Backbone analysis and algorithm design for the quadratic assignment problem

Transcription

1 Science in China Series F: Information Sciences 008 SCIENCE IN CHINA PRESS Springer wwwscichinacom infoscichinacom wwwspringerlinkcom Backbone analysis and algorithm design for the quadratic assignment problem JIANG He,, ZHANG XianChao, CHEN GuoLiang & LI MingChu School of Software, Dalian University of Technology, Dalian 66, China; Department of Computer Science, University of Science and Technology of China, Hefei 3007, China As the hot line in NP-hard problems research in recent years, backbone analysis is crucial for phase transition, hardness, and algorithm design Whereas theoretical analysis of backbone and its applications in algorithm design are still at a beginning state yet, this paper took the quadratic assignment problem (QAP) as a case study and proved by theoretical analysis that it is NP-hard to find the backbone, ie, no algorithm exists to obtain the backbone of a QAP in polynomial time Results of this paper showed that it is reasonable to acquire approximate backbone by intersection of local optimal solutions Furthermore, with the method of constructing biased instances, this paper proposed a new meta-heuristic biased instance based approximate backbone (BI-AB), whose basic idea is as follows: firstly, construct a new biased instance for every QAP instance (the optimal solution of the new instance is also optimal for the original one); secondly, the approximate backbone is obtained by intersection of multiple local optimal solutions computed by some existing algorithm; finally, search for the optimal solutions in the reduced space by fixing the approximate backbone Work of the paper enhanced the research area of theoretical analysis of backbone The meta-heuristic proposed in this paper provided a new way for general algorithm design of NP-hard problems as well quadratic assignment problem, NP-hard, backbone analysis, biased instance, meta-heuristic Introduction NP-hard problems [] have been the focus of computer science research for many years, with extensively wide applications arising in industrial control, transportation scheduling, network layout, very large scale integration (VLSI), etc According to the computational complexity theory [], there exists no exact algorithms to solve NP-hard problems in polynomial time unless P=NP Received October 7, 006; accepted July, 007 doi: 0007/s Corresponding author ( jianghe@dluteducn) Supported by the National Natural Science Foundation of China (Grant Nos and ), the Natural Science Foundation of LiaoNing Province (Grant No 00508), and the Gifted Young Foundation of Dalian University of Technology Sci China Ser F-Inf Sci May 008 vol 5 no

2 Therefore, exact algorithms are usually applied to small instances of NP-hard problems For larger instances, many heuristics offering very good near optimal solutions in a reasonable time have been developed Known heuristics can be classified into the following categories: ) simple heuristics, including greedy algorithm, local search, and tabu search [3] ; ) meta-heuristics, including ant colony algorithms [4,5], genetic algorithms [6], simulated annealing [7], neural networks [8], and quasi-physical and quasi-sociological algorithms [9] Backbone analysis [0] has been the hotline of research for NP-hard problems in recent years The backbone, the shared common part of all optimal solutions for an NP-hard problem instance, is important to evaluate the hardness and phase transition of NP-hard problems Monasson et al [] investigated the effect of the backbone on the hardness and phase transition of the satisfiability problem (SAT) in 998 Zhang [] analyzed the effect of the