Frequency Fitness Assignment: Optimization Without Bias for Good

Solution Outperforms Randomized Local Search on the Quadratic

Assignment Problem

Jiayang Chen

1 a

, Zhize Wu

1 b

, Sarah L. Thomson

2 c

and Thomas Weise

1 d

Institute of Applied Optimization, School of Artiﬁcial Intelligence and Big Data, Hefei University, Jinxiu Dadao 99, Hefei,

230601, Anhui, China

School of Computing, Engineering & the Built Environment, Edinburgh Napier University, 10 Colinton Road, Edinburgh,

EH10 5DT, U.K.

Keywords:

Quadratic Assignment Problem, Frequency Fitness Assignment, Randomized Local Search.

Abstract:

The Quadratic Assignment Problem (QAP) is one of the classical N P -hard tasks from operations research

with a history of more than 65 years. It is often approached with heuristic algorithms and over the years,

a multitude of such methods has been applied. All of them have in common that they tend to prefer better

solutions over worse ones. We approach the QAP with Frequency Fitness Assignment (FFA), an algorithm

module that can be plugged into arbitrary iterative heuristics and that removes this bias. One would expect

that a heuristic that does not care whether a new solution is better or worse compared to the current one should

not perform very well. We plug FFA into a simple randomized local search (RLS) and yield the FRLS, which

surprisingly outperforms RLS on the vast majority of the instances of the well-known QAPLIB benchmark set.

1 INTRODUCTION

The Quadratic Assignment Problem (QAP) is a

challenging and very important combinatorial opti-

mization problem (Koopmans and Beckmann, 1957;

Burkard et al., 1998; Loiola et al., 2007). Here, the

goal is to assign a set of n facilities to a set of n lo-

cations. Such an assignment can be represented as a

permutation s of the ﬁrst n natural numbers, where

s[i] speciﬁes the location where facility i should be

placed. For each QAP, a distance matrix A is given,

where A

speciﬁes the distance from location p to

location q, as well as a ﬂow matrix B, where B

i j

is the

amount of material ﬂowing from facility i to facility j.

The objective function f subject to minimization then

rates a permutation s as follows:

f (s) =

∑

i=1

∑

j=1

s[i]s[j]

i j

(1)

The QAP has a wide range of applications including,

e.g., building layout (Elshafei, 1977; C¸ ubukc¸uo

glu

https://orcid.org/0009-0004-2320-2980

https://orcid.org/0000-0001-7416-5711

https://orcid.org/0000-0001-6971-7817

https://orcid.org/0000-0002-9687-8509

et al., 2021; Krarup and Pruzan, 1978), key-

board layout (Burkard and Offermann, 1977), cir-

cuit design (Eschermann and Wunderlich, 1990),

wiring (Steinberg, 1961), and scheduling (Soroush,

2011). While there has been notable success in ap-

plying exact methods to the QAP (Drezner et al.,

2005), QAPs are N P -hard (Sahni and Gonzalez,

1976; Dr

eo et al., 2006) and thus are often solved

with heuristic algorithms such as simulated anneal-

ing (Thonemann and B

olte, 1994; Wilhelm and Ward,

1987), tabu search (Skorin-Kapov, 1990; Taillard,

1991; Misevi

cius, 2005; Misevi

cius, 2008), iter-

ated local search (St

utzle, 2006), evolutionary meth-

ods (Horng et al., 2000; Taillard and Gambardella,

1997), memetic algorithms (Fleurent and Ferland,

1993; Merz and Freisleben, 1999), estimation of dis-

tribution algorithms (Zhang et al., 2006), ant colony

optimization (Gambardella et al., 1999; Talbi et al.,

2001; Taillard and Gambardella, 1997), or even parti-

cle swarm optimization (Haﬁz and Abdennour, 2016).

All such heuristic approaches that have been ap-

plied to the QAP have one design principle in com-

mon: Their (iterative) search procedure is biased to-

wards good solutions. Regardless of whether they

employ diversity strategies or methods to increase ex-

ploration, on average over time, they do prefer (to

Chen, J., Wu, Z., Thomson, S. and Weise, T.

Frequency Fitness Assignment: Optimization Without Bias for Good Solution Outperforms Randomized Local Search on the Quadratic Assignment Problem.

DOI: 10.5220/0012888600003837

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 16th International Joint Conference on Computational Intelligence (IJCCI 2024), pages 27-37

ISBN: 978-989-758-721-4; ISSN: 2184-3236

exploit) better solutions (in terms of their objective

value) over worse ones. Indeed, this is maybe the

most fundamental concept of metaheuristic optimiza-

tion.

