DIMACS Series i n Discret e Mathematic s

and Theoretica l Compute r Scienc e

Volume 5

Reliability o f Compute r and Communicatio n Network s

Proceed ings o f a DIMAC S W o r k s h o p D e c e m b e r 2 - 4 , 198 9

Fred Robert s Frank Hwan g Clyde Monm a

Editors

NSF Scienc e an d Technolog y Cente r in Discret e Mathematic s an d Theoretica l Compute r Scienc e A consortiu m o f Rutger s University , Princeto n University ,

AT&T Bel l Labs , Bellcor e

https://doi.org/10.1090/dimacs/005

Th e DIMAC S Worksho p o n Reliabilit y o f Compute r an d Communicatio n Net -work s wa s hel d a t Rutger s University , Ne w Brunswick , Ne w Jersey , o n Decembe r 2 -4 , 1989 .

198 0 Mathematics Subject Classification (198 5 Revision). Primar y 05-06 , 05C99 , 68-06 , 68M10 , 68M15 , 68R10 , 90-06 , 90B25 , 94-06 , 94C15 .

Librar y of Congres s Cataloging-in-Publicatio n Dat a

Reliabilit y of compute r an d communicatio n networks : proceeding s of a DIMAC S Workshop , Decembe r 2-4 , 1989/Fre d Roberts , Fran k Hwang , Clyd e Monma , editors .

p. cm.—(DIMAC S serie s in discret e mathematic s an d theoretica l compute r science ; ISS N 1052-1798 ; v . 5)

Include s bibliographica l references . AMS : ISB N 0-8218-6592- 7 ACM ; ISB N 0-89791-387- 6 1. Compute r networks-Reliability-Congresses . I. Roberts , Fre d S . II. Hwang , Frank .

III. Monma , Clyd e L. IV . America n Mathematica l Society . V . Associatio n for Computin g Machinery . VI . Series . TK5105.5.R44 5 199 1 91-995 3 004.6-dc2 0 CI P

To orde r throug h AM S contac t the AM S Custome r Service s Department , P.O . Bo x 6248 , Providence , Rhod e Islan d 02940-624 8 USA . Fo r VIS A or MASTERCAR D order s cal l 1-800-321 -4AMS . Orde r cod e DIMACS/5 .

To orde r throug h AC M contac t AC M Orde r Department , P.O . Bo x 64145 , Baltimore , Mary -lan d 21264 . Phon e 301-528-4261 . Orde r numbe r 222912 .

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109 8 7 6 5 43 2 1 9 5 94 93 92 91

Contents

Preface v

Program i x

List o f Participant s x v

Robust Desig n o f Dynami c Routin g Network s GERALD R . AS H 1

Graph Searching , Path-Width , Tree-Widt h an d Relate d Problem s (A Survey)

DANIEL BIENSTOC K 3 3

On Residua l Connectednes s Networ k Reliabilit y F. BOESCH , A . SATYANARAYANA , AN D C . SUFFE L 5 1

Survivable Fibe r Networ k Desig n R. H . CARDWEL L 6 1

Decomposable Probabilisti c Influenc e Diagram s C. C . CHY U 7 9

Bounding Networ k Parameter s b y Approximatin g Graph s CHARLES J . COLBOUR N AN D EUGEN E I . LITVA K 9 1

The Optima l Multitermina l Cu t Proble m WILLIAM H . CUNNINGHA M 10 5

Polyhedral Approache s t o Networ k Survivabilit y M. GROTSCHEL , C . MONMA , AN D M . STOE R 12 1

A Surve y O n Doubl e Loo p Network s F. K . HWAN G 14 3

Quantitative Reliabilit y Analysi s o f Redundan t Multistag e Interconnection Network s

N I T A M . KINI , ANU P KUMAR , AN D DHARM A P . AGRAWA L 15 3

An Axiomati c Characterizatio n o f th e Reliabilit y Polynomia l RICHARD P . M C L E A N AN D DOUGLA S H . BLAI R 17 1

in

IV CONTENTS

Fault-Tolerant VLS I Architecture s Base d o n d e Bruijn Graph s (Galileo in the Mid Nineties )

