Parallel Quantum-inspired Genetic Algorithm for Combinatorial Optimization Problem

Transcription

1 Parallel Quantum-insired Genetic Algorithm for Combinatorial Otimization Problem Kuk-Hyun Han Kui-Hong Park Chi-Ho Lee Jong-Hwan Kim Det. of Electrical Engineering and Comuter Science, Korea Advanced Institute of Science and Technology(KAIST), 373-1, Kusong-dong Yusong-gu, Taejon, 35-71, Reublic of Korea Abstract- This aer rooses a new arallel evolutionary algorithm called arallel quantum-insired genetic algorithm (PQGA). Quantum-insired genetic algorithm(qga) is based on the concet and rinciles of quantum comuting such as qubits and suerosition of states. Instead of binary, numeric, or symbolic reresentation, by adoting qubit chromosome as a reresentation, QGA can reresent a linear suerosition of solutions due to its robabilistic reresentation. QGA is suitable for arallel structure because of raid convergence and good global search caability. That is, QGA is able to ossess the two characteristics of exloration and exloitation, simultaneously. The effectiveness and the alicability of PQGA are demonstrated by exerimental results on the knasack roblem, which is a well-known combinatorial otimization roblem. The results show that PQGA is suerior to QGA as well as other conventional genetic algorithms. 1 Introduction Evolutionary algorithms(eas) are rincially stochastic searches and otimization methods based on the rinciles of natural biological evolution. Comared to the traditional otimization methods, such as calculus-based and enumerative stragegies, EAs are robust, global and can be generally alied without recourse to domain-secic heuristics. But the characteristics of oulation diversity and selective ressure are not easy to be imlemented in EAs such as evolutionary rogramming(ep), evolution strategies(es) and genetic algorithm(ga), simultaneously. As selective ressure is increased, the search focuses on the to individuals in the oulation, but because of this exloitation genetic diversity is lost. The reason is that the reresentations of EAs are dened using deterministic values. The new evolutionary algorithm which uses stochastic reresentation was roosed in [1]. QGA are characterized by raid convergence and global search caability, simultaneously. QGA is based on the concet and rinciles of quantum comuting such as qubits and a linear suerosition of states. One individual of QGA can reresent many states at the same time, and there are weak relationshis between individuals since each individual is determined by current best solution and its robability, that is, the history of individual, u to date. Because of this reason, QGA is suitable for arallel structure. Recently, arallel evolutionary algorithms (PEAs) have been used to slove more difcult roblems which need a bigger oulation. This results in higher comutational cost. The basic motivation behind many early studies of PEAs was to reduce the rocessing time needed to reach an accetable solution [2]. This was accomlished by imlementing EAs on different arallel architecture. In addition, it was noted that in some cases the PEAs found better solutions than comarably sized serial EAs. There are several aroaches to arallelize the serial EAs [2, 3, 4, 5]. This aer offers a new arallel evolutionary algorithm called arallel quantum-insired genetic algorithm (PQGA). Esecially, a coarse-grained arallel scheme was alied to PQGA. PQGA can reduce the comutational time as comared with QGA. This aer is organized as follows. Section 2 and 3 describe the new evolutionary algorithm, QGA and the arallel quantum-insired genetic algorithm, resectively. Section 4 contains a descrition of the exeriment with GAs, QGAs and PQGAs for knasack roblems for comarison urose. Section 5 summarizes and analyzes the exerimental results. Concluding remarks follow in Section 6. 2 Quantum-insired Genetic Algorithm (QGA) QGA is based on the concets of qubits and suerosition of states of quantum mechanics. The smallest unit of information stored in a two-state quantum comuter is called a quantum bit or qubit [6]. A qubit may be in the 1 state, in the state, or in any suerosition of the two. The state of a qubit can be reresented as jψi = ffji + j1i; (1) where ff and are comlex numbers that secify the robability amlitudes of the corresonding states. jffj 2 gives the robability that the qubit will be found in state and jj 2 gives the robability that the qubit will be found in the 1 state. Normalization of the state to unity guarantees jffj 2 + jj 2 =1: (2) If there is a system of m-qubits, the system can contain information of 2 m states. However, in the act of observing a

