Gilmore-Lawler bound constraints

The Gilmore-Lawler bound (GLB) [37, 60] is one of the most famous bounding techniques for the quadratic assignment problem. Its mainstream awareness is not only reasoned in the simplicity of theGLB but also in its good performance. For an instance pA, B, Cqof (KBQAP) the Gilmore-Lawler bound relaxation is given by the following linear assignment

problem (LAP):

XPΠminⁿ xL`C, Xy, (6.1a)

where

L:“ pl_ijq with l_ij “ min

XPΠⁿ, Xij“1pAXBq_ij. (6.1b) The computation of the coefficients pl_ijqas permuted dot products reduces the overall complexity of theGLB toOpn³q, see [37]. Its low computational cost and the compara-tively good bounds are stimuli for us to incorporate the GLB into the considered SDP relaxations.

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By definition of L, we easily see that for each X PΠⁿ:

AXBX^T ědiag LX^T. (6.2)

This equation provides the opportunity to incorporate n additional linear constraints into the respective relaxation frameworks. The integration can be realized simply by adding the inequality condition

AY ědiag LX^T. (6.3)

Relaxations based on the vector lifting technique allow a deeper integration of the GLB conditions. By pB^T bAqvecpXq “vecpAXBq, we derive the identity

diagppB^T bAqvecpXqvecpXq^Tq “vecppAXBq ˝Xq, which, in turn, gives

diagppB^T bAqΥq ěvecpL˝Xq. (6.4) This vector inequality utilizes every single product of the termL˝X, whereas the identity diagpLX^Tq “ pL˝Xqe shows that (6.3) exploits solely the sums of the respective rows.

Apparently, (6.4) implies the validity of (6.3), thus leads to tighter relaxations. Since the additional computational costs are small compared to the overall efforts of VL₁, we suggest to use (6.4).

By incorporating the GLB based constraints into the respective SDP relaxations, we evidently obtain stronger bounds than with the plain Gilmore-Lawler bound procedure.

More specifically, by using Lemma 3.3 and Corollary 5.5, we show the following result.

Corollary 6.1. In respect of a given QAP instance pA, B, Cq, consider any of the previously discussed level-1relaxations, and additionally incorporate condition (6.3). The optimal objective value to this SDP relaxation is always greater than or equal to the optimal objective value to problem (6.1). Moreover, if the approximation tolerance ε is zero, the solution vector to the respective instance of problem (6.1) is unique, and the optimal objective values to both programs are identical, then their respective solution vectors correspond to the unique solution of the actual QAP.

Proof. The superiority of the respective SDP relaxation together with the incorporated GLB constraint over the plain GLB linearization is evident. For the remainder of the

proof, we assume identical objective values and uniqueness of the solution ˆX to problem (6.1). By uniqueness of ˆX, we have

@X P DⁿzΠⁿ: xL`C,Xy ą xL`C,Xy.ˆ Moreover, (6.3) implies

@X PDⁿ: xA, Yy ` xC,Xy ě xL`C,Xy.

Taken together, these inequalities necessitate unequal objective values whenever the feasible pointpX, . . .q to the given SDP relaxation does not correspond to an assignment.

By assumption, it is therefore X P Πⁿ. In this case, the validity of the conditional statement is an immediate consequence of Lemma 3.3 and Corollary 5.5.

For nonzero approximation thresholds ε ą 0, the respective level-1 relaxations do not inhere the characteristic stated in Corollary 5.5. We may restore this property by incorporating the following relaxation approach:

MPMinfⁿ, XPΠⁿ xM, Ey ` xC, Xy (6.5a)

s.t. M ěL˝X, (6.5b)

M ěU ˝X´U`AXB, (6.5c)

where Lis defined as in (6.1b) and

U :“ pu_ijq with u_ij “ max

XPΠ, Xij“0 pAXBq_ij. (6.5d) The above linearization for the QAP was introduced by Xia and Yuan [109, 111].¹ They extended the Gilmore-Lawler bounding procedure by a modified version of the Kaufman and Broeckx’s linearization [57], and proved that it inheres the desired property stated in Corollary 5.5, see [111, Theorem 3.7] and [109, Theorem 3].

The linearization approach by Xia and Yuan can be incorporated into the respective SDP relaxations by implementing

diagpAYq ě Me (6.6)

1The formula forpuijqgiven in [111, Eq. (29)] is incorrect. It was corrected by Xia in [109, Eq. (4)].

together with the constraints (6.5b) and (6.5c). The extended integration into relaxations based on vector lifting techniques is similarly straightforward. Additional to (6.4), one applies the vector inequality

diagppB^T bAqΥq ěvecpU ˝X´U `AXBq. (6.7)

Many different linearization techniques can be incorporated by a very similar procedure.

A typical approach to obtain tighter relaxations is the application of QAP reformulations.

The procedures proposed by Assad and Xu [3] as well as Carraresi and Mallucelli [20], among many other works such as [19, 31, 35, 55, 91], demonstrate the influence of these reformulations on the quality of the Gilmore-Lawler bound. On the basis of numerical tests, we observed that QAP reformulations which are suitable for the discussed SDP relaxations can be less practical for GLB based constraints. We deal with this circumstance by implementing the corresponding conditions in consideration of a different QAP reformulation.

Assume that the matrix ´L is constructed as in (6.1b), but for a specific reformulation instance pA,´ B,´ Cq´ whose parameter matrices satisfy

diagpA´q ” 0 and B´ “B`v´_be^T.

The Gilmore-Lawler bound linearizes the diagonal elements of these matrices in the same way as the considered relaxation frameworks. Moreover, adding a sum-matrix with constant columns toB has no effect on the bounding quality, see [34]. Hence, the presuppositions on the reformulated data matrices ´A and ´B serve just the purpose of simplicity and do not restrict the utility of the GLB conditions. The reformulated version of condition (6.3) is then

AY´ `AX´ v´_be^T ědiagLX´ ^T. (6.8) From the proof of Lemma 4.3, the adaptation of (6.4) is even more apparent:

diagppB´^T bA´qΥq ěvecpL´˝Xq. (6.9) Regarding the VL framework, there is no actual reason to choose different QAP reformu-lations for the objective function and the GLB inequalities. The consideration of this

case is nevertheless serviceable, because other constraints do benefit from different QAP reformulations.

Very similar inequalities can be derived for the off-diagonal elements of the matrix productAXBX^T. These may, for instance, be constructed by utilizing a slightly modified version of parameter L:

L:“ pl_ijq with l_ij “ min

XPΠ, Xij“0pAXBqij.

By definition, we then have AXBX^T ěoff LX^T, which may be used for additional cuts in the corresponding frameworks. Unfortunately, numerical tests have shown that the improvement of the resulting bounds is negligible, whereas the impact on the computational effort is strongly apparent. The picture for the corresponding extension to the vector lifting based frameworks is even worse. The introduction of n⁴´n² additional inequality constraints penalizes the computation times significantly and the bounding improvement seems to dissolve within the accuracy of the used SDP solver. With regard to the efficiency of the relaxation program, we therefore limit our concern on the presented diagonal element inequalities.

Another possibility to acquire more constraints out of the Gilmore-Lawler bound is to split the corresponding inequalities in the manner of the discussed matrix-splitting schemes. By [62, Theorem 3.2], however, it is clear that the deduction of additional GLB inequalities via matrix splitting is generally not recommendable.