Facility Location: Sequential and Distributed Approximation Algorithms

Facility Location: Sequential and Distributed Approximation Algorithms Peter Pietrzyk 04.05.2011

Overview • Facility Location • Definition • Properties • Sequential (anddistributed) Algorithms • GreedyAlgorithm • Primal-Dual Algorithm • AlgorithmcomprisingGreedyandPrimal-Dual approaches • Relevanceforour Project Group • External Dynamics • Controlled Dynamics

MetricFacility Location Problem 1 2 3 4 Facilities Completebipartite, weightedgraph Clients 3 2 4 5 6 1 • Nodes changeroles: • Facilities • Clients • openingcostsfor facility • distancebetween client and facility • areindicator variables Metric Clique

ApplicationsforlocalFacility Location algorithms • Computer Networks • Selected nodes provide a costly service • Latency corresponds to distance (shortest path metric) • Huge graphs require local algorithms • Changes in latency (External Dynamic) • Mobile Sensor Networks • Some robots provide services to other robots • Distance is represented by the Euclidean distance • High numbers of simple robots require local algorithms • Mobility of the robots (External Dynamic)

Properties oftheFacility Location Problem • Inherent locality (explainedlater) • NP-hard • Approximation (withoutmetric): • Approximation (withmetric): • factor is possible (byByrka [APPROX 2007]) • Lowerboundof 1.463 (by Guha et al. [J. ofAlgo 1999] (assuming NPDTIME) • Varioustypesofalgorithms: • Greedy • Primal-Dual • LP-Rounding • PLS (Polynomial Time Local Search)

What‘s an intuitive waytosolvetheproblem? • Open cheapfacilities • Open facilitieswithmanyclientscloseby • Do not open facilitiesthatareclosetoeachother 1 5 3 5 4 2 6 3 10 2 7 3 6 Find a balancebetweenthepropertiesabove

Radius Definition byMettuandPlaxton[FOCS 2000] • Radius of node : • Choose such that • istheradiusofnode • Small radius: • many nodes close by • cheap facility • What about nodes with small radius close to each other? • not considered in theradiusvalue

AlgorithmbyMettuandPlaxton[FOCS 2000] • Computetheradiusvalueforeachnode • Usetheradiustosortthenodes (ascendingorder) • Iterateoverthesortedlist • Open nodeifthereisnootheropenednode in times theradiusof • Runtime: where is the number of nodes • Approximation factor:

AsynchronousExecution Model • Computationateachnodeisfree • Atmostonenodeisactiveatanypoint in time • Runtimedefinedbynumberofrounds: • Nodes becomeactive in an arbitraryorder • A round ends when all nodes were active at least once • Locality: Nodes „close“ to each other can communicate

The distributedapproximationalgorithm Algorithm from a node’s point of view: • Gather information about neighbor nodes: • How far away are they? • Compute radius and round it to the next power of (only possible values for the radius) • In an endlessloop: • Wake up(becomeactive) • Check invariant (next slide) • Change role if necessary • Go back tosleep

Restoring the Invariant i ri ri 2ri 4ri Nodes choose their role in such a way that the following two conditions are satisfied: • If is a client, there must be a facility with and • If is a facility, there must not any other facility with and • Runtime: , where 𝑛 is the number of nodes • ProofIdea: • State ofthenodeswiththesmallestradiusdoes not change after 2 rounds • Once all nodeswithradiusstopchangingtheirstate, nodeswithradiuswill stopchangingthierstateswithin 2 rounds • Thereareonly different radiusvalues i

Distributed Execution Models • Asynchronous Model • Computationateachnodeisfree • Atmostonenodeisactiveatanypoint in time • Runtimedefinedbynumberofrounds: • Nodes becomeactive in an arbitraryorder • A round ends when all nodes were active at least once • Locality: Nodes „close“ to each other can communicate • SynchronousModel • Computation at each node is free • All nodes become active at the same time • LOCAL (unlimitedmessagesize) • CONGEST (limited messagesize)

-Approximation in rounds • Each nodecomputesitsownradius radiusclasses (setofnodeswith same radius) • Create a graphforeachradiusclass: • Nodeswithradiusbelongto • Edges: and • Compute a foreachgraph • All nodesnot in oneofthe‘s areclients • Rest ofthenodeschangestheirrolesuntil: • Eachclienthas a facilitycloseby • Foreachfacilitythereisnootherfacilitycloseby c c c c c c c c c