2 quantum state, it collases to a single state [7]. 2.1 Reresentation It is ossible to use a number of different reresentations to encode the solutions onto chromosomes in evolutionary algorithm. The classical reresentations can be broadly classied as: binary, numeric, and symbolic [8]. QGA uses a novel reresentation that is based on the concet of qubits. One qubit is dened with a air of comlex numbers, (ff; ), as» ff ; which is characterized by (1) and (2). And an m-qubits reresentation is dened as» ff 1 ff 2 ff m 1 2 ; (3) m where jff i j 2 + j i j 2 =1, i =1; 2; ;m. This reresentation has the advantage that it is able to reresent a suerosition of states. If there is, for instance, a three-qubits system with three airs of amlitudes such as " # ; (4) the state of the system can be reresented as 1 ji j1i 1 j1i 3 j11i (5) j1i 4 + j11i 1 3 j11i j111i The above result means that the robabilities to reresent the state ji, j1i, j1i, j11i, j1i, j11i, j11i, and j111i are 1, 3, 1, 3, 1, 3, 1, and 3, resectively. By consequence, the three-qubits system of (4) contains information of eight states. Evolutionary comuting with qubit reresentation has a better characteristic of diversity than classical aroaches, since it can reresent suerosition of states. Only one qubit chromosome such as (4) is enough to reresent eight states, but in classical reresentation at least eight chromosomes, (), (1), (1), (11), (1), (11), (11), and (111) are needed. Convergence can be also obtained with the qubit reresentation. As jff i j 2 or j i j 2 aroaches to 1 or, the qubit chromosome converges to a single state and the roerty of diversity disaears gradually. That is, the qubit reresentation is able to ossess the two characteristics of exloration and exloitation, simultaneously. 2.2 QGA The structure of QGA is described in the following. rocedure QGA t ψ initialize Q(t) make P (t) by observing Q(t) states evaluate P (t) store the best solution among P (t) while (not termination-condition) do t ψ t +1 make P (t) by observing Q(t 1) states evaluate P (t) udate Q(t) using quantum gates U (t) store the best solution among P (t) QGA is a robabilistic algorithm which is similar to genetic algorithm. QGA maintains a oulation of qubit chromosomes, Q(t) =fq t 1; q t 2; ; q t ng at generation t, where n is the size of oulation, and q t j is a qubit chromosome dened as q t j =» ff t 1 t 1 ff t 2 t 2 ff t m t ; (6) m where m is the number of qubits, i.e., the string length of the qubit chromosome, and j =1; 2; ;n. In the ste of initialize Q(t), ff t i and t i, i =1; 2; ;m, of all q t j, j = 1; 2; ;n,inq(t) are initialized with 1 2. It means that one qubit chromosome, q t j j t= reresents the linear suerosition of all ossible states with the same robability: X m 2 jψ q j i = 1 js ki; 2 m k=1 where S k is the k-th state reresented by the binary string (x 1 x 2 x m ). x i, i = 1; 2; ;m, is either or 1. The next ste makes a set of binary solutions, P (t), by observing Q(t) states, where P (t) =fx t ; 1 xt ; ; 2 xt ng at generation t. One binary solution, x t j, j = 1; 2; ;n, is a binary string of length m, and is formed by selecting each bit using the robability of qubit, either jff t i j2 or ji tj2, i =1; 2; ;m,of q t j. Each solution xt j is evaluated to give some measure of its tness. The initial best solution is then selected and stored among the binary solutions, P (t). In the while loo, one more ste, udate Q(t), is included to have tter states of the qubit chromosomes. A set of binary solutions, P (t), is formed by observing Q(t 1) states as with the rocedure described before, and each binary solution is evaluated to give the tness value. In the next ste, udate Q(t), a set of qubit chromosomes Q(t) is udated by alying some aroriate quantum gates 1 U (t), which is 1 Quantum gates are reversible gates and can be reresented as unitary oerators acting on the qubit basis states: U y U = UU y, where U y is the hermitian adjoint of U. There are several quantum gates, such as NOT gate, Controlled NOT gate, Rotation gate, Hadamard gate, etc.[6].