In (Weise et al., 2014), a mechanism called Fre-

quency Fitness Assignment (FFA) was proposed,

which was later shown to render optimization pro-

cesses invariant under all injective transformations of

the objective function value (Weise et al., 2021b) and,

as a result, removing the bias towards better solu-

tions (Weise et al., 2023). By replacing the objec-

tive value f (s) of a solution s with its encounter fre-

quency H[ f (s)] in all selection decisions of a heuris-

tic, FFA offers this new concept of optimization,

which breaks with the existing ideas upon which

all metaheuristics are built. The only algorithms

that have similar properties are random walks, ran-

dom sampling, and exhaustive enumeration – none

of which are ranked as good approaches to the QAP.

FFA has been shown to improve the performance of a

randomized local search (RLS) on the Max-Sat prob-

lem (Weise et al., 2021b; Weise et al., 2023), the

Job Shop Scheduling Problem (JSSP) (Weise et al.,

2021a; de Bruin et al., 2023), and on Traveling Sales-

person Problem (TSP) instances (Liang et al., 2022;

Liang et al., 2024).

However, whether it can improve algorithm per-

formance on a wide set of QAP instances has not yet

been studied. In this work, we do not aim to out-

perform any of the related heuristics listed above. It

instead is our goal to establish that FFA is indeed a

suitable technique for the QAP. Our ﬁrst contribu-

tion is to conduct the ﬁrst large experiment of FFA

on the QAP involving all instances from the QAP

benchmark set QAPLIB by (Burkard et al., 1997).

We publish all of our code, results, as well as the

scripts used for generating the tables and ﬁgures in an

immutable online archive at https://doi.org/10.5281/

zenodo.13324662. As a second contribution, we

show that, if plugged into a simple RLS, FFA yields a

signiﬁcant improvement in the quality of the discov-

ered results. We show that, despite using a compu-

tational budget 100 times smaller than in prior works

on FFA, this tangible improvement can be observed.

Finally, our third contribution is to provide lower

bounds m for the numbers M of possible different ob-

jective values for all instances of the QAPLIB. While

lower bounds lb for the objective function f ex-

ist (Peng et al., 2010; de Klerk and Sotirov, 2010;

Drezner et al., 2005), we are the ﬁrst to investigate m

on the QAPLIB instances. This lower bound m can

give us an impression about other aspects that may

be relevant for optimization and may be related to the

amount of neutrality present.

The rest of our work is structured as follows. In

Section 2, we discuss related works both on FFA and

the QAP before deﬁning the algorithms used in our

study in Section 3. In Section 4, we present the re-

sults of our experiment before concluding the paper

in Section 5 with a summary and outlook on future

work.

2 RELATED WORK

2.1 Related Works on the QAP

A wide variety of heuristics has been applied to

the QAP, which differ in their algorithmic design

philosophies, search strategies, operators, and param-

eters (Dr

eo et al., 2006). In this work here, we inves-

tigate whether the new paradigm FFA is applicable to

the QAP. Beating the state of the art is not our goal.

Nevertheless, it is important to at least provide a brief

overview of some of the diverse historical heuristic

solution ideas for the QAP.

(Wilhelm and Ward, 1987) studied the application

of simulated annealing to the QAP. They showed that

the simulated annealing algorithm produces good re-

sults but is sensitive to the setting of parameters and

tested the effect of several parameters on the perfor-

mance of the algorithm and CPU usage time.

(Taillard, 1991) developed a robust tabu search al-

gorithm for the QAP, which today still is considered

as competitive. It explores the neighborhood of the

current solution by pairwise exchanges. The aspira-

tion criterion allows forbidden moves if they produce

a solution better than the best so far one. A subset of

the QAPLIB instances with scales from 5 to 100 were

used to investigate the algorithm performance.

Soon thereafter, (Fleurent and Ferland, 1993) pre-

sented a hybrid genetic algorithm, which combines

the population-based evolutionary heuristic with lo-

cal search. In traditional genetic algorithms, the qual-

ity of individuals can only be improved by crossover,

mutation, and other operators. However, hybrid ge-

netic algorithms can improve the solution also by lo-

cal search or even tabu search. In experiments on

the sko-class of instances (Skorin-Kapov, 1990) with

scales up to 100, the hybrid algorithm outperformed

its component algorithms. (Merz and Freisleben,

1999) introduced a memetic algorithm (MA), which,

basically, is another hybrid evolutionary algorithm.

The experiment was based on another subset of the

QAPLIB instances and the MA outperformed several

other heuristics on all instances of practical signiﬁ-

cance (i.e., except for the randomly generated ones).

