Table Of ContentINTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
C 0 U R S E S AND L E C T U RES - No. 266
e
.
M
ANALYSIS AND DESIGN
OF ALGORITHMS
IN COMBINATORIAL OPTIMIZATION
EDITED BY
G. AUSIELLO
AND
M. LUCERTINI
IASI - CNR
ISTITUTO Dl AUTOMATICA
UNIVERSIT A' Dl ROMA
SPRINGER-VERLAG WI EN GMBH
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or similar means, and storage in data banks.
© 1981 by Springer-Verlag Wien
Originally publisbed by Springer Verlag Wien-New York in 1981
ISBN 978-3-211-81626-4 ISBN 978-3-7091-2748-3 (eBook)
DOI 10.1007/978-3-7091-2748-3
PREFACE
The School in Analysis and Desi/{11 of Algorithms in Combinatorial Optimization was
held in Udine in September 1979. It was financially supported by the National Research
Council of Italy (CNR), by the International Center ofM echanical Sciences (CISM) and by
the European Economic Community (CEE) and sponsored by GNASII-CNR, CSSCCA-CNR
and lstituto di Automazione of the University of Rome. The organizers are pleased to
express their thanks to the lecturers and participants who made the School stimulating and
fruitful. Special thanks are addressed to the organizing committee and in particular to
Angelo Marzollo, Alberto Marcbetti-Spaccamela and Paolo Serafini who provided their
friendly and valuable help making the school successful.
G. Ausiello
M. Lucertini
HTRODUCTION
The practical and theoretical relevance of problems to the NP-complete degree vf
complexity are widely known. From the practical point of view it is sufficient to remember
that in this class we find most of the combinatorial and optimization problems which b,,.·e
the widest range of applications, for example scheduling problems, optimization problems
on graphs, integer programming etc. As far as the theoretical relevance is concerned ;;.•e
should remember that one of the most outstanding problems in Computer Science, the
problem of deciding whether any NP-complete set can be recognized in polynomial time,
coincides with the problem of knowing whether the computation power of a
nondeterministic Turing machine which accepts a set in polynomial time is strictly stronger
than the power of ordinary polynomially bounded Turing machines or not. Until recen#y
the design of algorithms for finding exact approximate solutions to practical instances of
hard combinatorial and optimization problems was the main concern of experts in
Operations Research while the study of the complexity of these problems with respect to
various computation models and the analysis of general solution techniques was the main
interest of computer scientists.
In 1978 at the Mathematisch Centrum in Amsterdam a conference was held on
"Interfaces between Computer Science and Operations Research". In that occasion the
experience and scientific interest of experts in mathematical programming and of Computer
Scientists, interested in the design of algorithms and in the analysis of the complexity of
problems, met a'!d fruitfully interacted.
The School which was held in Udine in 1979 and which was followed by a School on
"Combinatorics and Complexity of Algorithms" held in Barcellona, Spain, in 1980) is in the
same framework. The program itself of the School was organized by alternating
methodological results with analysis of particular classes of problems. All lectures were
divided in two parts. In the first the lectures reviewed the current state of the subject and
in the second they dealed with advanced topics.
More in detail Ausiello introduced the basic concepts of machine models and
computation measures with special reference to non deterministic models. Afterwards he
presented a classification of many NP-complete optimization problems according to the
concepts of combiatorial structure and structure preserving reductions. Karp, after recalling
the main notions about NP-completeness tbeory, concentrated bis attention on the
probabilistical and statistical analysis of algorithms. Considering numerous examples be
pointed out what are the advantages ,md dis,zdvantages of his approach. johnson, after
r.':·icwing the subject of approximation algorithms for NP-complete problems, lectured
:•i,out strong NP-completeness and pseudopolynomiality. Then he presented the state of art
n.r tu:o problems: Bin packing and Graph colouring. Paz presented the study of NP-complete
optimi:ation problems according to a formalism introduced by himself and Moran.
Following this set-up he defined rigid, simple and p-simple problems. Exploiting this
cl.mification he gave necessary and sufficient conditions for the approximability and fully
.1.:•proximability of NP-complete problems. Finally he defined special types of polynomial
rt'.iuctions showing their properties. Sehnon gave lectures on different subjects. Firstly he
p1u~·ed lower bounds in the field of algebraic complexity using algebraic geometry. Then he
considered the so-called selftransformable combinatorial problems as a way to study the
relationship between the search problem and the decision problem, always in the field of
NP-completeness. Finally he presented some complexity properties of Boolean functions.
Luccio, after introducing basic data structures for combinatorial problems, illustrated
particular data structures and algorithms solving problems that occur in the field of
multidimensional memories. Lucertini showed the most important algorithmic techniques to
solve problems of mathematical programming. Then, he presented the group-theoretic
approach to the integer programming problem. Maffioli introduced matroid theory and its
applications to combinatorial optimization. Moreover he dealed with network design
problems and, in particular, classified the complexity of various kinds of the spanning tree
problem. Lawler presented the state of art and a generalization of network flow problems.
Finally he showed Lovasz 's recent algorithm for solving the linear 2-polymatroid problem.
