms ........................ 330
D. Bertsirnas, C-P. oeT and R. Vohra
Coloring Bipartite Hypergraphs .......................................... 345
H. Chen and A. Frieze
Improved Randomized Approximation Algorithms
for Lot-Sizing Problems .................................................. 359
C-P. oeT and D. Bertsimas
Session 10: Scheduling II
Minimizing Total Completion Time in a Two-Machine Flowshop:
Analysis of Special Cases ................................................. 374
J.A. Hoogeveen and T. Kawaguchi
A New Approach to Computing Optimal Schedules
for the Job-Shop Scheduling Problem ..................................... 389
P. Martin and D.B. Shmoys
Optimal On-Line Algorithms for Single-Machine Scheduling .............. 404
J.A. Hoogeveen and A.P.A. Vestjens
Session 11: Polyhedral Methods
The Strongest Facets of the Aeyclic Subgraph Polytope Are Unknown .... 415
M.X. Goernans and L.A. Hall
Transitive Packing ....................................................... 430
R. Miiller and A.S. Schulz
A Polyhedral Approach to the Feedback Vertex Set Problem .............. 445
M. Funke and .G Reinelt
Session 12: The Travelling Salesman Problem
Separating over Classes of TSP Inequalities Defined by 0 Node-Lifting
in Polynomial Time ...................................................... 460
R. Carr
Separating Maximally Violated Comb Inequalities in Planar Graphs ...... 475
L. Fleischer and E. Tardos
The Travelling Salesman and the PQ-Tree ................................ 490
R.E. Burkard, V.G. De~neko dna G.J. Woeginger
Author Index .......................................................... 505
Colourful Linear Programming
Imre B~r~ny 1 * and Shmuel Onn 2 **
I Mathematical Institute of the Hungarian Academy of Sciences, P.O.Box 127,
Budapest, 1364 Hungary
bsranyQmath-inst.hu
2 Department of Operations l~esearch, School of Industrial Engineering and
Management, Technion - Israel Institute of Technology, 32000 Haifa, Israel
[email protected] echulon.ac.il
Abstract. We consider the following Colourf~l generalisation of Lin-
ear Programming: given sets of points SI,'",St C ~d, referred to
as colours, and a point b E R a, decide whether there is a colourful
T : {sx, .... st) such that b ~ cony(T), and if there is, find one. Linear
Programming is obtained by taking h = d + 1 and 1$ = ... = Sa+l.
If k = d + 1 and b E N~+t 1 conv(Si) then a solution always exists, but
finding it is still hard. We describe an iterative approximation algorithm
for this problem, that finds a colourful T whose convex hull contains a
point e-close to ,b and analyse its Real Arithmetic and Taring complexi-
ties. We then consider a class of linear algebraic relatives of this problem,
and give a computational complexity classification for the related deci-
sion and counting problems that arise. In particular, Colourful Linear
Programming is strongly Aft-complete. We also introduce and discuss
the complexity of a hierarchy of (~1, ~2)-Matroid-Basis-Nonbasis prob-
lems, and give an application of Coloufful Linear Programming to the
algorithmic problem of Tverberg's theorem in combinatorial geometry.
1 Introduction
The so-called Cara~h~odory's Theorem allows to pose the problem of Linear
Programming as follows.
Linear Programming Problem. Given a finite set S C ~d and a point b E
~d Decide whether there is s subset T _C S of size at most d-~ 1 such that
b E cony(T), and if there is, Find one.
Carath~odory's Theorem admits a colourf~l generalization, due to the first au-
thor 3. To state it, we use the following terminology: given a family of sets
St--., Sk C R d, referred to as colours, a colourful set is a set T = ~sx,..., sk~
where si G Si for all i.
* Partially supported by Hungarian National Sdence Foundation no. 4296 and 016937.
~* Partially supported by the Alexander yon Humboldt Stiftung, the Fund for the
Promotion of Research at the Technlon, and Technlon VPR Fund no. 191-198.
Theorem 1. Coloufful Carath~odory's Theorem. If each of d -F 1" given colours
So,...,Sd C aR in d-space contains the point b in its convez hull, then b E
cony(T) for some eolourful set T = {so,..., sd).
A proof will be given in the next section.
The following algorithmic problem suggested by Theorem 1 is a natural gen-
eralization of Linear Programming.
Colourful Linear Programming Problem. Given colours $1,..-, kS C ~d
and a point b E ~a Decide whether there is a colourful T = {sl,..., sk} such
that b G cony(T), and if there is, Find one.
The specialization of this problem to Linear Programming is obtained by taking
S = $1 = ... = Sd+l.
In this article we study the complexity of this problem and its relatives
in linear algebra, matroids, and combinatorial geometry. We provide a rather
efficient approximation algorithm for the problem, and give a computational
complexity classification for a hierarchy of related decision and counting prob-
lems in linear algebra. In particular, Colourful Linear Programming is strongly
Aft-complete. We also introduce and discuss the complexity of a hierarchy of
(wl, w2)-Matroid-Basis-Nonbasis problems, and give an application of Colour-
ful Linear Programming to the algorithmic problem of Tverberg's theorem in
combinatorial geometry.
The article is organized as follows. In Sections 2 and 3 we study an approx-
imation algorithm for Colourful Linear Programming. We concentrate on the
case k = d§ and b= 0 E If~al+l i=l conv(Si) where a solution is guaranteed to
exist, but needs to be found. Given an e ~ ,0 the algorithm finds a colourful T
which is e-close to 0, that is, whose convex hull contains a point which is e-close
to 0. Interestingly, our algorithm specializes, in the case 1$ = .... Sd+l, to
an algorithm of yon Neumann for Linear Programming. Assuming that each ~S
contains at most n points, that the points are normalized, and that a ball B(0, p)
is contained in n I 1=~a+l conv(S~), we obtain the following results on the Real arith-
metic and Turing complexities of the algorithm, respectively (see Section 3 for
the precise statements):
Theorem 4: There is a positive constant c such that the number of real
-
arithmetic operations taken by the algorithm to find a colourful T which is
~-close to 0 is O k--~-log .
5: There is a positive constant c such that the running time of
- Theorem
the algorithm to find a colourful T which is e-close to 0, when applied to
rational data of bit size L, is O \ ~p
In Sections 4 and 5 we give a computational complexity classification of a hierar-
chy of related problems. Colourful Linear Programming is equivalent to deciding
if $1,..., kS admit a colourful T which is positively dependent. Replacing "pos-
itively" by "linearly" and "dependent" by "independent", we get four decision
