# VLSN Search Algorithms for Partitioning Problems Using Matching Neighbourhoods

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VLSN Search Algorithms for Partitioning Problems Using Matching NeighbourhoodsAuthor(s): T. ncan, S. N. Kabadi, K. P. K. Nair and A. P. PunnenSource: The Journal of the Operational Research Society, Vol. 59, No. 3 (Mar., 2008), pp. 388-398Published by: Palgrave Macmillan Journals on behalf of the Operational Research SocietyStable URL: http://www.jstor.org/stable/30132760 .Accessed: 28/06/2014 09:00

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Journal of the Operational Research Society (2008) 59, 388-398 C 2008 Operational Research Society Ltd. All rights reserved. 0160-5682/08 $30.00

www.palgravejou rnals.com/jors

VLSN search algorithms for partitioning problems using matching neighbourhoods T Oncanl, SN Kabadi2, KPK Nair2 and AP Punnen3*

1Galatasaray Universitesi, Ortakiy, Istanbul, Tiirkiye; 2University of New Brunswick, Fredericton, New Brunswick, Canada; and 3Simon Fraser University Surrey, Surrey, British Columbia, Canada

In this paper, we propose a general paradigm to design very large-scale neighbourhood search algorithms for generic partitioning-type problems. We identify neighbourhoods of exponential size, called matching neighbourhoods, comprised of the union of a class of exponential neighbourhoods. It is shown that these individual components of the matching neighbourhood can be searched in polynomial time, whereas searching the matching neighbourhood is NP-hard. Matching neighbourhood subsumes a well-known class of exponential neighbourhoods called cyclic-exchange neighbourhoods. Our VLSN algorithm is implemented for two special cases of the partitioning problem; the covering assignment problem and the single source transportation problem. Encouraging experimental results are also reported. Journal of the Operational Research Society (2008) 59, 388-398. doi:10.1057/palgrave.jors.2602356 Published online 3 January 2007

Keywords: integer programming; combinatorial analysis; VLSN search; partitioning problems; generalized assignment; local search

1. Introduction

Several problems of practical interest, well studied in the operations research literature can be formulated in a com- mon framework as partitioning problems. These include the capacitated minimum spanning tree problem (Ahuja et al, 2003), the graph partitioning problem (Kernighan and Lin, 1970), the k-constrained multiple knapsack problem (Ahuja and Cunha, 2005), the airline-crew scheduling problem (Hoff- man and Padberg, 1993), the problem of scheduling n jobs on m non-identical parallel machines so as to minimize the sum of completion times (Pinedo, 1995), and the p-centre and p-median problems (Drezner and Hamacher, 2004). Loosely speaking, a partitioning problem seeks a partition of the space of decision variables such that the 'cost of partitioning' is minimized. Depending on the way one defines the 'cost', partitioning problems are divided into various subclasses such as minmax problems, minsum problems etc. Minsum partitioning problems are perhaps the most well-studied class in the literature. In fact, except the p-centre problem, all the examples discussed above belong to the class of minsum partitioning problems.

A formal definition of the partitioning problem is as follows. Let N = {1,2,...,n} be a finite set and { F1, F2 ..., Fm) } be families of prescribed subsets of N. Let S = (S1,S2, 2 .. Smn) be an ordered partition of N, that is

Ui, Si = N and S, n S = 0 for i

: j. We say that the or-

dered partition S is feasible if and only if Si E Fi for all i. For each i = 1, 2 ..., m and X E Fi, a cost function ci(X) is also prescribed. Then the minsum partitioning problem is to find a feasible partition S such that f(S) = L1, ci (Si) is minimum. In this paper, we consider only minsum partition- ing problems and hereafter, a partitioning problem means a minsum partitioning problem.

When N is the node set of a complete edge-weighted graph G(N, E) and ci(X) is defined as the value of the smallest Hamiltonian cycle in the subgraph of G induced by X, the resulting partitioning problem is the well-known vehicle rout- ing problem. Later in this paper, we discuss other examples of partitioning problems.

