Variable neighborhood descent heuristic forcovering design problem

Nebojsa Nikolic a,1 Igor Grujicic a,2 -Dorde Dugosija b,3

a Faculty of Organizational Sciences, University of Belgrade, Jove Ilica 154, 11000 Belgrade, Serbia

b Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11 000Belgrade, Serbia

Abstract

In this paper we consider a variable neighborhood descent (VND) heuristic, a variantof variable neighborhood search, for the covering design problem. The neighborhoodstructures and local search are based on the systematic removing and adding blocksto the covering. We have tested VND search on the greedy coverings and established13 new best known upper bounds. The proposed approach is also applicable to anycovering design.

Keywords: covering design, variable neighborhood search, greedy.

1 Introduction

Let v, k, and t denote natural numbers where v ≥ k ≥ t. The family of k-subsets, called blocks, chosen from a v-set, such that each t-subset is contained

� This work is partially supported by the Serbian Ministry of Sciences, Project No. 174010.1 Email: sigma@fon.bg.ac.rs2 Email: grujicici@fon.bg.ac.rs3 Email: dugosija@matf.bg.ac.rs

Available online at www.sciencedirect.com

Electronic Notes in Discrete Mathematics 39 (2012) 193–200

1571-0653/$ – see front matter © 2012 Elsevier B.V. All rights reserved.

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in at least one of the blocks, is a (v, k, t) covering design, or just covering. Thenumber of blocks is the size of the covering. The covering number C(v, k, t)is the minimum size of a (v, k, t) covering. In other words, C(v, k, t) denotesthe smallest number of k-subsets of a v-set that cover all t-subsets.

The problem of determining C(v, k, t) is Covering design problem, or justcovering problem. It has a natural formulation as an NP-hard set-coveringproblem:

min 1T · xAx ≥ 1,

x ∈ {0, 1}(vk),

where A = [aij ] is(vt

)×

(vk

)matrix, and 1 is the vector of all 1’s with appro-

priate dimension. The element aij is equal to 1 if i-th t-subset is contained inj-th k-subset, otherwise it is 0.

Covering numbers have already been studied extensively, and numerouspapers have been published for particular values of v, k, and t. Nevertheless,the exact values of C(v, k, t) are known only when v, k, and t are small andin some special cases, such as C(v, 3, 2) and C(v, 4, 2). This is why a largenumber of papers deals only with lower and upper bounds on C(v, k, t).

The best general lower bound on C(v, k, t) comes from Schonheim [12]:

C(v, k, t) ≥⌈vk

⌈v−1k−1· · ·

⌈v−t+1k−t+1

⌉· · ·

⌉⌉.

Most of the papers are dedicated to determining the upper bounds oncovering numbers. Rodl [11] gives the best general upper bound on C(v, k, t).He defines the density of a covering as an average number of blocks containinga t-set, and shows that there exist coverings with density 1 + o(1) as v gets

large and k and t are fixed. More precisely: limv→∞

(kt

)(vt

) C(v, k, t) = 1.

Erdos and Spencer [2] give the bound:

(kt

)(vt

) C(v, k, t) ≤(1 + ln

(kt

)). Note

that this bound is weaker than the Rodl bound. However, unlike Rodl’s asymp-totic bound, it can be applied to all v, k, and t.

Most of the best known upper bounds can be found at the site [3]. Nu-merous best known upper bounds can also be found in [4,9,10].

Upper bounds are usually obtained by finding good coverings with thesmallest possible number of blocks. The majority of known methods forthe covering constructions are given in [4]: the greedy method, the induced-covering method, the dynamic programming, the constructions from finitegeometries, and other simple construction methods. Branch-and-Cut method

N. Nikolic et al. / Electronic Notes in Discrete Mathematics 39 (2012) 193–200194

for solving the covering design problem is presented in [6]. However, this ap-proach can be used only for small values of the parameters v, k, and t. Inorder to deal with larger values of the parameters, a few heuristic algorithmsare used. Besides the aforementioned greedy algorithm, Simulated annealing[8] and Tabu Search [1,9,10] were applied for the covering constructions.