DHIRAJ K . PRADHA N 18 3

The Use of Binomia l Moment s fo r Boundin g Network Reliabilit y ANDRAS PREKOPA , ENDR E BOROS , AN D KEH-WE I LI H 19 7

Boolean Decompositio n Scheme s and the Complexity o f Reliabilit y Computations

J. SCOT T PR O VAN 21 3

On Dynamic Ful l Access in Multistage Interconnectio n Network s C. S . RAGHAVENDR A AN D M . A . SRIDHA R 22 9

Algebraic Methods fo r Boundin g Network Reliabilit y DOUGLAS R . SHIE R 24 5

Preface

This volume contains original papers prepared following talks given at the DIMACS Worksho p o n Reliabilit y o f Compute r an d Communicatio n Net -works, held Decembe r 2-4 , 1989 , at Rutger s Universit y i n New Brunswick , New Jersey . Al l o f th e paper s (excep t fo r Chyu' s whic h wil l als o appea r elsewhere) ar e in final form.

Reliability problem s aris e wit h increasin g frequenc y a s ou r moder n sys -tems o f telecommunications , informatio n transmission , transportation , an d distribution becom e mor e an d mor e complex . Thi s workshop wa s designe d to analyz e th e discret e mathematica l method s whic h ar e relevan t t o thes e problems, to identify th e lates t trends an d importan t ope n problems , and t o survey potentia l practica l applications , wit h a n emphasi s o n compute r an d communication networks .

The worksho p analyze d bot h question s o f computatio n o f reliabilit y o f existing systems and questions dealing with the design o f highly reliable sys-tems to begin with . (Th e two question s ar e closel y related , sinc e on e need s to know how to compute reliability in order to design a reliable system.) Th e workshop als o analyzed th e closel y related notio n o f survivability . Networ k survivability mean s th e abilit y t o restor e "service " i n th e even t o f a catas -trophic failur e o f a network component . W e also discussed ho w to build re -dundancy int o a network, ho w to define an d measur e redundancy , an d ho w redundancy relate s to reliability . Th e workshop deal t wit h both singl e stag e and multistage, interconnected networks , which have been studied in connec-tion with computer networks, and in particular with fault tolerance and other reliability issue s i n mind . Th e worksho p emphasize d practica l application s through invite d speaker s fro m a variety o f companie s tha t ar e dealin g wit h practical networ k reliabilit y problems .

In all , ther e wer e 8 9 attendee s a t th e workshop , wit h th e averag e dail y attendance abou t 50 . Th e attendee s include d theoretica l mathematicians , computer scientists , and electrica l engineer s from academi a an d industry , a s well a s networ k practitioners , engineers , an d reliabilit y planner s fro m suc h companies as AT&T Bell Labs, Bellcore, Pacific Bell, GTE, Bolt, Beranek and Newman, MITRE , and IBM .

The worksho p starte d wit h a sessio n o n networ k reliability . Th e speak -ers i n tha t sessio n wer e Charle s Colbour n (Waterloo) , Scot t Pr o van (Nort h Carolina), an d Dougla s Shie r (Willia m an d Mary) . Thes e thre e speaker s summarized th e traditional proble m o f computin g the reliability polynomia l of a grap h whe n ther e ca n b e edg e failures . Colbour n discusse d th e us e o f graph transformations t o obtain bound s o n the reliability . Prova n discusse d

V

VI PREFACE

two types of approximation scheme s for reliability , one using the property of shellability and the other using series-parallel and delta-wye transformations . Shier discussed algebrai c methods fo r boundin g reliability .

A second session on network reliability featured talk s by Richard McLea n (Rutgers) an d Andras Prekopa (Rutgers) . McLean' s talk brought to the relia-bility problem the point of view of the economist, and in particular the game theorist, an d presente d axiom s whic h completel y characterize d th e reliabil -ity polynomial . Thi s poin t o f vie w wa s quit e nove l an d wa s receive d wit h great interest by the more traditional reliability theorists . Prekopa' s talk also presented a nontraditional approach , making use of some recently develope d lower and uppe r bounds fo r th e probability o f logica l functions o f event s to obtain lowe r bounds fo r th e reliability o f a network .

