an effective multilevel tabu search approach for balanced graph partitioning

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    Given an undirected graph G(V, E), where V and E denote setsof vertices and edges respectively, the balanced k-partitioning

    of cutting edges (i.e. edges whose endpoisubsets) isminimized. The particular partitito two is often called bisection. The class of p

    e rangata m

    roblemcase onstituoblem

    presented an improvement of the KL algorithm by introducing the

    gains. Additional KL-like heuristics are reported, for instance, in

    tabu search [7,23,3], simulated annealing [15], genetic andevolutionary algorithms [27,5,20,25,26].

    nitiallems.acing

    restrictions on which solutions the renement algorithm can visit.

    contribution reported in this paper is a perturbation-based iterated

    Contents lists available at ScienceDirect

    .e

    Computers & Oper

    Computers & Operations Research 38 (2011) 10661075with the two well-known partitioning packages METIS [16] andhao@info.univ-angers.fr (J.-K. Hao).tabu search procedure which is specially devised for balancedk-partitioning and used for partition renement of each coarsenedgraph. Integrated within the multilevel approach, our multileveliterated tabu search (MITS) algorithm competes very favorably

    0305-0548/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.

    doi:10.1016/j.cor.2010.10.007

    Corresponding author.E-mail addresses: benlic@info.univ-angers.fr (U. Benlic),concept of cell gain and an efcient data structure to calculate cell This work is based on the multilevel paradigm. The mainabout the same cardinality, one for each bisection part, and toimprove the quality of the bisection by swapping two vertices ofthe two subsets of the bisection. Fiduccia and Mattheyses [10]

    Basically, the approach allows one to approximate the iproblem by solving successively smaller (and easier) probMoreover, the coarsening helps lter the solution space by plefforts have been made in devising a number of efcient andpowerful heuristics.

    One of themost popular graph bisection heuristics is the famousKernighanLin (KL) algorithm [19] that iteratively renes anexisting bisection. The idea is to nd two subsets of vertices of

    coarsening phase, they differ greatly in theway the initial partitionis generated, and how the successive partitions are renedthroughout each uncoarsening phase.

    As illustrated in [29], the multilevel paradigm is a usefulapproach to solving combinatorial optimization problems.notable for its applicability to a widtions such as VLSI design [1,26], dsegmentation [24].

    The general graph k-partitioning premains NP-complete even for theTherefore, approximate methods cosolution approach to address this pre, such that the numbernts belong to differentoning casewhen k is setartitioning problems ise of important applica-ining [30], and image

    is NP-complete, and itf bisection (k2) [11].te a natural and useful. In recent years, many

    paradigm proves to be quite useful [2,8,18,28,21]. The basic idea ofa multilevel approach is to successively create a sequence ofprogressively smaller graphs by grouping vertices into clusters(coarsening phase). This phase is repeated until the size of thesmallest graph falls below a certain coarsening threshold.Afterward, a partition of the coarsest graph is generated (initialpartitioningphase), and then successively projected back onto eachof the intermediate graphs in reverse order (uncoarsening phase).The partition at each level is improved by a dedicated renementalgorithm beforemoving to the upper level. Even though almost allmultilevel algorithms employ the same or similar heuristics in theproblem consists in partitioning the vertex set V into k kZ2disjoint subsets of approximately equal siz

    To handle large and very large graphs, the so-called multilevelAn effective multilevel tabu search appr

    Una Benlic, Jin-Kao Hao

    LERIA, Universite dAngers, 2 Boulevard Lavoisier, 49045 Angers Cedex 01, France

    a r t i c l e i n f o

    Available online 17 October 2010

    Keywords:

    Balanced partitioning

    Multilevel approach

    Iterated tabu search

    a b s t r a c t

    Graph partitioning is one

    domains, such as VLSI desig

    k-partitioning problem con

    size, such that the number

    for balanced partition, wh

    periodic perturbations. Exp

    proposed approach not on

    METIS andCHACO, but also

    the literature.

    1. Introduction

    journal homepage: wwwach for balanced graph partitioning

    he fundamental NP-complete problems which is widely applied in many

    mage segmentation, data mining, etc. Given a graph G(V,E), the balancedts in partitioning the vertex set V into k disjoint subsets of about the same

    utting edges is minimized. In this paper, we present amultilevel algorithm

    integrates a powerful renement procedure based on tabu search with

    ental evaluations on awide collection of benchmark graphs show that the

    ompetes very favorably with the two well-known partitioning packages

    provesmore than two thirds of the best balancedpartitions ever reported in

    & 2010 Elsevier Ltd. All rights reserved.

