Combinatorial Optimization and Graph Algorithms: Communications of NII Shonan Meetings

126 Pages·2017·2.33 MB·English

by  Takuro Fukunaga

Takuro Fukunaga
Ken-ichi Kawarabayashi Editors
Combinatorial
Optimization
and Graph
Algorithms
Communications
of NII Shonan Meetings
Combinatorial Optimization and Graph Algorithms
Takuro Fukunaga Ken-ichi Kawarabayashi
(cid:129)
Editors
Combinatorial Optimization
and Graph Algorithms
Communications of NII Shonan Meetings
123
Editors
Takuro Fukunaga Ken-ichi Kawarabayashi
National Institute ofInformatics National Institute ofInformatics
Tokyo Tokyo
Japan Japan
ISBN978-981-10-6146-2 ISBN978-981-10-6147-9 (eBook)
DOI 10.1007/978-981-10-6147-9
LibraryofCongressControlNumber:2017948610
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Preface
Inaveryshorttimesincetheconceptof“P”and“NP”wasintroduced,algorithmic
aspect of mathematics and computer science has quickly gained explosion of
interest from the research community. Indeed, the problem of whether P = NP
(or P 6¼ NP) is one of the most outstanding open problems in all of mathematics.
A wider class of problems from mathematics, computer science, and perhaps
operations research has been known to be NP-complete, and even today, the
collection of NP-complete problems grows almost every few hours.
In order to understand NP-hard problems, the research community comes to
haveafairlygoodunderstandingofcombinatorialstructuresinthepast40years.In
particular, the algorithmic aspects of fundamental problems in graph theory (i.e.,
graph algorithm) and in optimization (i.e., combinatorial optimization) are two
of the main areas that the research community have some deep understandings.
Graph algorithm is an area in computer science that tries to design an efficient
algorithm for networks. Today, networks (i.e., graphs) are ubiquitous in today’s
world; the World Wide Web, online social networks, and search and query-click
logs can lead to a graph that consists of vertices and edges. Such networks are
growing so fast that it is essential to design algorithms to work for such large
networks.Sotheresearchcommunityworksontheoreticalproblems,aswellason
some practical problems that implementation of an algorithm for large networks.
Combinatorial optimization is an intersection of operations research and
mathematics,especiallydiscretemathematics,whichdealswithsomenewquestions
andnewproblemsfrompractice.Itistryingtofindanoptimumobjectfromafinite
set of objects. In order to tackle these problems, we require the development and
combination of ideas and techniques from different mathematical areas including
complexity theory, algorithm theory, matroids, as well as graph theory, and
combinatorics, convexand nonlinear optimization,discrete and convex geometry.
As mentioned before, combinatorial optimization and graph algorithms are the
areasthathaveattractedalotofattentionsrightnow.Thiscanbealsowitnessedby
the fact that there are many NII Shonan meetings in this area, including graph
algorithms and combinatorial optimization in February 2011 (which is the first
Shonan meetings) and combinatorial optimization in April 2016. Both meetings
v
vi Preface
have attracted many top researchers in combinatorial optimization and graph
algorithms, as well as many young brilliant researchers.
This volume is based on several Shonan meetings. We focus on theoretical
aspects of combinatorial optimization and graph algorithms, and we have asked
several participants to extend their lectures into written surveys and contribute to
this volume. We are fortunate to have five renowned researchers in this volume. It
covers variety of topics, including network designs, discrete convex analysis,
facility location and clustering problems, matching game and parameterized
complexity.
Thisvolume consists offivechapters. Letus look ateachchaptermore closely.
Facilitylocationisoneofthecentraltopicsinthecombinatorialoptimization.In
this problem, we are given a set offacilities and clients, and the problem demands
choosing subset offacilities to open and assigning every client to one of the open
facilities so that the opening costs offacilities and the distance from each client to
its assigned facility are minimized. The facility location problem can be seen as a
problem of clustering a given set of clients. By changing the problem setting
slightly, we can also define the k-median and k-center problems, which are
extensively studied problems in this field. Since these problems are all NP-hard,
approximation algorithms have been considered mainly. Indeed, studies on the
facility location and its variants lead to the discovery of many algorithmic
methodologies so far. In “Recent Developments in Approximation Algorithms for
Facility Location and Clustering Problems” by An and Svensson, they survey
recentprogressonapproximationalgorithmsfortheseproblems.Inparticular,they
focus on two algorithmic methodologies—local search and linear programming
based methods.