The first day concluded with a session on applications which featured talk s by A.G . (Sandy ) Frase r (AT& T Bel l Labs ) an d Pete r Kuba t (GT E Labora -tory). A thir d tal k i n thi s session , b y Mariann e Lep p (Bolt , Berane k an d Newman), wa s cancelled du e t o the speaker' s illness . Thi s first o f thre e ap -plied session s a t th e worksho p wa s intende d t o hav e practitioner s presen t both importan t use s of theoretical reliabilit y methods and new practical net -work reliability problems . Fraser' s talk deal t with congestio n an d instabilit y in data networks , and emphasize d th e new reliability problems posed by the large capacit y o f fiber optic connections . Ther e wa s a great dea l o f interac -tion between the speake r and th e audience , with man y suggestion s made fo r possible approaches to the problems posed, and with many ideas for potentia l future reliabilit y researc h bein g discussed . Afte r th e session , i t wa s agree d that th e talk was such a success that i t shoul d be followed u p by a more de-tailed presentatio n t o a DIMACS group by a Bell Labs scientist . On e of th e positive aspects of Marianne Lepp's cancellation was that we were able to al-low this part of the session to continue for one hour instead o f the scheduled half hour . Kubat' s tal k emphasize d a variety o f approache s t o the design o f reliable an d qualit y telecommunication s system s fro m th e poin t o f vie w o f future cost s and als o resulted i n a lively discussion .

The second day began with a session on reliability o f multistage network s and faul t tolerance . Th e sessio n wa s introduce d wit h a brie f summar y o f the topi c b y Anuja n Varm a (IBM) . Th e speaker s i n thi s sessio n wer e D.K . Pradhan (Massachusetts) , D.P . Agrawa l (Nort h Carolin a State) , an d C.S . Raghavendra (Souther n California) . Thi s sessio n feature d talk s b y electri -cal engineers/compute r scientists , an d wa s a very nic e complemen t t o bot h the mathematically oriente d talk s of the first day and the applied talks . On e of th e thing s whic h man y o f th e participant s pointe d t o a s a highligh t o f the meetin g wa s tha t w e go t togethe r s o man y divers e researcher s wh o d o not usuall y atten d th e sam e meetings . Pradha n talke d abou t a particula r multiprocessor networ k whic h i s o f grea t interes t today , th e d e Bruij n net -work. ( A later speaker, Jean-Claude Bermond, also talked about the de Bruijn network.) Agrawa l talke d abou t redundan t multistag e interconnectio n net -works (MIN's ) an d presente d a n algorith m fo r computin g thei r reliability . Raghavendra als o talked abou t MIN' s an d presente d a variety o f technique s for achievin g fault-tolerance .

PREFACE vn

The next sessio n deal t wit h networ k survivability , th e problem o f how t o design a network which had prescribed survivability characteristics, given the desire t o buil d suc h a networ k a t minimu m cost . Th e speaker s i n thi s ses -sion wer e Marti n Grotsche l (Augsburg ) an d Danie l Bienstoc k (Columbia) . Grotschel's talk was a beautiful mi x of theory and practice. I t described how he and Clyde Monma had worked with Bellcore scientists (including Richard Cardwell, a networ k practitioner ) t o convinc e th e regiona l Bel l Telephon e companies tha t survivabilit y coul d b e buil t int o a network wit h smal l addi -tional cost . I t discusse d ho w they interacted wit h the practitioner s t o defin e the precis e mathematica l problem , whic h the y the n solve d usin g polyhedra l methods. Whe n one of the attendees questioned th e seeming oversimplifica -tion i n the network survivabilit y mode l used by Grotschel , he was reassured by Cardwel l tha t thi s was exactly the appropriat e mode l t o use . Bienstock' s talk dealt with problems of searching a graph, seeking out an intruder by the use of guards. I t covered a variety o f important recen t development s havin g to do with such graphical parameters a s path-width an d tree-width .