backbone size on the asymmetric traveling salesman problem (ATSP) Moreover, backbone analysis has been currently an important way to design heuristics for NP-hard problems Schneider et al [3] and Zou et al [4] proposed multilevel reduction algorithms for the TSP by using the intersection of local optimal solutions as the approximate backbone, respectively Zhang et al [5] presented a backbone guided LK algorithm for the TSP Zhang et al [6], Dubois et al [7], and Valnir et al [8] gave backbone guided local search algorithms for the SAT, respectively Zou et al [9] developed the approximate backbone-guided fant (ABFANT) for the quadratic assignment problem (QAP) However, backbone analysis is still at the beginning state by experimental statistic methods Also, there are few theoretical analysis results of the backbone As to our knowledge, there is no report on algorithms designed by theoretical analysis results of the backbone The only theoretical result is that Kilby et al [0] proved that it is NP-hard to obtain the backbone of the TSP For more complex NP-hard problems, such as the QAP, there still lacks theoretical analysis results of the backbone For the QAP, there exist some interesting and important problems as follows Firstly, is there a polynomial time algorithm to obtain the backbone of the QAP? If so, we can use the backbone other than the approximate backbone for algorithm design Secondly, a serious disadvantage exists to those widely used methods, which approximate the backbone by intersection of local optimal solutions: When the number of optimal solutions is not unique, the size of the backbone is very small and the benefit by using the approximate backbone would be obviously limited Is there any method to settle the deficiency due to small size of the backbone for NP-hard problems? This paper solves the problems mentioned above as follows Firstly, we proved that there is no polynomial time algorithm to obtain the backbone of the QAP under the assumption that P NP In the proof, a method constructing the biased QAP instance is given Every QAP instance can be transformed into its corresponding biased instance, whose optimal solution is also optimal to the original instance Secondly, a new meta-heuristic, the biased instance based approximate backbone (BI-AB) for the QAP, is proposed It consists of five steps: Step Given a QAP instance, construct its biased QAP instance with unique optimal solution (ie, backbone) Step Obtain multiple local optimal solutions to the new constructed QAP instance by calling some QAP heuristic Step 3 Obtain the approximate backbone by intersection of local optimal solutions Step 4 Reduce the search space by fixing the approximate backbone Step 5 Search for solutions in the reduced search space by calling some existing QAP heuristic JIANG He et al Sci China Ser F-Inf Sci May 008 vol 5 no

3 It can be drawn that we solve the above problem of small size backbone, due to existence of multiple optimal solutions We used FANT [] as the subordinate algorithm and tested the BI-AB on the standard QAP benchmark QAPLIB [] Experimental results indicated that the new heuristic outperforms ABFANT and Restart FANT in terms of solution quality Work of this paper extends the research of the QAP The method by construction of the biased instance provides a direction for backbone analysis of other NP-hard problems On the other hand, the new heuristic being presented in this paper, that constructs the biased instance and then obtains the approximate backbone by intersection of local optimal solutions, throws a light on algorithm design for other NP-hard problems Preliminaries In this section, we shall give some related definitions and notations