Problem Definition 1 2 3 4 Facilities Completebipartite, metricgraph Clients 3 2 4 5 6 1 • openingcostsfor facility • distancebetween client and facility • areindicator variables Metric Clique

Duality (appliedtothefacilitylocationproblem) Primal • Duality Theorem: An optimal solutiontothe relaxed primalprogramhasthe same costsas an optimal solutiontothe dual program. Dual relaxed min. max. • s.t.s.t. primal slacknesscondition (1) (2) dual slacknesscondition (3) (4)

Duality & theComplemtarySlacknessCondition dual slacknesscondition (3) (4) primal slacknesscondition (1) (2) relaxed primalslacknesscondition (5) (6) optimal dual solution optimal primalsolution (relaxed) optimal primalsolution OPT/3 OPT 3 OPT Costs solution pair thatsatisfies relaxed slacknesscondition feasible dual solutions feasibleprimalsolutions

Intuition forthe dual program • Think of as the amount client is willing to pay: • (2) costsofeachopenedfacilityarecoveredby • (4) onlyclientsconnectedto pay for its opening • (1) means that is composed of the connection costs and the opening costs • Assume and are optimal solutions and contains only integers primal slacknesscondition (1) (2) dual slacknesscondition (3) (4)

Algorithmby Jain andVazirani [FOCS 1999] • Phase 1: • Infeasibleprimalsolution (all andaresetto) • Feasible dual solution (all and aresetto) • Increase all simultaneously • Once, startincreasing • Once, stopincreasing and of all clientswith. Also, open facility and connect clients to it with . • Phase 2 (explainedlater) Primal Dual

Algorithmby Jain andVazirani [FOCS 1999] (2) • We (potentially) opened way too many facilities in Phase 1. • Which facilities do we close? • Phase 2: • Weneedtoconstructthefollowingconflictgraph: • Facilitiesthathavemorethan 1 contributingclientarenode • If a clientcontributestofacility and , there is an edge between and • Compute an maximal independentset on • Close all facilitythatare not in the maximal independentset • Reconnecttheclientsthat „lost“ a facilitytoneighborsofin

ProofIdeaof Jain & Vazirani‘salgorithm • Must showthat: (since ) relaxed primalslacknesscondition (5) (6) • resp. time resp. opened Client was connectedto in thefirstphase

Distributed Version ofthe Jain andVaziraniAlgorithm [PODC 2009, Pandit & Pemmaraju] • Executed on a completebipartitegraphusingtheCONGEST model • Ensurepolylogarithmicruntimebyhandlingcheapfacilities (Initialization Phase) • 1. Phase: Increasingthevariable continuously is not possible multiply by in each step • UseLuby‘s MIS algorithmforthe 2. Phase • Proof: • Initialization Phase: canbeommited, sincefacilitycostsanddistancesare limited tobits • 1. Phase: Same asthe original version • 2. Phase: Need to check ifLuby‘salgorithmcanbecomputed on thebipartitegraph in theCONGESTmodel

The GreedyAlgorithm(with a Primal-Dual approach) • Authors: Jain, Madhian, Marakakis, Saberi, Vazirani • Journal ofthe ACM 2003 • Approximation factor: • Runtime: ( is the number of edges) • Analysis: Dual-fitting withfactorrevealing LP

The GreedyAlgorithm (Costefficient Stars) • A star consists of facility and a set of clients • Cost of a star is • The costofthecheapeststararoundisequaltoitsradius • Algorithm: • Foreach compute thecheapeststar • Open withtheoverallcheapeststarandconnect all to • Set andremovefrom • Repeat untilnoclientsleft in • In contrasttotheradius in thealgorithmbyMettuandPlaxtonthecostofthestarisupdatedafter a facilityhasbeenopened. • The algorithmabovecanbesimulatedby a „kindof“ primal-dual algorithm.