In this paper we present VND heuristic for covering design problem whichincludes a new local search procedure that we call Level Reduction. Basically,VND reduces the number of blocks of a given covering. We have applied it ongreedy lexicographic coverings. As a result, we got several best known upperbounds on covering numbers.

The paper is organized as follows. In Section 2 we give a greedy algorithmfor constructing covering designs as well as the details of our Level Reductionprocedure. Section 3 provides VND algorithm for covering design problemand computational results in 3.1. In Section 4 brief conclusions are indicated.

2 Greedy and level reduction algorithms

Greedy heuristics belong to the simplest constructive heuristic methods. Usu-ally, they are used for getting an initial solution for other more complex heuris-tics. Sometimes, greedy heuristics provide good, even optimal solutions. Gor-don et al. [4] presented the following algorithm for generating covering designs:

Algorithm 1. Greedy

1. Arrange the k-subsets of a v-set to a list.

2. Choose from the list the k-subset that contains the maximum number of t-sets that are still uncovered. In case of ties, choose the k-subset occurringearliest on the list.

3. Repeat Step 2 until all t-sets are covered.

The greedy algorithm was tested in a lexicographic, colex, gray, and ran-dom order, depending on the arrangement of the k-subsets in step 1. Onaverage, greedy lexicographic gave slightly better results than greedy colex,which appeared to be the second best. The third and forth heuristics, accord-ing to the quality result, were greedy gray and greedy random, respectively.

Greedy coverings are not optimal, but they have asymptotically good per-formances and reach Rodl’s bound [5]. Over 40% of the upper bounds fromthe paper [4] come from greedy coverings, out of which over half are equivalentto the best known upper bounds on C(v, k, t). Nevertheless, it could happenthat all t-subsets covered by a chosen k-subset A are also covered by the re-

N. Nikolic et al. / Electronic Notes in Discrete Mathematics 39 (2012) 193–200 195

maining k-subsets in the covering. This means that the block A is redundantand can be removed from the covering, i.e. initial covering can be reduced.

If a covering has more than one redundant block, there can exist t-subsetscovered only with redundant blocks. If all redundant blocks are removed, theset-subsets would become uncovered. One solution is to remove all redundantblocks at once, and then to repair covering, i.e. to cover uncovered t-subsetswith new blocks. Another possibility is to remove redundant blocks in someorder and to update the set of redundant blocks after each removal.

Described reduction methods are very simple but limited to the coveringsthat contain redundant blocks. Now, let us generalize the idea of reduction tothe coverings that do not contain redundant blocks.

Let C be some arbitrary (v, k, t) covering. To each block A of the givencovering, we assign value F (A), which represents the number of t-subsetscovered with A and not covered with remaining blocks from C. In other words,by removing block A from the covering C, exactly F (A) t-subsets remainuncovered. For example, if C is the set of all k-subsets, then F (A) = 0 for allblocks A. If C is Steiner system, then F (A) =

(kt

)for all blocks A. In general,

F (A) ∈{0, 1, . . . ,

(kt

)}for each block A.

The block A is redundant if and only if F (A) = 0. It follows that theabove described simple reduction is equivalent to removing all blocks A withproperty F (A) = 0 and covering uncovered t-subsets, if any.

If we replace condition F (A) = 0 with F (A) ≤ L, where L ∈{0, 1, . . . ,

(kt

)},

we get a new reduction. We have named it Level Reduction (LR). In LR al-gorithm below, the inputs are one (v, k, t) covering and a value of L.

Algorithm 2. Level reduction

1. For each block A of the given covering determine the value F (A).

2. Remove from the covering each block A with the property F (A) ≤ L .

3. Determine set T , of all t-subsets which are not covered with remainingblocks.

4. If T �= ∅, use Greedy lexicographic algorithm to cover t-subsets from T andadd new blocks to the covering.

In step 4, Greedy lexicographic algorithm is used for covering the uncov-ered t-subsets, because it gives (on average) better results than other greedyalgorithms. Clearly, instead of Greedy lexicographic algorithm, some othermethods (often heuristics) can be used.