A second session on network survivabilit y feature d talk s by William Cun -ningham (Carleto n University , Ottawa ) an d Fran k Boesc h (Stevens) . Cun -ningham discussed a variety of network attack problems, which are optimiza-tion problems in which one tries to minimize the difference between the effor t required by an attacker to destroy the edges in a subset o f a network and th e resulting damag e t o th e network . Boesc h gav e a very nic e summar y o f th e node reliability problem, the reliability problem where nodes fail, rather than edges.

The second da y ended wit h ou r secon d sessio n o n applications . Thi s ses-sion agai n featured practitioners , i n thi s case Gerald As h (AT& T Bell Labs) and Richar d Cardwel l (Bellcore) . A third talk , b y Nea l Crysta l o f Bellcore , had to be cancelled because the speaker was unable to attend. As h describe d the 198 7 AT& T switc h fro m hierarchica l routin g o f call s throug h it s net -work to dynamic nonhierarchica l routing . H e emphasized ho w much mone y AT&T had saved as a result and how much mor e reliable things had becom e as a result . I n answe r t o a question fro m th e audience , h e emphasize d th e important rol e tha t theoretica l development s ha d playe d i n th e decisio n t o make the switch to dynamic nonhierarchical routing , and in the design of the new system . H e cite d th e importan t contributio n o f a foundationa l pape r by Fan Chung , Ro n Graham , an d Fran k Hwang , and th e significanc e o f th e development o f th e linea r programmin g algorith m o f Karmarkar . As h als o discussed futur e routin g methodologies an d the technical problems involve d in implementing them. Cardwel l talked abou t alternative network topologie s for building survivable fiber networks. H e emphasized a ring topology which many of the participants found ver y interesting. Ther e was considerable dis-cussion abou t ho w t o balanc e of f th e initia l cos t saving s o f suc h a networ k with th e fac t tha t onc e th e networ k reache s capacity , i t i s very difficul t an d expensive t o modify . Man y interestin g an d importan t theoretica l question s arose from thi s discussion .

The thir d da y began wit h a session o n redundanc y an d reliability o f spe -cial networks . Th e fou r speaker s wer e Joel Cohe n (Rockefelle r University) ,

Vl l l PREFACE

Nicholas Maxemchuk (AT& T Bell Labs), Philip Boland (Universit y College , Dublin), and C.Y . Chyu (Berkeley) . Cohe n talked abou t calculatin g the reli -ability o f a random grap h o r digraph , an d als o abou t calculatin g the redun -dancy a s measured b y the expecte d numbe r o f spannin g trees . H e explore d the relationship s betwee n th e two . Maxemchu k discusse d a particula r net -work topology , th e Manhatta n stree t network , an d it s reliabilit y properties . Boland discusse d reliabilit y problem s i n whic h component s o f a system ar e arranged linearl y an d th e syste m fail s i f an d onl y i f k consecutiv e compo -nents fail. H e introduced positive dependence between adjacent component s and argued tha t th e system reliability i s a decreasing functio n o f thi s depen -dence for k large . Chy u talked about influence diagrams, structured digraphs which are used in applications to medical decisionmaking, reliability analysis, etc.

A secon d sessio n o n redundanc y an d reliabilit y o f specia l network s fea -tured tw o speakers , Jean-Claud e Bermon d (CNRS , Valbonne ) an d Fran k Hwang (AT&T Bell Labs). Bermon d continued the discussion o f specia l net -work topologie s b y discussin g reliabilit y propertie s o f d e Bruij n an d Kaut z graphs, an d by introducin g variou s generalizations o f thes e graphs . H e sur -veyed both recen t result s and ope n problems . Hwan g discussed doubl e loo p networks, whic h ar e directe d circulants , an d whic h als o hav e a potentiall y useful ne w networ k topology . H e describe d a nove l metho d fo r obtainin g the most reliable double loop networks. H e also gave a routing algorithm fo r such networks .