In the same year, (Gambardella et al., 1999) pre-

ECTA 2024 - 16th International Conference on Evolutionary Computation Theory and Applications

sented an ant colony system hybridized with a local

search. A comprehensive comparison experiment on

several QAPLIB instances with scales n between 19

and 90 showed that this algorithm performs especially

well on irregular problems (that is, instances whose

distance and/or ﬂow matrix contain disparate values)

and representative real-world instances.

(Horng et al., 2000) applied an evolutionary strat-

egy (ES) to the QAP. In order to prevent prema-

ture convergence to local optima, this method adds

the concept of clustering and family competition to

the population handling. The resulting higher diver-

sity leads to good performance on instances with n ∈

19. . . 90. In this work we take the alternative ap-

proach of FFA, which – different from the clustering-

based idea of that work – does not require any pop-

ulation. Also, diversity is often considered from the

search space perspective, whereas FFA tries to create

diversity in the objective space in the hope that this

induces diversity also in the genotypic representation

of the solutions.

As maybe the last of these historical research di-

rections to approach the QAP, (Haﬁz and Abdennour,

2016) proposed a discretization framework for parti-

cle swarm optimization. This continuous optimiza-

tion technique, too, can produce good results on the

QAP.

Some of the above algorithms, like tabu search or

the ES, introduce methods to increase the diversity

of the solutions under investigation. Thus, they have

components that try to prevent the algorithms from

converging to local optima. However, all of them pre-

fer better solutions over worse ones. In the following

section, we therefore discuss why FFA is a uniquely

different approach to diversity and optimization and

why investigating its performance on the QAP is nec-

essary.

It should be noted that in (Thomson et al., 2024),

we applied ﬁtness landscape analysis to FFA on the

taie27 set of 20 QAP instances of the same scale n =

27, which are not part of QAPLIB. In that paper,

our goal was to explain why and how FFA-based

search works. We presented visualizations of met-

rics for algorithm trajectories which substantiate the

good exploration ability of FFA-based algorithms.

The question of whether FFA is a suitable technique

for more general QAPs, however, was explicitly left

unanswered. We answer it now, by using many more

and entirely different instances. We also complement

the analysis with several new perspectives, such as an

analysis of the last improvement step or which kind of

instances FRLS can solve to optimality within a rea-

sonable computational budget.

2.2 Related Works Against

Convergence to Local Optima

The problem of premature convergence to local op-

tima is well-known in many ﬁelds of soft computing.

It occurs, for example, in k-means clustering (Shalev-

Shwartz and Ben-David, 2014; Arthur and Vassilvit-

skii, 2007) and the training of ANNs (Shalev-Shwartz

and Ben-David, 2014; Treadgold and Gedeon, 1998).

In optimization, it has been researched for a long

time (Weise et al., 2012; Weise et al., 2009).

Tabu Search (TS) (Glover and Taillard, 1993), one

of the most prominent methods to prevent premature

convergence, improves upon local search by declaring

solutions (or solution traits) that have been visited as

tabu, which prevents the algorithm from getting stuck.

It has found application in the QAP in several differ-

ent variants (Misevi

cius, 2008; Merz and Freisleben,

1999; Skorin-Kapov, 1990).

In the ﬁeld of Evolutionary Algorithms, the old

ideas of sharing, niching, and clearing (Mahfoud,

1997; Goldberg and Richardson, 1987; Deb and

Goldberg, 1989; P

etrowski, 1996) as well as cluster-

ing (Weise et al., 2011) combine density information

with the objective values into so-called ﬁtness values

to increase the diversity in the populations of can-

didate solutions. These methods only consider the

present populations and do not consider the history of

the search, whereas FFA incorporates and aggregates

knowledge over the whole course of optimization.

Methods that try to balance between solution qual-

ity and (population) diversity are today grouped under

the term Quality-Diversity (QD) algorithms (Cully

and Demiris, 2018; Gravina et al., 2019). QD algo-

rithms are mainly applied to games, maze solving,

and shape or robotics behavior evolution, but rarely

in the context of discrete or hard optimization tasks

from operations research.

Novelty Search (NS) (Lehman and Stanley, 2008;

Lehman and Stanley, 2011a) is an early QD algo-

rithm. NS is driven by a dynamic novelty met-

ric ρ measuring the mean behavior difference to the

k-nearest neighbors in the set of past solution “behav-

iors.” NS with Local Competition (NSLC) (Lehman

and Stanley, 2011b) combines the search for diverse

solutions with a local competition objective reward-

ing solutions that can outperform those most similar

to them.

In the QD method Surprise Search (SS) (Grav-

ina et al., 2016), a solution is rated by the differ-

ence between its observed behavior from the expected

At least the comprehensive paper QD paper list

by (Mouret and Cully, 2024) does not list a single work re-

ferring to the QAP or the TSP in its abstract.