Partitioning problems are inherently difficult combinato- rial optimization problems. As a result of the inadequacy of exact methods, heuristic and metaheuristic algorithms are widely used in practice. Very large-scale neighbourhood (VLSN) search algorithms (Ahuja et al, 2002) are known to be very powerful in solving partitioning problems. The concept of cyclic-exchange neighbourhoods (Thompson and Orlin, 1989; Thompson and Psaraftis, 1993) plays a vital role in the design of most of the well-studied VLSN search algorithms (Agarwal et al, 2004; Ahuja et al, 2004a,b). However, find- ing the best member in the cyclic-exchange neighbourhood is an NP-hard problem (Thompson and Orlin, 1989; Thompson and Psaraftis, 1993). Thus, VLSN search algorithms using the cyclic exchange neighbourhoods often employ heuristics or enumerative schemes to search the neighbourhood for an improving solution.

*Correspondence: AP Punnen, 17048, 57th Ave, Surrey, British Columbia, Canada V3S 8M9.

E-mail: apunnen@sfu.ca

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T Oncan et a--VLSN search algorithms for partitioning problems using matching neighbourhoods 389

In this paper, we introduce a closely related neighbour- hood, called the matching neighbourhood to design VLSN search algorithms for partitioning problems. Various prop- erties of the matching neighbourhood are established along with complexity results. The matching neighbourhood con- tains the cyclic-exchange neighbourhood and we observe that searching the matching neighbourhood is also NP-hard. Also the transition from one solution to another in the matching neighbourhood is achieved by means of one or more cyclic exchanges involving the movement of one or more items. In the case of a cyclic exchange neighbourhood, such a tran- sition is accomplished by one cyclic exchange, normally by moving one item. We show that a solution S is a local min- imum for a partitioning problem with respect to the match- ing neighbourhood if and only if it is a local minimum with respect to the cyclic exchange neighbourhood. In this sense, the two neighbourhoods are equivalent, although the path to a local minimum could be different in a VLSN search, es- pecially since an improving matching could be composed of several cyclic exchanges. Since both these neighbourhoods are NP-hard, heuristics employed to search the neighbour- hoods are different which could also lead to different search

paths resulting in termination of different approximate local optimal solutions. For a discussion of approximate local op- timal solutions, we refer to Orlin et al (2004). The matching neighbourhood is designed as the union of simpler neighbour- hoods.

Depending on the feasibility restrictions, each of these sim- pler neighbourhoods itself could be of exponential size. Inter- estingly, we show that these simpler neighbourhoods can be searched in polynomial time by solving a weighted bipartite matching problem (Lawler, 1976).

To illustrate our VLSN search algorithms for partition- ing problems, we consider two special cases: the covering assignment problem (CAP) and the single source transporta- tion problem (SSTP). Given n factories with non-negative demands al, a2,. an and m suppliers with non-negative supply quantities bl, b2 ..., bm, the classical transportation problem is to find a shipping plan so that the total shipping cost is minimized while supply and demand constraints are satisfied. The cost dij represents the shipping cost of one unit from supply point i to a demand point j. We assume that dij 0. A variation of the transportation problem is the SSTP where the demand points must be served by exactly one sup- plier. This problem can be mathematically formulated as

mi n min dijYij (1)

i=1 j=l n

s.t. Zyij bi i =1

.. m (2)

j=1

yij= aj j= 1

.... n (3)

i=1 (4) Yij = 0 or aj Vi, j (4)

where decision variable yij stands for the quantity supplier i supplies to factory j. Using the transformation Yij =ajxij and cij = dijaj, we get an alternative formulation of the SSTP as

inn

min Z c 1xij (5) i=1 j=1

n

s.t. Laixij

390 Journal of the Operational Research Society Vol. 59, No. 3

partition. The element cost functions ci(X) for X E Fi is defined as ci(X)= Cjexcij, i

= 1 ..., m. The major contributions of the paper can be summarized

as follows: A general algorithmic approach is introduced to develop VLSN algorithms for partitioning problems by means of matching neighbourhoods. It is shown that a restricted version of the matching neighbourhood and the well-studied cyclic exchange neighbourhood have the same set of local minima. In this sense, these two neighbourhoods are equiv- alent and searching each of these neighbourhoods is an NP- hard problem. The different combinatorial structure of these neighbourhoods leads to different heuristics to search these neighbourhoods. The matching neighbourhood could perform multiple cyclic exchanges in one move, whereas cyclic ex- change neighbourhood makes one cyclic exchange per move. This could potentially reduce the number of iterations in a VLSN search and may allow additional parallelization op- portunities. The matching neighbourhood is the union of a class of large neighbourhoods. It is shown that each of these component neighbourhoods can be searched for an improving solution in polynomial time. In addition to establishing desir- able theoretical properties of the matching neighbourhoods, the approach is tested on two classes of partitioning problems which produced promising experimental results.