In “Graph Stabilization: A Survey,” Chandrasekaran introduces graph
stabilization, which is an optimization problem motivated by the cooperative
matching game. In the cooperative game, a distribution of the profit to players
possessing some desirable property is called a core allocation. Stable graphs are
those graphs for which the matching game has non-empty core. The graph
stabilization seeks a smallest modification of a given graph to make it stable.
Besides the motivation from the game theory, the graph stabilization is interesting
becauseithasaconnectionwithmatchingtheory,integerprogramming,andlinear
programming. It has received considerable attention recently, and much progress
has been made in the research on it. Chandrasekaran surveys this progress.
Network design problems are optimization problems of constructing a
small-weight network and include many fundamental combinatorial optimization
problems such as the spanning tree, Steiner tree, and Steiner forest problems. In
“Spider Covering Algorithms for Network Design Problems,” Fukunaga surveys
spider covering algorithms, which are key technique for solving various network
design problems. Spider covering algorithms rely on the idea used in the analysis
of thewell-known greedy algorithm for theset cover problem. To extend this idea
to network design problems, a type of graph called a spider is used in spider
coveringalgorithms.Theyconstructanetworkbyrepeatedlychoosinglow-density
spiders.Theframeworkofthesealgorithmsareoriginallyinventedfordesigningan
Preface vii
approximation algorithm for the node-weighted Steiner tree problem and has been
sophisticated in a series of studies on numerous network design problems.
Fukunaga introduces the basic idea of the spider covering algorithms.
Discrete convex analysis is a theory of convex functions on discrete structures
and has been developed in the last 20 years. In “Discrete Convex Functions on
Graphs and Their Algorithmic Applications,” Hirai is a pioneer to explore the
theoryofdiscreteconvexanalysisongraphstructures.Hisresearchismotivatedby
combinatorial dualities in multiflow problems and the complexity classification of
facility location problems on graphs. In his chapter, the recent development of his
theory and algorithmic applications in combinatorial optimization problems is
discussed.
Theproblemoffindinganassignmentofauthorizeduserstotasksinaworkflow
insuchaway thatall business rulesaresatisfiedhasbeenwidelystudiedinrecent
years.Whathascometobeknownastheworkflowsatisfiabilityproblemisknown
to be hard, yet it is important to find algorithms that can solve the problem as
efficiently as possible, because it may be necessary to solve the problem multiple
timesforthesameinstanceofaworkflow.Hence,themostrecentworkinthisarea
has focused on finding algorithms to solve the problems in the framework of
“parameterizedcomplexity,”whichdealwithproblemsinwhichsomeparameterk
isfixed(wecall“FixedParameterTractable”).In“ParameterizedComplexityofthe
Workflow Satisfiability Problem” by D. Cohen et al., they summarize our recent
results.
We are very grateful to all our contributors and the editors of Springer Nature.
Finally, we would like to thank the organizers of the corresponding Shonan
meetings and their participants. Special thanks also go to NII for organizing the
Shonan workshops, which now becomes the Asian “Dagstuhl.”
Tokyo, Japan Takuro Fukunaga
July 2017 Ken-ichi Kawarabayashi
Contents
Recent Developments in Approximation Algorithms for Facility
Location and Clustering Problems.. .... .... .... .... .... ..... .... 1
Hyung-Chan An and Ola Svensson
Graph Stabilization: A Survey..... .... .... .... .... .... ..... .... 21
Karthekeyan Chandrasekaran
Spider Covering Algorithms for Network Design Problems.. ..... .... 43
Takuro Fukunaga
Discrete Convex Functions on Graphs and Their Algorithmic
Applications... .... .... .... ..... .... .... .... .... .... ..... .... 67
Hiroshi Hirai
Parameterized Complexity of the Workflow Satisfiability Problem. .... 101
D. Cohen, J. Crampton, G. Gutin and M. Wahlström
ix
Recent Developments in Approximation
Algorithms for Facility Location
and Clustering Problems
Hyung-ChanAnandOlaSvensson
Abstract Wesurveyapproximationalgorithmsforfacilitylocationandclustering
problems,focusingontherecentdevelopments.Inparticular,wereviewtwoalgo-
rithmicmethodologiesthathavesuccessfullyleadtothecurrentbestapproximation
guaranteesknown:localsearchandlinearprogrammingbasedmethods.
1 Introduction
Thefacilitylocationproblemisoneofthemostfundamentalproblemswidelystudied
inoperationsresearchandtheoreticalcomputerscience:see,e.g.,Balinski[7],Kuehn
andHamburger[32],Manne[37],andStollsteimer[40].Inthisproblem,wearegiven
asetoffacilitiesF andclientsD,alongwithadistancefunctionc:F ×D →R+
andopeningcosts fi ∈R+associatedwitheachfacilityi ∈F.Giventhisinput,the
goaloftheproblemistochooseasubsetoffacilitiesto“open”andtoassignevery
clienttooneoftheopenfacilitieswhileminimizingthesolutioncost,wherethecost
isdefinedasthesumoftheopeningcostsofthechosenfacilitiesandthedistance
fromeachclienttoitsassignedfacility.