Frequency Fitness Assignment: Optimization Without Bias for Good Solution Outperforms Randomized Local Search on the Quadratic

Assignment Problem

Algorithm 1: RLS( f : S 7→ N).

sample s

from S u.a.r.; z

← f (s

);

for 10

− 1 times do ▷ our termination criterion

← swap 2 values in s

u.a.r.; z

← f (s

);

if z

≤ z

then s

← s

; z

← z

;

return s

, z

Algorithm 2: FRLS( f : S 7→ N).

H ← (0, 0, · ·· , 0); ▷ H-table initially all 0s

sample s

from S u.a.r.; z

← f (s

);

← s

; z

← z

; ▷ best may otherwise get lost

for 10

− 1 times do ▷ our termination criterion

← swap 2 values in s

u.a.r.; z

← f (s

);

if z

< z

then s

← s

; z

← z

;

H[z

] ← H[z

] + 1; H[z

] ← H[z

] + 1;

if H[z

] ≤ H[z

] then s

← s

; z

← z

;

return s

, z

▷ return preserved best

behavior. A history of discovered solution behav-

iors is maintained and used to predict the behavior

of new solutions. (Gravina et al., 2019) combine SS

and NSLC.

Finally, the MAP-Elites algorithm by (Mouret and

Clune, 2015) combines a performance objective f and

a user-deﬁned space of features that describe candi-

date solutions. MAP-Elites searches for the highest-

performing solution in each cell of the discretized fea-

ture space.

Sharing techniques require a population and all

the other methods discussed above were designed as

optimization algorithms themselves. FFA, however,

can be plugged into a wide range of optimization al-

gorithms as long as their objective functions are dis-

crete. Instead of using the objective values z com-

puted by the objective function f (s) = z when com-

paring solutions s, FFA prescribes using their ob-

served encounter frequencies H[z]. This makes FFA

invariant under all injective transformations of the ob-

jective function value, a property further distinguish-

ing it from all related techniques (Weise et al., 2021b;

Weise et al., 2023).

3 OUR APPROACH

The pure randomized local search algorithm RLS is

illustrated in Algorithm 1. This algorithm starts by

sampling a solution s

from the set S of all permuta-

tions of the ﬁrst n natural numbers uniformly at ran-

dom (u.a.r.). It evaluates the objective function f and

obtains the quality z

of s

. In a loop, it then cre-

ates a copy s

of s

in which two values are swapped,

u.a.r.. The quality z

= f (s

) of s

is computed. If s

is better than or equally good as s

, it will replace s

The loop is repeated until the termination criterion is

met, which, in our case, is the consumption of a total

of 10

objective function evaluations (FEs, including

the evaluation of the random initial solution).

We plug FFA into this algorithm and obtain the

FRLS in Algorithm 2. While RLS accepts the new so-

lution s

if its objective value z

is not worse than the

objective value z

of the current solution s

, FRLS ac-

cepts s

if the encounter frequency H[z

] of z

in the

selection decision is not higher than the encounter fre-

quency H[z

] of z

. For this purpose, it begins by ﬁll-

ing the frequency table H with zeros at the beginning

of the algorithm. In each iteration, H[z

] and H[z

]

are both incremented by one and then replace z

and

in the selection decision. This means that FRLS is

not biased towards better solutions and will replace s

with a worse s

if its objective value z

is encountered

less than or equally often as z

. Therefore, instead of

returning s

and z

at the end, FRLS must remember

the best-encountered solution and objective value in

additional variables s

and z

, respectively.

4 EXPERIMENTS AND RESULTS

The QAPLIB by (Burkard et al., 1997) is a com-

monly used and continuously updated database of

QAP benchmark instances and their solutions. It con-

tains both real-life instances and randomly generated

instances. In our experiments, we use all 134 in-

stances of the latest version of the QAPLIB at the time

of this writing, which is maintained by (Hahn and An-

jos, 2018) and was last updated in 2018. From this

resource, we also take the lower bounds lb of the ob-

jective functions f . For each instance, we perform

3 independent runs which, together with the many in-

stances, are already sufﬁcient to observe very clear

differences in performance. The instances have the

following properties:

• burn* (Burkard and Offermann, 1977), 8 in-

stances, n = 26, all optima known

• chrn* (Christoﬁdes and Benavent, 1989), 14 in-

stances, n ∈ {12, 15, 18, 20, 22, 25}, all optima

known

• els19 (Elshafei, 1977), 1 instance, n = 19, opti-

mum known

• escn* (Eschermann and Wunderlich, 1990), 19 in-

stances, n ∈ {16, 32, 64, 128}, all optima known

• hadn (Hadley et al., 1992), 5 instances, n ∈

{12, 14, 16, 18, 20}, all optima known

ECTA 2024 - 16th International Conference on Evolutionary Computation Theory and Applications

