Distributed Memory Auction Algorithms for the Linear Assignment Problem

Distributed Memory Auction Algorithms for the Linear AssignmentProblemLibor BuˇsDepartment of Computer Science and Engineering Czech Technical UniversityPrague,Czech Republicxbus@fel.cvut.czPavel Tvrd´ıkDepartment of Computer Science and Engineering Czech Technical UniversityPrague,Czech Republictvrdik@fel.cvut.czABSTRACTThe Linear Sum Assignment Problem(LSAP)consists infinding a minimum cost assignment of objects to per-sons given a cost matrix.We have developed a suite of distributed memory parallel auction algorithms(AA)forsolving the LSAP and implemented them in MPI-1on a PC cluster with a Myrinet network.The suite includes for-ward,reverse,and forward-reverse AAs,all with or without scaling the bidding increment.The subset of free persons acting in parallel in the auction can be chosen adaptivelyduring the computation and this feature,called block-size-adaptivity,allows to tune the parallel AAs for optimum per-formance.Our algorithms allow to solve large-scale LSAPinstances by allocating enough processors with sufficient total capacity of RAM memory,so that the effect of mem-ory swapping is eliminated.KEY WORDSDistributed Algorithm,Linear Sum Assignment Problem,Auction Algorithm,Linear Optimization1.IntroductionOne of the classic combinatorial optimization problems is the linear assignment problem.The task is tofind an opti-mal assignment of persons to objects.A specific type of the assignment problem depends on the interpretation of the optimality measure.In this paper,we consider the Linear Sum AssignmentProblem(LSAP).Its inputs are two distinct sets of per-sons and objects and a cost matrix,where is the cost of the assignment of object to person.TheLSAP output is a permutationsuch that is minimum.The LSAP can be also formulated using graph the-ory terminology as a minimum weight matching problem in bipartite graphs.It can also be defined as a0-1linear programming problem,i.e.,Ethernet.They used columnwise mapping of the cost ma-trix.The paper does not give any specific performance re-sults.Our understanding is that the results were disappoint-ing due to the slow interconnect and expensive broadcast and reduce operations and led the authors to a conclusion that a distributed memory implementation is not worth con-sidering.They have not considered the swapping issues for large instances of problems.In this paper,we describe a suite of truly distributed implementations of parallel AAs on a homogeneous dis-tributed memory parallel computer.We focus on large-scale and full dense LSAPs that would notfit into the main memory of a single computer.We propose several opti-mizations of the communication among processors.2.Auction algorithmsIn this section,we give a brief description of auction algo-rithms.For more details,we refer the reader to book[5]. The LSAP is defined as a minimization problem,but the AAs are usually described in the context of solving maxi-mization problems.We also use the maximization version, which is closer to real-world auctions.Both approaches are computationally equivalent.One can be transformed to an-other one by a simple transformationfor all.2.1Forward auction algorithmThe input of a forward auction algorithm(FAA)is a cost matrix and the output is vectorof object-person assignments.We also need array to store for each object its current price satisfying-complementary slackness(-CS)condition:for all.Initially,each object is unassigned and its price is.The main parameter of an FAA is the minimal increment in one bid denoted by.It has been shown in book[5]that for input instances with integer costs,the as-signment found by an FAA withare passed to the next phase,and the bidding resumes with empty assignment and old prices.That means,if,then the threshold size of the queue becomes zero,i.e.,the complete assignment is computed,since the theory guarantees that the assignment is optimal.The AAs that change the bidding increment in this way are called-scaled,those using constantProcessor stores locally submatrices such thatmod .The other data structures,i.e.,assign-ment vectorsand ,price vector ,profit vector ,and the queues of free persons and of free objects ,are replicated in local memories of allprocessors.iFigure 1.The cyclic diagonalwise mapping of among 4processors in our FRAA.The number in each cell determines the number of the processor in whose memory is stored.4.Experimental ResultsWe have written the algorithms in C ++using Myricom MPICH-GM implementation of MPI-1.The code was compiled with g++using the option -O2.4.1Computing environmentWe have evaluated the performance of our implementation of AAs on a 16node Linux cluster with a Myrinet network.Each node had Intel Pentium III processor at 733MHz with 256MB memory.Because the problem size of a sequen-tial AA on these nodes is limited to,we have evaluated the sequential algorithms on a Pentium III 1GHz processor with 1GB memory.To relate different frequen-cies of processors,we have multiplied the times achieved by the 1GHz processor by experimentally found factor 1.2.4.2Test cases and methodologyWe have tested the algorithms on the following types of dense input problems:RAND:–Each is a random integer from the inter-val ,where is