The final sessio n o f th e worksho p wa s th e thir d sessio n o n applications . This feature d a tal k b y Yin g Chen g (AT& T Bel l Labs) . A secon d talk , b y Ralph Evan s (privat e consultant) , wa s cancelle d du e t o th e illnes s o f th e speaker. Cheng' s tal k deal t wit h AT M (asynchronou s transfe r mode ) net -works. H e use d algebrai c an d combinatoria l method s t o develo p a n erro r correction/error detectio n schem e in suc h networks .

The workshop was a success because i t fostered s o many new interaction s between researcher s an d between researcher s an d practitioners . I t ha s als o already ha d a very visibl e an d dramati c impact . A number o f participant s from suc h companies as Pacific Bel l were so excited by the potential value of the methods fo r designin g highly survivable fiber optic networks which were discussed at the meeting, that they suggested that Bellcore arrange a follow-up meeting fo r Bellcor e employee s an d employee s o f th e Bel l system operatin g companies. A s a direc t resul t o f ou r DIMAC S activity , suc h a workshop , entitled "Telecommunication s Networ k Survivabilit y Symposium, " wa s or -ganized fo r Bellcor e an d th e Bellcor e clien t companies . On e o f us , Clyd e Monma, wa s a co-organizer. Th e respons e t o the workshop was very enthu -siastic: ther e were over 20 0 participants an d th e feedback fro m participant s was that th e workshop wa s timely and important .

The editor s hop e tha t th e foundation s lai d b y th e workshop , it s follow -up workshop , an d thi s volume wil l lead t o continuin g development s i n thi s exciting and importan t field.

Frank Hwan g Clyde Monm a Fred S . Roberts

Program

Friday, December 1 , 1989

6:30-8:30 pm Registratio n

Saturday, December 2, 1989

7:30-8:30 Breakfas t 8:30-9:00 Registratio n an d Coffe e 9:15-9:30 Openin g Chair Fre d S. Roberts, Rutgers Universit y

Daniel Gorenstein , Director , DIMAC S Clyde Monma, Bellcore, for th e Organizin g Committee

SESSION O N NETWOR K RELIABILITY , PAR T 1

Chair 9:30-10:00

10:00-10:10 10:10-10:40

10:40-10:50 10:50-11:20 11:20-11:50

11:50-12:00 12:00-1:15

Michael Ball , University o f Marylan d Charles Colbourn, University o f Waterlo o Bounding Network Reliability by Graph Trans-formations Discussion J. Scot t Provan, University o f North Carolin a Approximating Two-Terminal Reliability in Graphs Discussion Coffee Douglas Shier, College of William an d Mar y Algebraic Methods for Bounding Reliability Discussion Lunch

Chair

SESSION O N NETWOR K RELIABILITY , PAR T 2

Endre Boros, Rutgers Universit y

x PROGRA M

1:15-1:45 Richar d McLean, Rutgers Universit y A Characterization of the Reliability Polyno-mial for Polyhedral Network Problems

1:45-1:55 Discussio n 1:55-2:25 Andra s Prekopa, Rutgers Universit y

The Use of Binomial Moments for Approxi-mate Network Reliability Calculation

2:25-2:35 Discussio n 2:35-3:05 Coffe e

SESSION O N APPLICATIONS , PAR T 1

Chair Michae l Rothkopf , Rutger s Universit y 3:05-3:35 A . G. Fraser, AT&T Bell Laboratorie s

Congestion and Instability in Data Networks 3:35-3:45 Discussio n 3:45-4:15 Mariann e Lepp, Bolt, Beranek and Newma n

Reliability in Internet Routing 4:15-4:25 Discussio n 4:25-4:55 Pete r Kubat, GTE Laborator y

Network Reliability and Cost Effective Telecommunication System Design

4:55-5:05 Discussio n 5:15-6:15 Win e and chees e part y 6:15-7:15 Dinne r

Sunday, December 3, 1989

SESSION O N RELIABILIT Y O F MULTISTAG E NETWORK S

AND FAUL T TOLERANC E

Chair Anuja n Varma, IBM 7:30-8:30 Breakfas t 8:45-9:15 D . K. Pradhan, University o f Massachusett s

The de Bruijn Multiprocessor Network: A Fault-Tolerant Versatile Parallel Processing Network