    [17] for bisection and in [8,18,28] for k-partitioning. Other well-known algorithms are based on popular metaheuristics such as

    lsevier.com/locate/caor

    ations Research

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    many of these best partitions were obtained by previous algorithms

    benchmark graphs. Finally, in Section 6 we investigate twoimportant features of the proposed MITS algorithm.

    to nd a balanced partition with the minimum sum of weights ofcutting edges E .

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    set ofphase.graph.hat nol timet leastto thetching

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    U. Benlic, J.-K. Hao / Computers & Operations Research 38 (2011) 10661075 1067c

    3. Multilevel tabu search graph partitioning

    3.1. General procedure

    Our multilevel tabu search approach follows the general multi-level paradigm [8,4,2].

    Let G0(V0,E0) be the initial graph, and let k denote the numberof partition subsets. The multilevel paradigm can be summarizedby the following steps.

    (1) Coarsening phase: The initial graph G0 is transformed into asequence of smaller graphs G1, G2,y,Gm such that jV0j4jV1j4 jV2j4 4 jVmj. This phase stops when jVmj reaches axed threshold (coarsening threshold).

    (2) Partitioning phase: A k-partition Pm of the coarsest graphGm(Vm,Em) is generated.

    (3) Uncoarsening phase: Partition Pm is progressively projected2. Problem description and notations

    Given an undirected graph G(V, E), V and E being the set ofvertices and edges respectively, and a xed number k, ak-partitioning ofG can be dened as amapping (partition function)p : V-f1,2, . . . ,kg that distributes the vertices of V among k disjointsubsets S1 [ S2 [ . . . [ Sk V of roughly equal size.

    The partition function p induces a new graph Gp GpS,Ec,where S {S1, S2,y,Sk} and an edge fSx,SygAEc exists if there aretwo adjacent vertices u,vAV such that uASx and vASy. The set Eccorresponds to the set of cutting edges of G induced by thepartition. Vertices u and v of a cutting edge fu,vgAEc are calledborder vertices of the partition.

    Throughout this paper, the initial input graph G is supposed tohave a unit cost weight for both vertices and edges. However, asexplained in Section 3.2, the multilevel approach generatesintermediate (coarsened) graphs where vertices and edges mayhave different weights (see Section 3.2). As a general rule, theseweights increase during the coarsening phase, and decrease duringthe uncoarsening phase. Let jvj denote the weight of a vertex v.Then, the weightWSi of a subset Si is equal to the sum of weightsof the vertices in Si, WSi

    PvASi

    jvj.The optimizationobjective f of our balancedpartitionproblem isusing a signicantly longer running time (up to one week).The rest of the paper is organized as follows. In the next section,

    we provide the formal graph partitioning description, as well assome notations used throughout this paper. In Section 3, wedescribe the multilevel paradigm, and detail the phases of theproposed multilevel algorithm. In Section 4, we present theperturbation-based tabu search algorithm, which is the keypartition renement mechanism in our multilevel approach. InSection 5, we present computation results and comparisons on theprolcurritioning archive reveal that MITS is able to nd better partitionsh fewer cuttingedges) thanMETIS andCHACO for a largeportionese benchmark graphs. Furthermore, when the running time isonged up to 1 h, MITS even improves some two thirds of theent best balanced partitions ever reported in the literaturewhilesetpartCO [8]. Indeed, extensive experiments performed on the wholeof benchmark graphs from the University of Greenwich graphCHAback to each intermediate Gi (im1,m2,y,0). Before each thence we have obtained a sequence of smaller graphs in thersening phase, the next step is to generate an initial partition ofusing an edge contraction heuristic called heavy-edge matching(HEM), which has OjEj time complexity [17]. This methodconsiders vertices in random order, matching each unmatchedvertex vwith its unmatched neighbor u, if any, such that theweightof edge {u,v} is maximum among all the edges incident to v.

    3.3. Initial partition and its renementtime. If a vertex is not incident to any edge of subset G, it is scopied over to Gi+1.

    When vertices v1,v2AVi are collapsed to form a vertex vcAthe weight of the resulting vertex vc is set equal to the sweights of vertices v1 and v2. The edge that is incidentbecomes the union of all the edges incident to v1 and v2, minedge fv1,v2gAEi. Therefore, during the coarsening process, bovertex and edge weight increases. At any level of the coarsphase, the