Thefacilitylocationproblem,initsessence,isaclusteringproblem:weaimat
findingalow-costpartitioningofthegivensetofclientsintoclusterscenteredaround
open facilities. While the facility location problem penalizes trivial clusterings by
imposingopeningcosts,mostoftheotherversionsofclusteringproblemsspecifya
hardboundonthenumberofclustersinstead.Alongwiththishardbound,different
objectivefunctionsgiverisetodifferentclusteringproblems.Thek-medianproblem,
for example, minimizes the total distance between every client and its assigned
facility,subjecttotheconstraintthatatmostkfacilitiescanbeopened(butopening
B
H.-C.An( )
DepartmentofComputerScience,YonseiUniversity,Seoul03722,Korea
e-mail:[email protected]
O.Svensson
SchoolofComputerandCommunicationSciences,ÉcolePolytechnique
FédéraledeLausanne,1015Lausanne,Switzerland
e-mail:ola.svensson@epfl.ch
©SpringerNatureSingaporePteLtd.2017 1
T.FukunagaandK.Kawarabayashi(eds.),CombinatorialOptimization
andGraphAlgorithms,DOI10.1007/978-981-10-6147-9_1
2 H.-C.AnandO.Svensson
costs do not exist). The facility location problem in fact is closely related to the
k-medianproblemastheformercanbeconsideredasaLagrangianrelaxationofthe
latter.Thek-centerproblemisabottleneckoptimizationvariant,whereweminimize
thelongestassignmentdistanceratherthanthetotal.Finally,theobjectivefunction
ofthek-meansproblemisthesumofthesquaredassignmentdistances.Aswewill
seesoon,techniquesdevisedinoneoftheseproblemsoftenextendtoothers,letting
ustoconsidertheseproblemsafamily.
All of these problems have been intensively studied to lead to the discovery of
many algorithmic methodologies; however, one of the common characteristics of
theseproblems,namelythatanopenfacilitycanbeassignedanunlimitednumber
ofclients,canbecomeanissuewhenweapplytheseproblemstopractice.Inorder
toaddressthisdifficulty,thecapacitatedfacilitylocationproblemwasproposed.In
this problem, each facilityi ∈F is associated with a maximum number of clients
Ui ∈Z+itcanbeassigned,calledcapacity.Thispracticallymotivatedvariant,nat-
urally,alsoexistsfortheother(hard-bounded)clusteringproblems.Unfortunately,
thesecapacitatedvariantsseemtomakefundamentaldifferenceintermsoftheprob-
lems’difficulty:manyalgorithmictechniquesdevisedinthecontextofuncapacitated
problemsdidnotextendtothecapacitatedones.
Most of the problems we have discussed above are NP-hard, and we therefore
do not expect that efficient algorithms exist for these problems. Faced with NP-
hard problems, we often overcome the difficulty by pursuing an approximation
algorithm which trades off the solution quality for fast computation. Formally, a
ρ-approximationalgorithm(foraminimizationproblem)isdefinedasanalgorithm
that runs in polynomial time to produce a solution whose cost is no worse than
ρ times the optimum. As the facility location problem in its most general form is
SetCover-hard, we often focus on the case where c is metric, i.e., c satisfies the
triangleinequality.Moreover,inthek-meansproblem,weusuallyconsiderthecase
wheretheclientslieinanEuclideanspaceandfacilitiescanbechosenfromthesame
continuousspace.
Inthisarticle,wewillsurveyapproximationalgorithmsforfacilitylocationand
otherclusteringproblems,puttingemphasisonrecentdevelopments.Inparticular,
wewilldiscusstwomainalgorithmtechniquesthathavebeensuccessfulindevising
suchalgorithms:localsearchandlinearprogramming(LP)basedmethods.
2 LocalSearchAlgorithms
Inthelocalsearchparadigm,wedefineanadjacencyrelationbetweenfeasiblesolu-
tions to the given problem, where this adjacency usually corresponds to a small
change in the solution. The algorithm is then to start from an (arbitrary) feasible
solutionandtorepeatedlywalktoanadjacentimprovedsolutionuntilwearriveat
a local optimum. The analysis, accordingly, is in two parts: first, we establish that
any local optimum is a globally good approximate solution; second, we show that
wearriveatalocaloptimumwithinpolynomialtime.Sometimesthesetwopartsare