Definition Given n facilities, represented by the set F = { f, f,, f n }, and n locations represented by the set L= { l, l,, l n }, the quadratic assignment problem (QAP) is to determine to which location each facility must be assigned at minimal cost Mathematically, let A= ( a ) nxn be a matrix where a represents the flow between facilities fi and f j Let B = ( ) be a matrix where instance b represents the distance between locations li and l j A feasible solution of a QAP n n j i i j) QAP( A, B ) is an assignment :{,,, n} {,,, n} whose cost function is defined as g ( A, B) = = = a b( ) ( In the QAP, the goal is to find a feasible solution * :{,,, n} {,,, n} that minimizes the cost, ie, g * ( A, B) = min({ g ( A, B) Π }), where Π is the set of all permutations of{,,, n} The QAP is a classic combinatorial optimization problem with extensively wide applications, such as hospital layout, keyboard arrangement, and VLSI Sahni et al [] have shown that the problem is NP-hard, so exact algorithms are currently only effective for instances of size n 30 Therefore, several heuristics have been proposed for finding near-optimum solutions to large QAP instances, including ant colony [3,4], genetic algorithm [5], simulated annealing [6], neural networks [7], tabu search [8,9], and GRASP [30] Definition Given the flow matrix, let m = max({ a i, j n} {}) Given the distance matrix A B = ( b ) nxn, let mb = max({ b i, j n} {}) The biased matrix of A is defined as ˆ in + jn A= ( a + ξ ) n n, where ξ = (3mB ), and the biased matrix of B is defined as ˆ in+ j B = ( b + δ ) n n where δ = (3mA ) In this paper, the biased QAP instance of QAP( A, B) is denoted as QAP( Aˆ, Bˆ ) Definition 3 Given a QAP instance optimal solutions to QAP( A, B), where backbone of QAP( A, B) P( ) = {( i, ( i)) i n} is defined as QAP( A, B), let q b nxn Π = {,,, } be the set of all * q = Π represents the number of optimal solutions The bone( A, B) = P( ) P( ) P ( * ), where q 478 JIANG He et al Sci China Ser F-Inf Sci May 008 vol 5 no

4 It is significant to obtain the backbone bone( A, B) for algorithm design According to Definition 3, a heuristic cannot obtain an optimal solution to a QAP instance, unless it obtains the whole backbone bone( A, B ) On the other hand, if the bone( A, B) is obtained, the search space could then be effectively reduced by assigning facilities in the backbone to some fixed locations Boese [3] observed that a local optimal solution tends to have 80% common edges shared with an optimal solution Such observations have also led to the big valley structure, which suggests that extensively many local optimal solutions form a single cluster around the optimal solutions Similar structures are also found in the graph partitioning problem [3], and the flow shop scheduling problem [33] Since it is hard to obtain the optimal solutions, many researchers usually approximate the backbone by intersection of local optimal solutions, according to the relationship between local optimal solutions and optimal solutions Definition 4 Given a QAP instance QAP( A, B), the approximate backbone is defined as a- bone(,,, k ) = P( ) P( ) P( k ), where,,, k represent local optimal solutions to the QAP( A, B) 3 Theoretical analysis for backbone Without loss of generality, we assume that the entries a, b in the matrices A and B are nonnegative integers, ie, a, b Z + { 0} for i, j n In Lemma and Lemma, we shall study the relationship between the solution of a QAP instance and that of the biased QAP instance Lemma Given the matrices A= ( a ) nxn ( a Z + {0} ) and B = ( b ) nxn ( b Z + {0} ), if g ( AB, ) < g ( AB, ) for any two distinct solutions, to the QAP instance QAP( A, B), then g ( AB ˆ, ˆ) < g ( AB