Rinnooy Kan lectured about advanced techniques in mathematical programming with
Khachian 's algorithm for solving linear programming and gave some complexity results of
relevant problems of production planning. Finally Lenstra presented a detailed analysis of
the most important results in scheduling theory and surveyed some topics in the field of
routing problems.
This volume contains papers presenting developments of some of the topics discussed
by the lecturers during the School. These contributions were collected after the School with
the twofold aim of giving an outlook of this important research area and of providing more
detailed treatments of some specific subjects.
Giorgio Ausie//o
Mario Lucertini
LIST OF CONTRIBUTORS
G. AUSIELLO IASI-CNR, Istituto di Automatica, Universit:l di Roma.
A. D'ATRI Istituto di Automatica, Universidt di Roma.
V. FERRARI IASI-CNR, Istituto di Automatica, Universita di Roma.
M.R. GAREY Bell Laboratories, Murray Hill, New Jersey.
S. GIULIANELLI IASI-CNR, Istituto di Automatica, Universita di Roma.
D.S. JOHNSON Bell Laboratories, Murray Hill, New Jersey.
E.L. LAWLER Computer Science Division, University of California, Berkeley.
J .K. LENSTRA Mathematisch Centrum, Amsterdam.
F. LUCCIO Universit:l di Pisa.
M. LUCERTINI IASI-(NR, Istituto di Automatica, Universita di Roma.
F. MAFFIOLI Istituto di Elettrotecnica ed Elettronica, Politecnico di Milano.
S. MORAN Dept. of Mathematics, Technion, Israel Institute of Technology.
A. PAZ Dept. of Computer Science, Technion, Israel Institute of
Technology.
M. PROTASI Istituto di Matematica, Universita deli'Aquila.
A.H.G. RONNOOY KAN Erasmus University, Rotterdam.
CONTENTS
Preface . . . . . . . . I
Introduction . . . . Ill
List of Contributors . V
Non-Deterministic Polynomial Optimization Problems and Their Approximation
by A. Paz, and S. Moran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
A Characterization of Reductions Among Combinatorial Problems
by G. Ausiello, A. D'Atri, M. Protasi . . . . . . . . . . . . . . . . . . . . . . . . . . 7
A Recursive Approach to the Implementation of Enumerative Methods
by j.K. Lenstra and A.H.G. Rinnooy Kan ........................ 65
Data Structures for Combinatorial Problems
by F. Luccio ....................................... 85
Data Structures for Bidimensional Memory
by F. Luccio . . . . . . . . . . . . . . . . .... 97
Complexity of Optimum Undirected Tree Problems: A Survey of Recent Results
by F. Maffioli ......................................1 07
An Introduction to Polymatroidal Network Flows
by E.L. Lawler ................. . . ...........1 29
Approximation Algorithms for Bin Packing Problems: A Survey
by M.R. Garey and D.S. johnson ............... . . .147
Additional Constraints in the Group Theoretical Approach to Integer Programming
by V. Ferrari, S. Giulianelli, M. Lucertini ........................1 73
NON-DETERMINISTIC POLYNOMIAL OPTIMIZATION
PROBLEMS AND THEIR APPROXIMATION
* **
A. PAZ and S. MORAN
*
Dept. of Computer Science
TECHNION-Israel Institute of
Technology
**
Dept. of Mathematics, TECHNION
Israel Institute of Technology
1. INTRODUCTION
NP-problems are considered in this paper as recognition
*
problems over some alphabet L, i.e. A C L is an NP problem
if there exists a NDTM (non-deterministic Turing machine)
recognizing A in polynomial time. It is easy to show that the
following theorem holds true.
THEOREM 1. Let A be a set in NP. Then there exists a
NDTM MA which recognizes A such that MA = M oM o MA , where
\JA TIA 1
1) The operation hoh is defined as follows: M1 oM2 (x) is
M1 (M2 (x)); M1,M2 are Turing machines and xis an input
tape.
2) MA is a polynomial time deterministic encoding machine.
1
Its task is to encode an input a E A in some proper way
to be denoted by a•.
2 A. Paz and S. Moran
3) M is a NDTM which choses some permutation n(a1) out of
nA
a possible subgroup of the group of all permutations of
the encoded input tape a• in polynomial time.
4) M is a polynomial time DTM which computes a number
llA
ll(n(a')).
11 ( n (a i) ) < ka (min problem)
5) a E A iff {
ll(n(ai)) > k (max problem)
- a
where ka is a number compu·ted in polynomial time by the
machine MA (ka is part of the encoding of a).
1
Thus every NP problem can be represented as an optimiz~
tion problem and the recognition process can be split into
three stages where the non-deterministic stage (the machine
M ) is separated from the other stages.
nA
2. NP OPTIMIZATION PROBLEMS
The cqnjecture that P ~ NP is widely believed to be
true. This conjecture prompted many researchers to develop
and study polynomial approximations for problems in NP. when
considered as optimization problems. See e.g. [Jo 73] or
[Sa 76] .
The previous section points toward the possibility of
a new approach to the study of NP problems and NP optimiza
tion problems. In what follows, an attempt is made to develop
that new approach. The results achieved so far are promising.
These results provide some new insight into recently proved
approximation results and it is hoped that they will serve
as a basis for a more extensive theory of combinatorial ap-