9:15-9:25 Discussio n 9:25-9:55 D . P. Agrawal, North Carolin a Stat e Univer -

sity Reliability Evaluation of Redundant Multistage Interconnection Networks

PROGRAM

9:55-10:05 10:05-10:35 10:35-11:05

11:05-11:15

Discussion Coffee C. S. Raghavendra, Universit y o f Souther n California Reliability and Fault-Tolerance in Multi-Stage Interconnection Networks Discussion

SESSION O N NETWOR K SURVIVABILITY , PAR T 1

Chair Danie l Kleitman , MI T 11:15-11:45 Marti n Grotschel , University o f Augsburg

Polyhedral Approaches to Network Survivability 11:45-l 1:55 Discussio n 11:55-12:25 Danie l Bienstock , Columbi a Universit y

Graph Searching, Tree-Width and Path-Width 12:25-12:35 Discussio n 12:35-2:00 Lunc h

SESSION O N NETWOR K SURVIVABILITY , PAR T 2

Chair Richar d Va n Slyke, Polytechnic Institute o f New York

2:00-2:30 Willia m Cunningham , Carleto n Universit y Network Attack Problems

2:30-2:40 Discussio n 2:40-3:10 Fran k Boesch , Stevens Institute o f Technolog y

The Residual Node Reliability Problem 3:10-3:20 Discussio n 3:20-3:50 Coffe e

SESSION O N APPLICATIONS , PAR T 2

Chair Dvor a Tzvieli, AT&T Bel l Laboratorie s 3:50-4:20 Geral d Ash , AT&T Bell Laboratorie s

Dynamic Routing in Telecommunications Net-works: Improved Robustness at Lower Cost

4:20-4:30 Discussio n 4:30-5:00 Richar d Cardwell , Bellcor e

Survivable Fiber Network Design

Xll PROGRAM

5:00-5:10 Discussio n 6:00-7:00 Dinne r

Monday, December 4, 1989

SESSION O N REDUNDANC Y AN D RELIABILIT Y

OF SPECIA L NETWORKS , PAR T 1

Chin-Lin I, AT&T Bel l Laboratorie s Breakfast Joel Cohen, Rockefeller Universit y Reliability and Redundancy of Anisotropic Random Graphs and Random Directed Graphs Discussion Nicholas F . Maxemchuk, AT& T Bell Laborato-ries Failure Mechanisms in the Manhattan Street Network Discussion Coffee Philip Boland, University College , Dublin Linear Dependence in Consecutive k out ofn : F Systems Discussion C. C. Chyu, University o f California , Berkele y Decomposable Probabilistic Influence Diagrams Discussion Lunch

SESSION O N REDUNDANC Y AN D RELIABILIT Y

OF SPECIA L NETWORKS , PAR T 2

Wen-Ching Winni e Li, Pennsylvania Stat e University Jean-Claude Bermond, CNRS, Valbonne Reliability ofde Bruijn and Similar Networks Discussion Frank Hwang, AT&T Bel l Laboratorie s Reliabilities of and Fault Tolerant Routing for Double Loop Networks Discussion Coffee

Chair 7:30-8:30 9:00-9:30

9:30-9:40 9:40-10:10

10:10-10:20 10:20-10:50 10:50-11:20

11:20-11:30 11:30-12:00

12:00-12:10 12:10-1:15

Chair

1:15-1:45

1:45-1:55 1:55-2:25

2:25-2:35 2:35-2:55

PROGRAM xm

SESSION O N APPLICATIONS , PAR T 3

Chair Joh n Healey, Bellcor e 2:55-3:25 Yin g Cheng, AT&T Bell Laboratorie s

An Error Correction/Detection Scheme in ATM Network

3:25-3:35 Discussio n 3:35-3:40 Closin g

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List of Participants

AGRAWAL, Dharma P .

ASH, Geral d AYANOGLU, Ende r BAGCHI, Ansuma n

BALL, Michae l

BAUER, Dougla s

BENZAKEN, Claud e

BERMOND, Jean-Claud e BIENSTOCK, Danie l

BITAR, Youaki n

BLAIR, Dougla s

BLOOM, Gar y

BOESCH, Francis T .