ˆ, ˆ) for the biased QAP instance QAP( A ˆ, B ˆ ) Proof Firstly, we shall prove that max({ b i, j n}) 0 and max({ a i, j n}) 0 Otherwise, if max({ a i, j n }) = 0, then we have a = 0 for i, j n It follows that g ( A, B) = g ( A, B) = 0 which contradicts with the assumption of Lemma Similarly, we have max({ b i, j n}) 0 Thus, m = max({ a i, j n} {}) = max({ a i, j n}), A m = max({ b i, j n}) Secondly, we shall show that B g ( AB ˆ, ˆ) < g ( AB ˆ, ˆ) holds if g ( AB, ) < g ( AB, ) for any distinct feasible solutions, to the QAP instance QAP( A, B) + By the assumption of Lemma, we have g ( AB, ) g ( AB, ) Z By definition of the biased QAP instance, for any feasible solution, we have ˆ n n g ( A, Bˆ ) = ( a + ξ )( b + δ ) δ j= i= () i ( j) () i ( j) n n n n n n n n ab j i () i ( j) a δ j i () i ( j) ξ j i b() i ( j) ξ = = = = = = j= i= () i ( j) = JIANG He et al Sci China Ser F-Inf Sci May 008 vol 5 no

5 n n n n n n (, ) δ j i ( i) ( j) ξ j i ( i) ( j) ξ j i δ = = = = = = ( i) ( j) = g A B + a + b + Let Δ Then we have and n n n n n n = aδ() i ( j), Δ = ξb() i ( j), Δ = ξδ j= i= j= i= and 3 j= i= () i ( j) n n n n ( in ) + ( j) n n ( in ) + ( j) a δ j i () i ( j) a j i ma m j i A m = = = = = = A Δ = = ( ) ( ) n n () in+ ( j) n n in+ j =, 3 j= = i= 3 j= = i= 3 < n n + n 3 n n n n in + jn n n in + jn ξ b j i () i ( j) b j i () i ( j) mb m j i B m = = = = = = B Δ = = ( ) ( ) n n in + jn n n in+ j j= i= j= i= n n + n = < = <, n n n n in + jn + () i n+ ( j) ξδ j i () i ( j) mm = = j= i= A B Δ = = (9 ) n n in + jn + ( i) n+ ( j) n n ( i) n+ ( j) j= i= j i A B mm = = A B = < 9mm 9 n n in+ j = 9mm j= = i= 9 n n n A B mm + < A B 9 By the definition, we have in+ j 3 a 0, b 0, in + jn ξ = (3m ) > 0 and B 3 7 δ = (3mA ) > 0, so Δ 0, Δ 0, Δ 0, and 0 Δ +Δ +Δ 9 If g ( AB, ) < g ( AB, ) for any two distinct feasible solutions,, then we have 3 3 g ˆ ˆ ˆ ˆ ( A, B) g ( A, B) = ( g (, ) (, )) ( ) ( ) A B g A B + Δ +Δ +Δ Δ +Δ +Δ have if then Since g ( A, B) g ( A, B) g ( Aˆ, Bˆ) g ( Aˆ, Bˆ) > 0 and 7 Δ +Δ +Δ Δ +Δ +Δ ) 7, 3 3 ( ) ( 9 9 and so this lemma is proved Lemma Given the matrices A= ( a ) nxn ( a Z + {0} ) and B = ( b ) nxn ( b Z + {0} ), g ( AB, ) = g ( AB, ) for any two distinct solutions, to the QAP instance QAP( A, B), g ( AB ˆ, ˆ) g ( AB ˆ, ˆ) for the biased QAP instance QAP( A ˆ, B ˆ ) Proof By the assumption of Lemma, we have g ( A, B) = g ( A, B) Therefore, 3 3 g ˆ ˆ ˆ ˆ ( A, B) g ( A, B) = ( Δ +Δ +Δ ) ( Δ +Δ + Δ ) We need only to 3 3 prove that ( Δ +Δ +Δ ) ( Δ +Δ +Δ ) 0 Since, so there must exist i ( i n) subject to () i () i Let hi (, j, ) = in + jn + () in+ ( j), D = { hi (, j, ) hi (, j, ) hi (, j, ), i, j n}, and D = { hi (, j, ) hi (, j, ) hi (, j, ), i, j n}, then we have D D Let m = max( D D ) Therefore, we D D =, 480 JIANG He et al Sci China Ser F-Inf Sci May 008 vol 5 no

6 and 3 n n ( a δ () ( ) () ( ) () ( )) j i i j ξb i j ξδ = = i j = i, j n aδ() i ( j) + ξb() i ( j) + ξδ() i ( j) h (,, ) m Δ +Δ +Δ = + + ( ) + ( a δ + ξ b + ξ δ ), i, j n ( i) ( j) ( i) ( j) ( i) ( j) h (,, ) >m 3 n n ( a δ () ( ) () ( ) () ( )) j i i j ξb i j ξδ = = i j = i, j n aδ( i) ( j) + ξb( i) ( j) + ξδ( i) ( j) h (,, ) m Δ +Δ +Δ = + + By the definition of ( ) + ( a δ + ξ b + ξ δ ) i, j n () i ( j) () i ( j) () i ( j) h (,, ) >m, m for any i, j n, if hi (, j, ) = hi (, j, ), and ( i ) = ( i ), ( j ) = ( j ) Hence, Thus, we have > m (,, ) >, we have hi j m i, j n () i ( j) () i ( j) () i ( j) h (,, ) = ( a δ + ξ b + ξ δ ) i, j n () i ( j) () i ( j) () i ( j) h (,, ) > m 3 3 Δ +Δ +Δ ( Δ +Δ +Δ ) = ( a δ + ξ b + ξ δ ) i, j n ( i) ( j) ( i) ( j) ( i) ( j) h (,, ) m ( a δ + ξ b + ξ δ ) ( a δ + ξ b + ξ δ ) i, j n () i ( j) () i ( j) () i ( j) h (,, ) m () in+ ( j) in + jn in + jn + () i