• kran* (Krarup and Pruzan, 1978), 3 instances, n ∈

{30, 32}, all optima known

• lipan* (Li and Pardalos, 1992), 16 instances, n ∈

{20, 30, 40, 50, 60, 70, 80, 90}, all optima known

• nugn* (Nugent et al., 1968), 15 instances, n ∈

{12, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 27, 28, 30},

all optima known

• roun (Roucairol, 1987), 3 instances, n ∈

{12, 15, 20}, all optima known

• scrn (Scriabin and Vergin, 1975), 3 instances, n ∈

{12, 15, 20}, all optima known

• skon* (Skorin-Kapov, 1990), 13 instances,

n{42, 49, 56, 64, 72, 81, 90, 100}, all optima

unknown

• ste36* (Steinberg, 1961), 3 instances, n = 36, all

optima known

• tain* (Taillard, 1991; Tail-

lard, 1995), 26 instances, n ∈

{12, 15, 17, 20, 25, 30, 35, 40, 50, 60, 64, 80, 100,

150, 256}, optima of 16 instances unknown

• thon (Thonemann and B

olte, 1994), 3 instances,

n ∈ {30, 40, 150}, only optimum of tho30 known

• wiln (Wilhelm and Ward, 1987), 2 instances, n ∈

{50, 100}, no optimum known

We implement our algorithms using the

moptipy (Weise and Wu, 2023) framework and

run the experiments on a Windows 10 machine using

Python 3.10 and the numba JIT.

Table 1 shows the arithmetic mean of the best ob-

jective values achieved by RLS and FRLS over the

3 runs per QAPLIB instance. The last row, # best,

tells us that FRLS achieved the best average result

113 times, while RLS did this only 35 times. The

average result of FRLS hits the lower bound lb, i.e., is

optimal 73 times. Its best-of-3-runs results (not tabu-

lated) reach it 78 times. RLS achieves this feat only

14 respectively 20 times. In other words, not only

does FRLS outperform RLS on 74% of the QAPLIB

instances in terms of its average result, it also solves

58% of them to optimality.

In (Liang et al., 2022), it was found that the per-

formance of FRLS may strongly depend on the num-

ber M of different objective values that an optimiza-

tion problem exhibits. The good performance of

FRLS on the escn problems may be caused by the

many zeros in their ﬂow matrix resulting in few dif-

ferent possible object values.

Exactly determining M for the QAPLIB instances

would be another N P -hard problem in itself. There-

fore, we do not have the exact values of this mea-

sure available. However, we can approximate it us-

ing the estimate, or better, a lower bound m: Each

run of FRLS maintains its own frequency table H and

we collect these tables in our log ﬁles. We also log

all improving moves that any algorithm makes, so we

additionally have at least the strictly monotonous se-

quence of visited f -values for RLS. Finally, the web-

site of the QAPLIB offers the best-known or even op-

timal solutions for all instances, which are better than

our results on 42% of the instances. Therefore, by

setting m to be the size of the joint set of all of these

values of all runs, we can get a lower bound for M.

When m is much smaller than our total computational

budget over all runs of FRLS (for which we collect the

complete H-tables), i.e., where m ≪ 3 ∗ 10

, it should

be a reasonable estimate of M. Otherwise, at least it

informs us whether M is probably small or large. We

therefore also include it in the tables.

Revisiting the results of both algorithms in Table 1

and considering them from the perspective of m con-

ﬁrms the ﬁndings by (Liang et al., 2022). If m of an

instance is small, FRLS tends to solve the instance to

optimality (and hit the lower bound lb), even if the

scale n is not small (e.g., at lipa50a). Vice versa,

the tables also show that FRLS is outperformed by

RLS even on small problems if their m is large, see,

e.g., tai15b. The comparatively good performance of

FRLS on the taina instances versus the tainb instances

is also interesting because the former are usually con-

sidered as harder (Ochoa and Herrmann, 2018).

A remarkable piece of evidence of the exploration

power of FRLS, which discovers most of the encoun-

tered objective values, are the high m-values for many

instances. FRLS contributed 215 196 721 values to

the estimation m = 215 196 971 for tai30b. Since we

conducted only 3 runs at 10

FEs each, this means

that 71% of all the solutions that these FRLS runs

have sampled had unique objective values. If all solu-

tions on a problem instance would have unique objec-

tive values, then FRLS would always accept the new

solution s

and hence become a random walk. But

this does not seem to be the case: On tai20b, FRLS

encountered 173 058 828 different objective values –

and outperformed RLS by a margin of over 10%.