BOLAND, Phili p

Department o f Electrica l an d Computer Engineering , Nort h Carolina Stat e University, Raleigh , NC

AT&T Bell Laboratories, Holmdel , N J AT&T Bell Laboratories, Holmdel , NJ RUTCOR, Rutger s University, Ne w

Brunswick, N J College of Business Management ,

University o f Maryland , Colleg e Park, M D

Department o f Mathematics , Steven s Institute o f Technology, Hoboken , NJ

University o f Scienc e and Medicine , Grenoble, Franc e

CNRS, Valbonne, Franc e Department o f Operations Research ,

Columbia University , New York, NY

Department o f Compute r Science , Princeton University , Princeton , N J

Department o f Economic s an d RUTCOR, Rutger s University , New Brunswick, N J

Department o f Mathematics , Cit y University o f New York, New York, NY

Stevens Institute o f Technology , Hoboken, N J

Department o f Mathematics , University College , Belfield, Dublin , Ireland

xviii LIS T O F

PREKOPA, Andra s

PROVAN, J . Scot t

RAGHAVENDRA, C . S.

REIBMAN, Andre w ROBERTS, Fred S .

ROTHBLUM, Urie l

ROTHKOPF, Michae l

SACCOMAN, Joh n

SAKAI, Denise

SALIZKIY, Olga SERPANOS, D.

SHA, Hsing-Mea n

SHAMIR, Ro n

SHAPIRO, Joh n SHULMAN, A. SHIER, Dougla s

SQUIER, Richar d

SUFFEL, Charle s

TEWARI, Raji v

TONG, Yun g L.

PARTICIPANTS

RUTCOR, Rutger s University, Ne w Brunswick, N J

Department o f Operations Research , University o f North Carolina , Chapel Hill , NC

Department o f Electrica l Engineerin g Systems, University o f Souther n California, Lo s Angeles, CA

AT&T Bell Laboratories, Holmdel , NJ DIMACS, Rutgers University , Ne w

Brunswick, N J RUTCOR, Rutger s University, Ne w

Brunswick, N J RUTCOR, Rutger s University , Ne w

Brunswick, N J Stevens Institute o f Technology ,

Hoboken, N J RUTCOR, Rutger s University , Ne w

Brunswick, N J Hofstra University , Hempstead , N Y Department o f Compute r Science ,

Princeton University , Princeton , N J Department o f Compute r Science ,

Princeton University , Princeton , N J DIMACS, Rutgers University , Ne w

Brunswick, N J Fordham University , Bronx , NY GTE Laboratories Inc. , Waltham, M A Department o f Mathematics , Colleg e

of William an d Mary , Williamsburg , VA

Department o f Compute r Science , Princeton University , Princeton , N J

Stevens Institute o f Technology , Hoboken, N J

Department o f Compute r Science , Rutgers University , New Brunswick , NJ

School of Mathematics , Georgi a Institute o f Technology , Atlanta , GA

LIST OF

TONG, X .

TRALDI, Lorenz o

TZVIELI, Dvor a VAN SLYKE, Richar d

VARMA, Anuja n

WANG, Ch i

WANG, Hui-Y u

WILLIAMS, Alber t

WINKLER, Pete r ZHENG, Maoli n

ZHOU, Ming-Kun g

ZIMMERMAN, Gu y

PARTICIPANTS xi x

Department o f Compute r Science , Rutgers University , Ne w Brunswick , NJ

Department o f Mathematics , Lafayett e College, Easton, PA

AT&T Bell Laboratories, Holmdel , N J Polytechnic Institut e o f New York,

Brooklyn, N Y IBM, T. J. Watson Research Center ,

Yorktown Heights , NY RUTCOR, Rutger s University , Ne w

Brunswick, N J Department o f Compute r Science ,

Rutgers University , New Brunswick , NJ

RUTCOR, Rutger s University, Ne w Brunswick, N J

Bellcore, Morristown, N J RUTCOR, Rutger s University, Ne w

Brunswick, N J City University o f New York, New

York, NY Michigan Stat e University, Eas t

Lansing, MI

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