n+ ( j) i, j n ( a (3mA ) b() i ( j) (3mB ) + (9mAmB )) h (,, ) m = + = () in+ ( j) in + jn in+ jn+ () in+ ( j) ( a (3m ) b (3m ) (9m m ) ) + + i, j n A () i ( j) B A B h (,, ) m m m ( ( i) n+ ( j)) m ( in + jn ) m h( i, j, ) (9mAmB )) i, j n (3mBa + 3mb A () i ( j) + h (,, ) m m m ( ( i) n+ ( j)) m ( in + jn ) m (,, h i j ) i, j n (3mBa + 3mAb() i ( j) + ) h (,, ) m By the definition, we have m = max( D D ) D D, ie, m D or D that In the following proof, we only consider the case that m D, Assume that we can prove it in a similar way * 3 * m = h( i, j, ) = i n + j n + ( i ) n+ ( j ), 3 3 Δ +Δ +Δ ( Δ +Δ +Δ ) * 3 * in + jn B mab ( i ) ( j ) ( i ) n+ ( j ) m h( i, j, ) m D For the case m m ( ( i) n+ ( j)) m ( in + jn ) m (,, h i j ) = (9mAmB )) i, j n (3mBa + 3mAb() i ( j) + ) hi (, j, ) < m + (3m a + + ) then ) JIANG He et al Sci China Ser F-Inf Sci May 008 vol 5 no

7 m ( ( i) n+ ( j)) m ( in + jn ) m h( i, j, ) i, j n (3ma B + 3mb A () i ( j) + ) h (,, ) < m m m ( ( i) n+ ( j)) m ( in + jn ) m h( i, j, ) = (9mm A B )) i, j n (3ma B + 3mb A () i ( j) + ) h (,, ) < m * 3 * in+ jn B mb A ( i ) ( j ) ( i ) n+ ( j ) + (3ma + + ) m ( ( i) n+ ( j)) m ( in + jn ) m (,, h i j ) i, j n (3ma B + 3mb A () i ( j) + ) h (,, ) < m If hi (, j, ) < for i, j n,, then 3ma,, and m B m ( ( i) n+ ( j)) m h(, i j, ) 3mb If A () i ( j) m ( in + jn ) * 3 * in+ jn are even integers, so are 3ma B and + ( ) ( ) A ( i ) ( j ) 3 mb i n j ( ( ) ( )) hi (, j, ) < for i, j n, then m ma i n+ j m h(, i j, ),, and 3mb m m ( in + jn ) A are even integers Therefore, () i ( j) m ( ( i) n+ ( j)) m ( in + jn ) m h( i, j, ) i, j n (3ma B + 3mb A () i ( j) + ) h (,, ) < m * 3 * in+ jn ( i ) n+ ( j ) + (3ma B + 3mb A + ) ( i ) ( j ) ( in + jn ) m h( i, j, ) i, j + h (,, ) Hence, m ( ( i) n+ ( j)) m n (3ma B 3mb A () i ( j) < m 3 3 ( Δ +Δ +Δ ) ( Δ +Δ +Δ ) 0 B and so this lemma is proved + ) 0 Theorem Given the matrices A= ( a a Z + ) nxn ( { 0} ) and B = ( b ) nxn ( b Z + { 0}), there is a unique optimal solution to the biased QAP instance QAP( Aˆ, Bˆ ) Proof Otherwise, there must at least be two distinct optimal solutions to the QAP( Aˆ, Bˆ ) Let, be among these optimal solutions subject to We distinguish three cases, according to the relationship between g ( AB, ) and g (, ) AB Case g ( AB, ) = g ( AB, ) By Lemma, we have g ( ˆ, ˆ) ( ˆ, ˆ AB g AB), a contradiction Case g ( AB, ) < g ( AB, ) By Lemma, we have g ( Aˆ, Bˆ) < g ( Aˆ, Bˆ), a contradiction Case 3 g ( AB, ) < g ( AB, ) By Lemma, we have g ( ˆ, ˆ) ( ˆ, ˆ AB < g AB), a contradiction Thus, the theorem is proved Corollary Given the matrices A= ( a ) nxn ( a Z + {0}) and B = ( b ) nxn ( b Z + {0} ), the optimal solution to the biased QAP instance QAP( Aˆ, Bˆ ) is also optimal to 48 JIANG He et al Sci China Ser F-Inf Sci May 008 vol 5 no

8 the instance QAP( A, B) Proof By Theorem, there exists a unique optimal solution denoted by * to the biased QAP instance QAP( Aˆ, Bˆ ) Then, the solution * must be optimal to the instance QAP( A, B), * otherwise there exists a solution to the instance QAP( Aˆ, B ˆ ) which is subject to g ( AB, ) < g * ( AB, ) However, by Lemma, we have g ( AB ˆ, ˆ) < g * ( AB ˆ, ˆ) The corollary is proved Theorem There exists no polynomial time algorithm to obtain the backbone of the QAP unless P = NP Proof Otherwise, there must exist an algorithm denoted by Λ which is able to obtain the backbone of the QAP in polynomial time Given any arbitrary flow matrix A= ( a ) nxn ( a 0 ) and distance matrix B = ( b ) nxn b 0), the instance QAP( A, B) could be transformed to its biased QAP instance ( by a polynomial time algorithm denoted by Γ Let Ο ( ) be the time complexity of the algorithm Γ Since the QAP( Aˆ, Bˆ ) is also a QAP