The strong ability to explore and keep improving

of FRLS is further illustrated in Figure 1. Here, we

plot the average life index of the objective function

evaluation (FE) where the last improving move was

made over the problem scale n. In other words: Each

run of an algorithm on a given problem instance even-

tually stops improving its best-so-far solution. It may

or may not have discovered the optimal solution by

then, but after that, no more improvement is made

(within the provided computational budget, at least).

The index of the algorithm step when, for the last time

in a run, a new (better) best-so-far solution is discov-

Frequency Fitness Assignment: Optimization Without Bias for Good Solution Outperforms Randomized Local Search on the Quadratic

Assignment Problem

Table 1: The average result over 3 runs of the RLS and the FRLS on the 134 QAPLIB instances, in comparison with the lower

bound lb of f and the number m of observed and known objective values as a lower bound for the number of possible different

objective values. The best result is marked in boldface.

instance lb m RLS FRLS instance lb m RLS FRLS

bur26a 5 426 670 1 480 802 5 442 929 5 434 256 lipa60a 107 218 4 915 108 368 107 461

bur26b 3 817 852 1 021 194 3 838 077 3 818 291 lipa60b 2 520 135 456 660 3 016 957 3 005 080

bur26c 5 426 795 1 384 071 5 440 307 5 428 857 lipa70a 169 755 6 880 171 358 170 429

bur26d 3 821 225 945 677 3 833 028 3 821 540 lipa70b 4 603 200 651 696 5 569 556 5 642 958

bur26e 5 386 879 1 579 830 5 405 301 5 389 526 lipa80a 253 195 7 772 255 351 254 606

bur26f 3 782 044 1 111 729 3 793 182 3 782 454 lipa80b 7 763 962 940 457 9 423 095 9 650 856

bur26g 10 117 172 2 672 208 10 145 555 10 127 889

lipa90a 360 630 9 976 363 412 362 571

bur26h 7 098 658 1 876 059 7 141 228 7 101 399 lipa90b 12 490 441 1 277 577 15 173 637 15 617 417