The GreedyAlgorithm (Description) • Imagine to be the amount client wants to pay • Each client increases until it is connected to an open facility • Once client starts paying for facility • Once facility is opened ( are unconnected clients) and all contributing clients are connected to Solution costs: • Only paid facilities are opened • Clients contribute only to a single facility • Clients contribute after the connection costs are paid 1 2 facilities client j

Infeasibilityofthe dual solution • The computedsolutionis not a feasible dual: • Open if ( are unconnected clients) • should be openedearlier! • Can not usedualitytoprovetheappr. factor (yet!) Dual

Dual Fitting • Find , such thatis a feasible dual solution • istheapproximationfactor • Whatistheminimumthatworksfor all possibleinstances? feasibleprimalsolutions cost of the primal solution () optimal solution feasible dual solutions

FactorRevealing LP Lemma 1: Foreverytwoclients and a facility we have: • Find theminimumforwhich • Same asfindingthemaximumratioof • are variables • The constraintsof Lemma 1 and 2 must bemet metricproperty Lemma 2: Foreveryclient and facility : nofacilityis over-paid

Parallel ExecutionoftheGreedyAlgorithm • CONGEST Model (synchronous, limited messagesize) • Eachclientincreases in discrete time stepsbymuliplyingitwith, • Onceforfacility, isincreasedin the same way • Once, facilityisopenedand all contributingclientsareconnectedwithit… Problem: Whathappensifmanyfacilitiesareeligibleforopeningin the same step?

Dealingwithfacilityselection 1 2 3 4 Facilitiesthatarefullypaid andcouldbeopenednow Clients payingforthesefacilities 3 2 4 5 6 1 • Compute a Maximal Independent Set in rounds with Luby’salgo. • Open Facilities in theMIS • MIS guarantees: • Openedfacilitygets all itsclients • Clientassignedtoatmostonefacility Facility Graph 1 2 3 4

The Problem! • Set containsfullypaidfacilities • Payment of is in • Need toguarantee: • Facility is open in the next round, • orthepaymentforis • Thatis not alwaysthecase! Lemma 2: Foreveryclient and facility : nofacilityis over-paid Modified Version: 1 2 3 4 • Ifpaymentof not openedfacilities, thentheirpaymentmightbenextround • Lemma 2 violated 3 2 4 5 6 1

Dealingwithfacilityselection (2) • MIS does not presevetheapproximationfactor • Rajagopalan & Vaziranigive a 2-approximation in [J. ofComp 1998] • Pandit & Pemmarajugive a constantfactorapproximation in expectation in [PODC2010] • Bycomputing an independentsetinterativelyonecanget a approximation (but in time)

Overview • Facility Location • Definition • Properties • Sequential (anddistributed) Algorithms • GreedyAlgorithm • Primal-Dual Algorithm • AlgorithmcomprisingGreedyandPrimal-Dual approaches • Relevanceforour Project Group • ExternalDynamics • Controlled Dynamics

Howtomodeland deal withExternal Dynamics • Modeling External Dynamics • Input Stream (distances and costs change over time) • Only small changes to the problem instance in a single step • Analyzing External Dynamics • Competitive Analysis • Quiescence Analysis • Stop changing the instance at time • At time the algorithm computed a good solution • What is )? • Is the entire solution or just a limited area affected? • Are all nodes affected? • …

The distributedGreedyAlgorithmbyMettuandPlaxtondealingwithExternal Dynamics The algorithm from a node’s point of view: • Sleep (inactive state, most of the time) • Weak Up (active state)! • Gather information about neighbor nodes: • What is their radius? • What is their role (facility/client)? • How far away are they? • Compute radius • Check invariant (change role if necessary) • Go back to sleep (1.) We can guarantee that: • Role changes take place in constant distance of an event (violation of the invariant) • Number of affected nodes is O(log2(n)) (euclideancase) • Amount of role changes per node is limited to 2

Overview • Facility Location • Definition • Properties • Sequential (anddistributed) Algorithms • GreedyAlgorithm • Primal-Dual Algorithm • AlgorithmcomprisingGreedyandPrimal-Dual approaches • Relevanceforour Project Group • ExternalDynamics • Controlled Dynamics

Controlled Dynamics • Algorithms not onlycanchangetheroleofnodes, but also thedistancesbetweenthenodes (in a limited way) 1 3 5 6 4 3 • What should the costs of our new solution be compared to and how? • Challenge: Find a reasonable quality measure for algorithms that use Controlled Dynamics. 7 3 6

Thank you for your attention! Heinz Nixdorf Institute & Computer Science Institute University of Paderborn Fürstenallee 11 33102 Paderborn, Germany Tel.: +49 (0) 52 51/60 64 66 Fax: +49 (0) 52 51/62 64 82 E-Mail: toon@upb.de http://www.upb.de/cs/ag-madh

Facility Location: Sequential and Distributed Approximation Algorithms