The main step in LR is the selection of blocks that will be removed. Instep 2, all blocks with the property F (A) ≤ L can be removed either all at

N. Nikolic et al. / Electronic Notes in Discrete Mathematics 39 (2012) 193–200196

once or in some order with updating the values of F (A) after each removal.Note that for all L such that L < min

A∈CF (A) there are no removed blocks from

the covering and that, as L increases, the number of removed blocks does notdecrease.

We can say that LR is successful if the number of removed blocks is greaterthan the number of blocks added to the covering. Under the assumption thatthe number of blocks needed for covering some t-subsets is proportional to thenumber of t-subsets, it is advisable to uncover the smallest possible number oft-subsets by removing a certain (small) number of blocks from the covering.Therefore we remove a certain number of blocks A for which the value F (A)is the lowest possible. In LR algorithm, this is accomplished by F (A) ≤ L,where L is a certain (small) value. An additional reason to assign a smallvalue to L is the computational complexity of LR algorithm, which is directlyproportional to the complexity of the method from step 4 (in our case Greedyalgorithm), as well as to the number of uncovered t-subsets. Since increasingof L increases the number of uncovered t-subsets, the complexity of algorithmalso increases.

3 Variable neighborhood descent

Variable neighborhood search (VNS) is a metaheuristic proposed in [7], basedon a simple and effective idea: a systematic change of the neighborhood withina local search algorithm. A variant of VNS, where changes of neighborhoodis performed in a deterministic way, is called Variable neighborhood descent(VND).

In the other local search heuristics for solving covering design problem,a search space is a set of B blocks (usually, B is slightly less than the bestknown upper bound). The neighborhood is defined by choosing one or moreelements of a block and replacing these by other elements that do not belongto this block. The size of such neighborhood is relatively small. In order toenlarge the neighborhood, we enable the replacement of one or more blocksfrom the covering by the ones that do not belong to the covering. In our case,the search space for covering design problem with parameters v, k, and t isthe space of all possible solutions, i.e. the set of all (v, k, t) coverings. The sizeof the neighborhood can now be relatively large and the local search shouldbe reduced. Therefore, instead of searching neighborhoods exactly, we use LRprocedure (section 2). Hence, in our VND implementation on covering designproblem, the local search procedure on neighborhood Nk is defined by LRprocedure for certain value of L. More precisely, the local search in N1(C) is

N. Nikolic et al. / Electronic Notes in Discrete Mathematics 39 (2012) 193–200 197

LR for Lmin = minA∈C

F (A), the local search in N2(C) is LR for Lmin + 1, etc.

In VND algorithm below, the input is (v, k, t) covering C.Algorithm 3. VND

1. Set k ← 0 and find Lmin = minA∈C

F (A).

2. Repeat the following sequence until k = kmax.

(a) Apply LR for L = Lmin + k on the covering C.(b) If the covering thus obtained C′ is better than C, set C ← C′, k ← 0, andupdate Lmin. Otherwise, set k ← k + 1.

Exploration of neighborhood Nk(C), k = 1, . . . , kmax is performed in thestep 2(a). If LR have found better (improved) covering C′ in neighborhoodNk(C) (covering with a smaller number of blocks), we set C ← C′ and restartthe process. If an improved solution could not be found, the process continuesfor k ← k + 1. Algorithm stops if no further improvement are possible, i.e. ifk = kmax.

3.1 Computational results

Our VND can be applied to any (v, k, t) covering. In this paper, we appliedVND on greedy lexicographic coverings for v > k > t ≥ 2, v ≤ 32, k ≤ 16,and t ≤ 8. Note that approximately 20% of these 1631 coverings is alreadyequal to the best known. Parameter kmax is set to 4, and LR is performedwith updating the values of F (A), A ∈ C after each removing of blocks.