chr12a 9 552 33 801 14 899 9 552 nug12 578 232 606 578

chr12b 9 742 33 627 14 589 9 742 nug14 1 014 366 1 037 1 014

chr12c 11 156 33 377 14 939 11 156 nug15 1 150 432 1 182 1 150

chr15a 9 896 52 353 16 015 9 896 nug16a 1 610 532 1 673 1 610

chr15b 7 990 53 657 11 952 7 990 nug16b 1 240 483 1 297 1 240

chr15c 9 504 50 900 14 913 9 504 nug17 1 732 608 1 813 1 732

chr18a 11 098 66 156 18 142 11 098 nug18 1 930 650 1 978 1 930

chr18b 1 534 3 083 1 648 1 534 nug20 2 570 826 2 681 2 570

chr20a 2 192 8 429 3 325 2 192 nug21 2 438 970 2 510 2 438

chr20b 2 298 8 307 3 556 2 335 nug22 3 596 1 531 3 759 3 596

chr20c 14 142 91 709 31 659 14 142 nug24 3 488 1 254 3 608 3 488

chr22a 6 156 16 932 6 824 6 156 nug25 3 744 1 277 3 950 3 744

chr22b 6 194 16 846 6 861 6 215 nug27 5 234 1 860 5 470 5 234

chr25a 3 796 21 052 6 509 3 796 nug28 5 166 1 728 5 417 5 166

els19 17 212 548 30 545 903 25 266 593 18 821 866 nug30 6 124 2 018 6 439 6 124

esc16a 68 34 68 68 rou12 235 528 58 475 248 938 235 528

esc16b 292 22 292 292 rou15 354 210 97 118 378 899 354 210

esc16c 160 73 160 160 rou20 725 522 175 690 759 802 725 522

esc16d 16 36 16 16 scr12 31 410 28 833 33 079 31 410

esc16e 28 29 28 28 scr15 51 140 53 073 56 646 51 140

esc16f 0 1 0 0 scr20 110 030 120 453 126 571 110 030

esc16g 26 35 26 26 sko42 15 332 4 201 16 351 15 812

esc16h 996 272 996 996 sko49 22 650 5 802 23 909 23 403

esc16i 14 37 14 14 sko56 33 385 8 202 35 337 34 467

esc16j 8 20 8 8 sko64 47 017 10 379 49 509 48 524

esc32a 130 253 151 130 sko72 64 455 13 439 67 707 66 378

esc32b 168 124 183 168 sko81 88 359 17 353 92 575 91 107

esc32c 642 194 642 642 sko90 112 423 20 859 117 639 115 853

esc32d 200 117 205 200 sko100a 143 846 25 618 153 965 152 557

esc32e 2 50 2 2 sko100b 145 522 26 389 156 111 154 557

esc32g 6 37 6 6 sko100c 139 881 25 903 151 014 148 430

esc32h 438 175 467 438 sko100d 141 289 25 616 151 863 150 203

esc64a 116 124 116 116 sko100e 140 893 26 623 151 569 149 795

esc128 64 192 65 64 sko100f 140 691 25 266 151 695 149 570

had12 1 652 228 1 665 1 652 ste36a 9 526 14 213 10 213 9 526

had14 2 724 394 2 753 2 724 ste36b 15 852 96 128 17 766 15 852

had16 3 720 478 3 815 3 720 ste36c 8 239 110 7 804 921 8 970 338 10 698 219

had18 5 358 622 5 413 5 358 tai12a 224 416 64 051 240 311 224 416

had20 6 922 856 6 969 6 922 tai12b 39 464 925 60 287 923 45 156 248 39 492 474

kra30a 88 900 8 200 94 843 88 900 tai15a 388 214 94 668 400 575 388 214

kra30b 91 420 8 581 95 967 91 420 tai15b 51 765 268 35 623 423 51 943 701 52 001 756

kra32 88 700 8 828 93 730 88 700 tai17a 491 812 122 615 523 634 491 812

lipa20a 3 683 435 3 795 3 683 tai20a 703 482 178 309 751 881 704 195

lipa20b 27 076 10 778 31 241 27 076 tai20b 122 455 319 173 058 953 143 287 002 129 766 839

lipa30a 13 178 1 088 13 442 13 178 tai25a 1 167 256 252 308 1 231 845 1 174 603

lipa30b 151 426 57 162 178 015 151 426 tai25b 344 355 646 202 832 378 384 043 042 395 447 601

lipa40a 31 538 1 976 32 042 31 538 tai30a 1 706 855 319 665 1 918 997 1 853 616

lipa40b 476 581 184 489 563 999 476 581 tai30b 637 117 113 215 196 971 710 795 743 721 008 038

lipa50a 62 093 3 296 62 902 62 093 tai35a 2 216 627 397 009 2 559 439 2 509 553

lipa50b 1 210 244 348 151 1 438 601 1 308 415 tai35b 269 532 400 125 920 739 317 695 376 334 904 454

tai40a 2 843 274 468 546 3 307 957 3 281 287 tai100b 1 151 591 000 152 455 325 1 240 769 163 1 543 004 655

tai40b 608 808 400 165 765 853 693 265 760 806 047 127 tai150b 441 786 736 43 282 106 509 821 471 612 165 283

tai50a 4 390 920 612 489 5 152 389 5 245 447 tai256c 44 095 032 5 758 252 44 940 419 48 194 560

tai50b 431 090 700 129 307 016 504 050 091 590 061 093 tho30 149 936 63 292 157 237 149 936

tai60a 6 325 978 752 294 7 553 963 7 717 479 tho40 226 490 102 223 251 221 240 708

tai60b 592 371 800 137 418 134 643 368 525 791 205 717 tho150 7 620 628 918 879 8 319 988 8 855 816

tai64c 1 855 928 1 691 310 1 860 059 1 861 098 wil50 48 121 6 760 49 465 48 835

tai80a 11 657 010 1 019 112 14 030 598 14 523 961 wil100 268 955 26 012 275 203 273 622

tai80b 786 298 800 120 691 529 873 374 711 1 075 394 622 # best 35 113

tai100a 17 853 840 1 268 760 21 828 809 22 720 707

ECTA 2024 - 16th International Conference on Evolutionary Computation Theory and Applications

50 100 150 200 250

mean life (log-scaled)

RLS

FRLS

Figure 1: The average life index of the objective function

evaluation (FE) where the last improving move was made,

plotted in log-scale over the problem scale n.

ered, averaged over the runs, is presented as life.

We ﬁnd that the time during which the RLS can

keep improving increases slightly with n. However,

it remains roughly in the range of at most a few

100 000 FEs. Over almost all problem scales, FRLS

can keep improving for, basically, the complete avail-

able budget of 10

FEs. This strongly indicates that

if we had allocated not 10

FEs but 10

, as it was

done in (Weise et al., 2021b; Weise et al., 2023; Liang

et al., 2022; Liang et al., 2024), we very likely would

have seen several more instances solved to optimality.