3 formed by using the binary solutions P (t) and the best stored solution. The aroriate quantum gates can be designed in comliance with ractical roblems. Rotation gates such as» cos( ) sin( ) U ( ) = ; (7) sin( ) cos( ) where is a rotation angle, will be used for knasack roblems in the next section. This ste makes the qubit chromosomes converge to the tter states. The best solution among P (t) is selected in the next ste, and if the solution is tter than the best stored solution, the stored solution is relaced by this best solution. The binary solutions P (t) are discarded at the of the loo. In QGA, the oulation size, i.e., the number of qubit chromosomes is always ket constant. This is due to conservation of qubits based on quantum comuting. QGA with the qubit reresentation can have better convergence with diversity than conventional GAs which have xed and 1 information. 3 Parallel Quantum-insired Genetic Algorithm (PQGA) To nd more otimized solution in shorter comuting time, arallelization of EAs is indisensable. The basic motivation behind many early studies of PEAs was to reduce the rocessing time needed to reach an accetable solution. This was accomlished by imlementing EAs on different arallel architectures. Using many rocesses shortens the comuting time. In the exeriments reorted in [3], Using n rocessers shortens the comutation time by a factor whose value is limited to n. In the shifting balancing theory roosed by Sewall Wright [3], the evolution seed of small suboulations with loose connection is faster than that of large oulation. In addition, it was noted that in some cases the PEAs found better solutions than comarably sized serial EAs, because the migration of individuals between suboulations introduces the ossibility of global search. The arallelization of EAs has been imlemented in several ways, like globally PEAs, coarse-grained PEAs and negrained PEAs. These are classication according to the connection toology of multi-rocessors [2, 3]. Also, the hybrid structure can be alied for arallelization [9]. The coarsegrained arallel scheme is considered here. The imortant characteristics of coarse-grained algorithms are the use of few relatively large demes and the introduction of a migration oerator. The whole oulation is divided into some number of oulations, each suboulation evolves indeently, and the exchange of individuals, called migration, occurs after some generations. This scheme can be easily imlemented and even if there is no arallel comuter available, it is easy to simulate one with a network of workstation or even in a single rocessor machine. Utilizing migration and using some number of demes revent converging to local otima. The erformance of CGPEAs mostly des on the migration rate, migration eriod and the selection of migrated individuals. The methods of migration scheme are random migration, imrovement insertion, worst deletion, best migration and crossover migration. One individual of QGA can reresent many states at the same time, and there are weak relationshis during individuals since each individual is determined by current best solution and its robability. Because of this characteristic, QGA is better than EAs such as EP, GA and ES for imlementation of a arallel architecture. The best migration scheme was considered for the method of migration in this aer. 4 Exeriment The knasack roblem, a kind of combinatorial otimization roblem, is used to investigate the erformance of PQGA. The knasack roblem can be described as selecting from among various items those items which are most rotable, given that the knasack has limited caacity. The -1 knasack roblem is described as: given a set of m items and a knasack, select a subset of the items so as to maximize the rot f (x): subject to f (x) = i x i ; w i x i» C; where x =(x 1 x m ), x i is or 1, i is the rot of item i, w i is the weight of item i, and C is the caacity of the knasack. In this section, some conventional GA methods used in the -1 knasack roblem are described, and this is followed by the detailed algorithms of QGA and PQGA for the knasack roblem. 4.1 Conventional GA methods Three tyes of conventional algorithms are described and tested: algorithms based on enalty functions, algorithms based on reair methods, and algorithm based on decoder [1]. In all the algorithms based on enalty functions, a binary string of the length m reresents a chromosome x for the roblem. The rot f (x) of each string is determined as f (x) = i x i Pen(x); where Pen(x) is a enalty function. There are many ossible strategies for assigning the enalty function [11, 12]. Two tyes of enalties are considered, such as logarithmic enalty and linear enalty: Pen 1 (x) = log 2 (1 + ρ ( P m w ix i C)) ;