The paper is organized as follows. In Section 2, we in- troduce the matching neighbourhood and investigate its prop- erties. We also introduce a general VLSN search algorithm for the partitioning problems using the matching neighbour- hoods. Section 3 deals with application of the VLSN search algorithm to the special cases of the CAP and the SSTP. Com- putational results are given in Section 4 followed by conclud- ing remarks in Section 5.

2. The matching neighbourhood

Let us now define the matching neighbourhoods for the partitioning problem. Our work on matching neighbourhoods is inspired by the work of Punnen (2001) in the context of the travelling salesman problem. For any subset X of N, let ci(X) = jexcij and a(X)

= Zjexaj. Throughout

this paper, we assume that summation over the empty set is zero. Let S = (SI, S2 ..., S,,n) be a given ordered partition of N. Then the cost of S, denoted by C(S), is given by C(S) = L7i ci(Si). For each set Si in S, choose a subset li. Note that

I/ could be the empty set. The ordered collection

I = (II, I2 . I .,

7,) is called an ejection vector of S. Given S and an ejection vector I, we construct a bipartite graph G, called the improvement graph, as follows. The generic bipar- tition of its vertex set is V1 U V2 where VI {=1, 2,..., m} and V2 = {Z1,, Z

.. Zmnl}. Node i in VI represents the subset Si

and node zj in V2 represents the subset Il. A node i in VI is connected to a node zj in V2 by an arc (i, zj) if (Si U Ij)\Ii e

Fi. For example, in the case of SSTP verifying this condition is equivalent to testing if a(Si) - a(l) + a(Ij) bi, that is, node i in V1 is connected to a node zj in V2 by an arc if and

only if a(Si) - a(li) + a(I) o bi. The edge (i, zoj) signifies adding Ij to Si while i is ejected from Si. The cost of (i, z j) is given by

fij- =

c(i(j) - ci(li)

Note that for each i = 1,2 ... m the arc (i, zi) is in G and

/3i =0.

Let M= {(i, zL(i))i = 1, 2 ..., m} be any perfect matching in G. (G contains at least one perfect matching.) Then S =

(SI, S2 ....

S,,,) is a feasible ordered partition where Si = IUi U Si \Ii, i 1, 2, ...,

m. The operation of building S from S is called an M-matching exchange. The simple matching neighbourhood of S with respect to a given ejection vector

1= (11, 12, . . .

In,}, denoted by MI(S), is the collection of all feasible ordered partitions each of which can be obtained by a M-matching exchange from S using some perfect matching M of G.

For example, consider the SSTP with cost matrix

13 14 14 17 13 13 15 14 17 15 17 19 19 13 11 11 20 20 12 14 19 12 15 12 16 14 15 20 13 20 12 15 13 15 13 15 17 17 11 11

supplies (al, a2z,..., alo)=(3, 2, 2, 1, 3, 3, 2, 1,2, 1) and de- mands (bl, b2, b3, b4)=(10, 6, 8, 3). Let S= (SI, S2, S3, S4) where S1 = {1,3,4, 6}, S2 = {2, 5}, S3 = {7, 8, 9, 10}, and S4 = 0 be a given feasible ordered partition. Consider the ejection vector I = {I1, 12, 13, 4} where II - {4, 6}, 12 = {5}, 13 = {10}, and 14 = 0. The resulting improvement graph G is given in Figure 1.

The minimum cost perfect matching is marked by thick lines. The cost of this matching is -13 and the resulting improved partition is given by S1 = {1, 3, 5}, S2 = (2, 4, 6}, S3 = {7, 8, 9} and S4 = {10}.

Lemma 1 There is a cost preserving one-to-one correspon- dence between elements of the simple matching neighbour- hood Mi(S) of S and perfect matchings in G.

The proof of this lemma is straightforward and hence omitted. The minimum cost perfect matching in G can be computed in O(m3) time (Ahuja et al, 1992; Lawler, 1976).

The size of the matching neighbourhood Mi(S) for a given I could be as large as O(m!) and the best member in this neighbourhood can be identified in O(m3) time (Ahuja et al, 1992; Lawler, 1976) by solving a weighted matching prob- lem on G. Let us now consider a generalization of Mr (S). Let M(S) = U1 Mi(S), that is M(S) is the union of simple matching neighbourhoods corresponding to all possible ejec- tion vectors I. We call M(S) the matching neighbourhood of S. For other ways of generating large neighbourhoods,...

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