an input constant,GEOM –First,random points in 2D squareare generated and each is assigned the inte-ger Euclidean distance between points and .All random values were generated by function using different random seeds for each run.4.3ResultsFirst,we have experimented with strategies for determin-ing the block size.The results were dependent on the ver-sion of the AA and also on the characteristics of input data.Our results show that for RAND input instances,the block size adaptivity based on the linear interpolation described in Section 3.2works quite well.An example of a dependency of the parallel time of FAA with RAND input data on the block size is shown in Table 1.All values of parallel times in the following text are in seconds.We have run the FAA on a 4processorcluster,the problem size was,the maximum cost ,and the block size for the threshold queue size was set to .1001665.6057.6859.8012850.58algorithmRAND 128100RAND 1650GEOM 16100GEOM450Table 2.Parameters for computing block size used infurther experiments.We have used in our code function MPIBidwith three components:the maximal value ,the second maximal value ,and the index of the best object for the current person in case of FAA.Our reduction oper-ation combines two structures (,,),(,,)to (,,)where is the maximal and the second max-imal value from the values ,,,,and is ifand otherwise.If block size is greater than1,MPIBid structures which is at least by one order of magnitude faster on our parallel clus-ter than a sequence of MPIBid structures in Gauss-Seidel versions.For example,MPIBid is only10times slower than MPIBid structure on our system.This is the main reason why the block-size-adaptive AA is faster than the pure Gauss-Seidel algorithm.The next set of experiments was to examine the speedups of various AAs for both kinds of input instances. Table3gives the running times of-scaled and unscaled AAs for RAND instances.We have not included results for unscaled FAA,because its performance strongly depends on the maximal cost and for large it works poorly.On the other hand,the-unscaled FRAAs for RAND instances outperforms both-scaled versions of FAA and FRAA.For sequential FRAAs,this observation was reported in pa-per[6].-scaledFAA FRAA187.4668.524310.3560.3816.836315.3895.9626.2412298.6493.9419.71Table3.Parallel times of AAs for RAND instances with size and maximal cost.Table4contains the running times of-scaled FAA and-scaled FRAA for GEOM input instances.From this table,we see that GEOM instances are much harder for the AAs.Because larger cost ranges may lead to longer’price wars’,-scaling is necessary to obtain reasonable perfor-mance and we do not consider-unscaled versions of AAs on GEOM data.The-scaled FAA achieves much worse speedups than-scaled FRAA,more specifically,there is no speedup for,but it is significantly faster.The speedups of the best methods for both instance types i.e.,-unscaled FRAA for RAND instances and-scaled FAA for GEOM instances,is depicted in Fig.2.We suppose that the speedup worsening for5processors com-pared to4processors is caused by dependency of the col-lective communication performance on.Table5summarizes the results of previous experi-ments in the context of varying problem size.We have al-ways chosen the minimal number of processors for which the datafit to their local memories and no swapping is needed.Table6gives the running times of the-scaled FAA with RAND data that do notfit into the memory of given-scaledFRAA 593.11211.25506.87608.98681.45613.28586.30RAND/FRAA2500 6.321155.20 750021.474211.25 1250028.0781088.04 1750083.61152062.77 Table5.Running times of the best AAs on instances of LSAP which stillfit into the RAM memories of nodes.300MB4300MB1102.25–21691.0223.883490.55865.54is of order or more,the communication and com-putation complexity can be comparable and this could pro-vide up to50%time improvement.However,this would require nonblocking collective communication operations, currently not available in the MPI library.Other future work will consist in investigating the best strategies for block size adaptivity.References[1]R.E.Burkard and E.C¸ela,Linear Assignment Prob-lems and Extensions,in P.M.Pardalos and D.-Z.Du(Ed.),Handbook of Combinatorial Optimization (Dordreck:Kluwer Academic Publishers,1999)75–149.[2]D.P.Bertsekas and D.A.Casta˜n on,Parallel syn-chronous and asynchronous implementations of the auction algorithm,Parallel Computing,17(6–7), 1991,707–732.[3]Ch.Sch¨u tt and J.Clausen,Parallel Algorithms for theAssignment Problem-an Experimental Evaluation of Three Distributed Algorithms,in P.M.Pardalos, M.G.C.Resende and K.G.Ramakrishnan(Ed.),Par-allel Processing of Discrete Optimization Problems (American Mathematical Society,DIMACS,1995) 337–351.[4]M.Brady,K.K.Jung,H.T.Nguyen,R.Raghavan andR.Subramonian,The Assignment Problem on Paral-lel Architectures,in D.S.Johnson and C.C.McGeoch (Ed.),Networkflows and matching(American Math-ematical Society,DIMACS,1993)469–517.[5]D.P.Bertsekas and J.N.Tsitsiklis,Parallel and Dis-tributed Computation:Numerical Methods(New Jer-sey:Prentice-Hall Inc,1989).[6]D.A.Casta˜n on,Reverse Auction Algorithm for As-signment Problems,in D.S.Johnson and C.C.Mc-Geoch(Ed.),Networkflows and matching(American Mathematical Society,DIMACS,1993)407–429.。