instance, its backbone can be obtained by Λ in polynomial time which is denoted by QAP( Aˆ, Bˆ ) Ο ( ) Also, by Lemma, the backbone is not only the unique optimal solution to the QAP ( Aˆ, Bˆ) itself, but also optimal to the QAP( A, B) Hence, any QAP instance can be exactly solved in Ο ( ) +Ο( ) time by Γ and Λ Obviously, such conclusion contradicts with the result that the QAP is NP-hard by Sahni [] Thus, this theorem is proved 4 Biased instance based approximate backbone algorithm In this section, we shall propose a new meta-heuristic for the QAP biased instance based approximate backbone (BI-AB), by constructing the biased QAP instance used in the backbone analysis Firstly, we review the background and disadvantages of the ABFANT After that, we give our algorithm BI-AB 4 Approximate backbone guided FANT Among those researchers who applied the backbone for algorithm design of NP-hard problem, Zou et al [9] proposed the ABFANT for the QAP in 005 Algorithm: ABFANT Input: QAP instance QAP(A, B) Output: Solution 0 Begin Find an initial solution 0 by using FANT for the QAP instance QAP(A, B) Do the following: Find k different solutions,,, k by FANT Obtain the approximate-backbone a- bone( 0,,, k ) of 0,,, k JIANG He et al Sci China Ser F-Inf Sci May 008 vol 5 no

9 3 Fix a- bone( 0,,, k ) to obtain a new search space 4 Run FANT to obtain the solution k 5 If g ( AB, ) < g ( AB, ), then set 0 = k k 0 S * ( QAP( A, B)) 3 Return 0 End The ABFANT achieved better performance than the FANT, in the way that reduces the search space by fixing the approximate backbone The theoretical basis of the AB-FANT is the big valley structure as mentioned above However, the effect of the approximate backbone on the QAP is not as prominent as that on the TSP, for which Zou et al [4] designed a multilevel reduction algorithm using the approximate backbone The reason for this is, that the backbone size is usually equal to the size of the TSP instances from the TSPLIB, ie, there exists unique optimal solution to such TSP instance Slaney et al [0] generated -D integer Euclidean TSP problems with 0 cities randomly placed in a square and found that the backbone size quickly approximates n as the length of the square is increased Zhang [] observed that the backbone of random generated asymmetric TSP instances rises to n quickly, as the precision of intercity distances increases However, the property of solutions to the QAP is more complex than that of the TSP Firstly, the backbone size of the QAP sharply varies from instance to instance For example, many instances from the QAPLIB have more than one optimal solution By enumeration method, we obtained the statistic results of instances with facilities from the QAPLIB (see Table ) We can find that only an optimal solution exists to every QAP instance of chrb, chrc, roua, and taia, whose backbone size is the same as the instance size here Whereas, the instances scra and nuga have more than one optimal solution, ie, the backbone size is 0 Here, the backbones of the instances scra and nuga are not significant for algorithm design Table Backbone size and optimal solutions of instances from the QAPLIB Instance Instance size Optimal solutions Backbone size chrb chrc roua scra taia nuga In the mean time, the size of the QAP approximate backbone obtained by methods of Zou et al is far smaller than that of the TSP, so is the average probability of the QAP approximate backbone appearing in the optimal solutions For typical instances from the QAPLIB, Zou et al found by statistic method that the proportion of the backbone in intersection of local optimal solutions 484 JIANG He et al Sci China Ser F-Inf Sci May 008 vol 5 no