The single downward rectangular slot in both curves

in the diagram is caused by esc128, at which both al-

gorithms converge earlier (FRLS to the optimum, af-

ter which no further improvement is possible). The

next larger instances are at n = 150 where the trend

resumes.

We now plot the progress of the two algorithms

in terms of the best-so-far objective value divided by

the lower bound lb of the objective function f over

time measured in FEs and averaged over all the runs

and instances in each of the 15 groups of QAPLIB.

Instance esc16f with lb = 0 is omitted. From Figure 2,

it is visible that FRLS ﬁnds better average end result

qualities on all groups except sten and tain. Even on

these groups, it would have probably overtaken RLS if

we had given more runtime. In most of the diagrams,

RLS is initially faster and then stagnates, while FRLS

steadily and continuously keeps improving.

5 CONCLUSIONS

In the past, Frequency Fitness Assignment (FFA) has

led to surprisingly good results on several N P -hard

optimization problems, including Max-Sat (Weise

et al., 2021b; Weise et al., 2023), the JSSP (Weise

et al., 2021a; de Bruin et al., 2023), and the

TSP (Liang et al., 2022; Liang et al., 2024). In this

work, we conclusively showed that FFA can achieve

this on one more of these classical hard tasks from

operations research: the Quadratic Assignment Prob-

lem (QAP).

We ﬁnd that the FFA-based randomized local

search FRLS does not just ﬁnd better solutions than

the objective-guided RLS algorithm on the vast ma-

jority of the QAPLIB instances, it also keeps improv-

ing its current best solution for the complete computa-

tional budget of 10

FEs that we assigned to the runs.

With this budget, it can discover the optimal solutions

of over 58% of the QAPLIB instances. Had we as-

signed a larger budget – (Liang et al., 2022; Liang

et al., 2024; Weise et al., 2021b; Weise et al., 2023)

use 10

FEs – we would likely have seen even more

instances solved.

We furthermore conﬁrm the remarkable ability

of FFA to discover very diverse solutions (at least

from the perspective of the objective function). It is

known that on the QAP, many solutions tend to have

the same objective values (Tayarani-N. and Pr

ugel-

Bennett, 2015). Yet, on some of the instances, more

than half of the objective values discovered by FRLS

were unique.

The QAP is strongly related to the TSP (Dr

et al., 2006). (Liang et al., 2022; Liang et al., 2024)

found that the FFA performance strongly depends on

the number M of possible different objective values.

We are the ﬁrst to report a lower bound and estimate m

of M for each of the QAPLIB instances. We con-

ﬁrm that, indeed, if m is high, then the performance

of the FRLS declines in comparison to the objective-

guided RLS, adding to our understanding of the per-

formance of this algorithm.

(Liang et al., 2022; Liang et al., 2024) showed

that the performance of the FRLS can signiﬁcantly

be improved if it is hybridized with RLS sharing the

budget in a round-robin fashion and if simulated an-

nealing (SA) is used as a basic algorithm. Investi-

gating plugging FFA in other algorithms on the QAP,

such as the SA by (Wilhelm and Ward, 1987), the

tabu search by (Taillard, 1991), the hybrid evolution-

ary algorithms by (Fleurent and Ferland, 1993; Merz

and Freisleben, 1999), or the ant colony optimization

method by (Gambardella et al., 1999), is therefore an

important branch of our future work.

ACKNOWLEDGEMENTS

The authors acknowledge support from the Project

of National Natural Science Foundation of China

62406095, the Project of Natural Science Foundation

of Anhui Province 2308085MF213, the Key Research

Frequency Fitness Assignment: Optimization Without Bias for Good Solution Outperforms Randomized Local Search on the Quadratic

Assignment Problem

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

bur

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

chr

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

els

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

esc

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

had

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

kra

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

lipa

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

nug

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

rou

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

scr

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

sko

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

ste

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

tai

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

tho

1.0

1.1

1.2

1.3

1.4

1.5

time in FEs

scaled f

RLS

FRLS

wil

Figure 2: The progress in terms of the best-so-far objective value divided by the lower bound lb of f averaged over all runs

and instances of an instance group and plotted over the time measured in FEs (log-scaled). Instance esc16f is omitted from

this statistic (the esc group) due to having a lower bound of 0. On the chrn instances, RLS is off the scale.

Plan of Anhui Province 2022k07020011, the Univer-

sity Scientiﬁc Research Innovation Team Project of

Anhui Province 2022AH010095, as well as the Hefei

Specially Recruited Foreign Expert program and the

Hefei Foreign Expert Ofﬁce program.

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12.

Frequency Fitness Assignment: Optimization Without Bias for Good Solution Outperforms Randomized Local Search on the Quadratic

Assignment Problem