Described VND has made improvements in 483 cases (29.61% of initialcoverings). In 13 cases, obtained coverings are with size equal to, while in 13cases too, they are better than the best known upper bounds. Parametersrelated to these coverings are given in Table 1.

Notice that the most of the coverings equal to the best known are obtainedfor the small values of parameter t (t = 2 or t = 3), while coverings whichare better than the best known are achieved for larger values of t (t = 7 ort = 8). This can be explained with the fact that the other methods were lesssuccessful in solving large size coverings problems such as those with t ≥ 7.

All the successful improvements were found in reasonable time. AverageCPU time for all the improvements was 38.66 minutes.

All obtained coverings, including computational time, are presented at theweb site http://www.math.fon.bg.ac.rs/vnd.

N. Nikolic et al. / Electronic Notes in Discrete Mathematics 39 (2012) 193–200198

Table 1Coverings with size equal to or better than the best known upper bounds

equal to the best known better than the best known

covering’s covering’sv k t v k t

size size

12 5 2 9 32 7 6 15359517 4 2 26 29 8 7 25805920 9 2 7 30 8 7 33460122 7 2 13 31 8 7 42405724 11 2 7 32 8 7 52463327 12 2 7 29 9 8 64277328 13 2 7 30 9 8 87497730 14 2 7 30 10 8 26134032 15 2 7 31 9 8 11675678 5 3 8 31 10 8 35069420 14 3 6 32 9 8 152316221 13 3 9 32 10 8 46579011 9 8 25 32 11 8 164593

4 Conclusions

In this paper, we have proposed variable neighborhood descent (VND) searchalgorithm for solving covering design problems. Our VND algorithm is basedon the level reduction (LR) procedure. LR improves a solution by destroyingand repairing the solution, i.e. by systematic removing and adding blocks tothe covering.

Computational experiments in solving covering design problem show thatthe VND outperforms simulated annealing and tabu search, two existingheuristics in the literature. First, VND can be applied to any (v, k, t) cov-ering. We have found the solution for all values v ≤ 32, k ≤ 16, and t ≤ 8.On the other hand, the author in [8] note that there is a point (v > 13) whenthe space and time requirements make the usage of simulated annealing in-tractable or even impossible, while, due to computer memory requirements,tabu search tests had to be limited to problems where v ≤ 20 [1]. Conse-quently, our number of instances was 1631, which is much larger than 168 inthe case of simulated annealing and 156 in the case of tabu search. Secondly,VND is less time consuming than simulated annealing and tabu search in solv-ing covering design problem. For instance, we spent around 51 CPU hours tosolve our longest problem ((32,9,8) covering with 1523162 blocks), while usingmultilevel tabu search [1], it was spent around 147 CPU hours to solve thelongest problem ((17, 8, 4) covering with 53 blocks).

N. Nikolic et al. / Electronic Notes in Discrete Mathematics 39 (2012) 193–200 199

Our approach is especially successful for values of v which are close to 32,t ≥ 7, and k ≤ t + 3. New upper bounds have been found for 13 problemswith such values of parameters. In addition, in the other 13 cases, obtainednew coverings are with size equal to the best known upper bounds. Besidesdeterministic choosing of blocks that can be removed, future work may includestochastic approaches in VNS heuristic for solving covering design problem.

References

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[4] Gordon, D. M., G. Kuperberg, and O. Patashnik, New constructions forcovering designs, Journal of Combinatorial Design 3 (1995), 269–284.

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[8] Nurmela, K. J., and P. R. J. Ostergard, Upper bounds for covering designs bysimulated annealing, Congressus Numerantium 96 (1993), 93–111.

[9] Nurmela, K. J., and P. R. J. Ostergard, Coverings of t-sets with (t + 2)-sets,Discrete Applied Mathematics 95 (1999), 425–437.

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