10 to the intersection itself is comparatively low Moreover, the size of intersection of local optimal solutions decreases quickly as the number of local optimal solutions being intersected increases (When the number of local optimal solutions increases across 3, the intersection size of most QAP instances decreases to 0) Table gives the statistic results [9] of intersection of local optimal solutions to typical instances from the QAPLIB Table Statistic results of approximate backbone (AB) Instance Instance size Approximate Average probability of AB backbone size appearing in the optimal solutions (%) Iterations times chr5a tai30b tai40b lipa40a tai50b chrb tai80b Biased instance based approximate backbone algorithm In this subsection, we give the new meta-heuristic BI-AB under the analysis of subsection 4 The basic idea of the BI-AB is to transform a QAP instance to its biased QAP instance with unique optimal solution, ie, the backbone, by constructing biased matrices of the original matrices A= ( a ) nxn and B = ( b ) nxn The optimal solution to the new instance is also optimal to the original one, so we need only to solve the new instance After the transformation, we use some existing algorithm (denoted by Λ) as the subordinate algorithm to obtain k local optimal solutions to the new instance and fix the approximate backbone by intersecting such k solutions to obtain smaller search space which is further searched for new and better solutions Algorithm: BI-AB Input: QAP instance QAP(A, B), subordinate algorithm Λ Output: Solution 0 Begin Construct the biased QAP instance QAP( Aˆ, B ˆ) for the instance QAP( A, B) Find an initial solution 0 by using Λ for the QAP( Aˆ, Bˆ ) 3 Do the following: 3 Find k- different solutions,,, k to the QAP( Aˆ, Bˆ ) by Λ 3 Obtain the approximate-backbone a- bone( 0,,, k ) of 0,,, k * 33 Fix a- bone( 0,,, k ) to obtain a new search space S ( QAP( Aˆ, B)) ˆ 34 Run Λ to obtain the solution k 35 If g ( AB ˆ, ˆ) < g ( AB ˆ, ˆ), then set 0 = k k 4 Return 0 End The run time of the BI-AB can be estimated as follows: 0 JIANG He et al Sci China Ser F-Inf Sci May 008 vol 5 no

11 Step Takes Ο( n ) time to construct the biased QAP instance Step Let the run time of Λ to solve the QAP( Aˆ, B ˆ) be Ο ( ) Step 3 Takes ( k ) Ο ( ) time Step 3 Takes Ο ( kn) time to obtain the approximate backbone Step 33 Takes O(n) time Step 34 Takes Ο ( ) time Step 35 Takes O(n ) time Step 4 Takes O() time Summarizing the above, the time complexity of the BI-AB is Ο( n ) + kο ( ) + Ο( kn) There are many advantages of the BI-AB Firstly, the BI-AB provides a framework which can employ any existing algorithm as the subordinate algorithm Λ Secondly, it is really simple to construct a biased QAP instance, though the proof of unique optimal solution is overloaded with details Finally, the size of backbone is remarkably improved by constructing the biased QAP instance, and the effect of the approximate backbone is also enhanced 5 Experiments and analysis Sahni et al [] showed in 976 that there is no ε approximation algorithm for the QAP unless P=NP (ε > 0) Hence, it is the only way to evaluate heuristics by experimental methods The meta-heuristic BI-AB provides a framework, which could employ tabu or greedy algorithm as the subordinate algorithm Since we mainly investigate the application of the approximate backbone in algorithm design, the FANT is incorporated as the subordinate algorithm in order to compare the BI-AB with ABFANT Performance comparison results about FANT, genetic algorithms and tabu search can be found in ref [34] We implement the BI-AB (denoted by BI-ABFANT) in C++ language For performance comparison, we also implement ABFANT and random restart FANT (denoted by Re-FANT) In the experiments, we take the parameter k= The iteration times of the FANT is set to be 000 The Re-FANT runs 3 times FANT and chooses the best solution as the final result Our experiments are conducted on a PC with one P4 4G CPU and 5 MB of main memory running Redhat Linux 90 Experimental results for 0 runs of typical instances from the QAPLIB are given in Table 3, where g opt denotes cost of the optimal solution, t mean the average runtime in seconds, τ min the minimum percentage excess over the optimal solution, and τ mean the average percentage excess over the optimal solution, respectively Table 3 illustrates that the ABFANT, Re-FANT, and BI-ABFANT achieve good performance on most instances from the QAPLIB in nearly same time The ABFANT, Re-FANT, and BI-ABFANT can find optimal solutions to 8 instances out of the total 0 instances in 0 runs It means that the subordinate algorithm FANT is efficient and effective The BI-ABFANT outperforms the ABFANT on average solution quality for instances Conversely, the ABFANT outperforms the BI-ABFANT on average solution quality for instance The BI-ABFANT is better for 9 instances, worse for 3 instances than the Re-FANT on average solution quality, respectively For the remainder instances, both the BI-ABFANT and the Re-FANT always obtain the optimal solutions Summarizing the above, the BI-ABFANT significantly outperforms the ABFANT and the Re-FANT in terms of solution quality 486 JIANG He et al Sci China Ser F-Inf Sci May 008 vol 5 no

12 Table 3 Results of ABFANT, Re-FANT, BI-ABFANT on instances from the QAPLIB (0 runs) ABFANT Re_FANT BI-ABFANT Instance g opt τ min τ mean τ t min τ mean τ (%) (%) mean t min τ mean (%) (%) mean (%) (%) t mean bur6a chr5a chr5b chra chrb chr5a lip40a lip50a nug6a nug30a rou0a scra scr0a ste36b taia taib tai40b tai50a tai80b tho30a Conclusions In this paper, we performed a theoretical analysis for the backbone of the QAP By constructing the biased QAP instance, we proved that it is NP-hard to obtain the backbone of the QAP, so it is reasonable to obtain the approximate backbone by intersection of local optimal solutions The biased instance based approximate backbone algorithm is then proposed Work of this paper provides a case study of theoretical analysis for the backbone of NP-hard problems Moreover, the way of designing algorithms based on the biased instance, casts a light on algorithm design for other NP-hard problems by using the backbone Garey M R, Johnson D S Computers and Intractability: A Guide to the Theory of NP-completeness San Francisco: W H Freeman, Sahni S, Gonzalez T P-complete approximation problems J ACM, 976, 3(3): Glover F, Laguna M Tabu Search Boston: Kluwer Academic Publishers, Dorigo M, DiCaro G The ant colony optimization meta-heuristic In: Corne D, Dorigo M, Glover F, eds New Ideas in Optimization London: McGraw Hill, UK, Duan H B Ant Colony Algorithms: Theory and Applications (in Chinese) Being: Science Press, Chen G L, Wang X F, Zhuang Z Q Genetic Algorithm and its Application (in Chinese) Being: Posts & Telecom Press, Kang L S, Xie Y, You S Y, et al Non numeric Parallel Algorithms: Simulated Annealing (in Chinese) Being: Science Press, Zhou Z H, Cao C G Neutral Networks and Applications (in Chinese) Being: Press of Tsinghua University, Huang W Q, Xu R C Introduction to Modern Computational Theory: Background, Perspective, and Algorithms Research of NP-hard Problems (in Chinese) Being: Science Press, Slaney J, Walsh T Backbones in optimization and approximation In: Proc 7th Intern Joint Conf Artif Intell (IJCAI-0) San Francisco: Morgan Kaufmann Publishers, JIANG He et al Sci China Ser F-Inf Sci May 008 vol 5 no

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