4 Pen 2 (x) =ρ ( P m w ix i C) ; where ρ is max m f i =w i g. In algorithms based on reair methods, the rot f (x) of each string is determined as f (x) = i x i; where x is a reaired vector of the original vector x. Original chromosomes are relaced with a 5% robability in the exeriment. The two reair algorithms considered here differ only in selection rocedure, which chooses an item for removal from the knasack: Re 1 (random reair): The selection rocedure selects a random element from the knasack. Re 2 (greedy reair): All items in the knasack are sorted in the decreasing order of their rot to weight ratios. The selection rocedure always chooses the last item for deletion. A ossible decoder for the knasack roblem is based on an integer reresentation. Each chromosome is a vector of m integers; the i-th comonent of the vector is an integer in the range from 1 to m i +1. The ordinal reresentation references a list L of items; a vector is decoded by selecting aroriate item from the current list. Dec 1 (random decoding): The build rocedure creates a list L of items such that the order of items on the list corresonds to the order of items in the inut le which is random. 4.2 QGA for the knasack roblem The algorithm of QGA for the knasack roblem is based on the structure of QGA roosed and it contains a reair algorithm. The algorithm can be written as follows: rocedure QGA t ψ initialize Q(t) make P (t) by observing Q(t) states reair P (t) evaluate P (t) store the best solution b among P (t) while (t <MAX GEN) do t ψ t +1 make P (t) by observing Q(t 1) states reair P (t) evaluate P (t) udate Q(t) store the best solution b among P (t) A qubit string of length m reresents a linear suerosition of solutions as in (6) to the roblem. The length of a qubit string is the same as the number of items. The i-th item can be selected for the knasack with robability j i j 2 or (1 jff i j 2 ). Thus, a binary string of length m is formed from the qubit string. For every bit in the binary string, we generate a random number r from the range [::1]; ifr>jff i j 2, we set the bit of the binary string. The binary string x t j, j =1; 2; ;n, of P (t) reresents a j-th solution to the roblem. For notational simlicity, x is used instead of x t j in the following. The i-th item is selected for the knasack iff x i =1, where x i is the i-th bit of x. The reair algorithm of QGA for the knasack roblem is imlemented as follows: rocedure reair (x) knasack-overlled ψ false if P m w ix i >C then knasack-overlled ψ true while (knasack-overlled) do select an i-th item from the knasack x i ψ if P m w ix i» C then knasack-overlled ψ false while (not knasack-overlled) do select a j-th item from the knasack x j ψ 1 if P m w ix i >C then knasack-overlled ψ true x j ψ P m The rot of a binary solution x is evaluated by ix i, and it is used to nd the best solution b after the udate of q j, j =1; 2; ;n. A qubit chromosome q j is udated by using the rotation gate U ( ) of (7) in this algorithm. The i-th qubit value (ff i ; i ) is udated as»»» ff i cos( i ) sin( i ) f = : (8) sin( i ) cos( i ) i i In this knasack roblem i is given as s(ff i i ) i. The arameters used are shown in Table 1. For examle, if the condition, f (x) f (b), is satised and x i and b i are 1 and, resectively, we can set the value of i as :25ß and s(ff i i ) as +1, 1,or according to the condition of ff i i so as to increase the robability of the state j1i. The value of i has an effect on the seed of convergence, but if it is too big, the solutions may diverge or have a remature convergence to a local otimum. The sign s(ff i i ) determines the direction of convergence to a global otimum. The looku table can be

5 s(ff i i ) x i b i f (x) i ff i i ff i i ff i i f (b) > < = = false true 1 false 1 true :5ß 1 +1 ±1 1 false :1ß 1 +1 ±1 1 true :25ß +1 1 ±1 1 1 false :5ß +1 1 ±1 1 1 true :25ß +1 1 ±1 Table 1: Looku table of i, where f ( ) is the rot, s(ff i i ) is the sign of i, and b i and x i are the i-th bits of the best solution b and the binary solution x, resectively. Figure 1: The connection structure for 4 rocessors. used as a strategy for convergence. This udate rocedure can be described as follows: rocedure udate (q) i ψ while (i <m) do i ψ i +1 determine i with the looku table obtain (ff i ; i ) as: [ff i i ]T = U ( i )[ff i i ] T q ψ q The udate rocedure can be imlemented using various methods with aroriate quantum gates. It des on a given roblem. 4.3 PQGA for the knasack roblem The structure of each rocessor was same as the rocedure of QGA as mentioned reviously. The coarse-grained arallel scheme and the best migration method were used in these exeriments. In QGA, the best solution migrated from other rocessor cannot cause raid convergence to ossibly local otimum, since QGA has a robabilistic reresentation including the meaning of history of evolved individual. To reduce the load of communication between each rocessors, new structures were used as Figure 1 and 2. Figure 1 and 2 show the connection structures for 4 rocessors and 16 rocessors, resectively. Figure 2: The connection structure for 16 rocessors. 5 Results In all exeriments strongly correlated sets of unsorted data were considered: w i = uniformly random[1; 1) i = w i +5: Average knasack caacity given by: C = 1 2 was used. The oulation size considered for all the six conventional genetic algorithms (CGAs) was equal to 1. As in [1], robabilities of crossover and mutation were xed as.5 and.1, resectively. The oulation size of QGA1 was equal to 1, and the oulation size of QGA2 was equal to 48, this being the only difference between QGA1 and QGA2. The number of rocessors of PQGA1 and PQGA2 were equal to 4 and 16, and the suboulation sizes in each rocessor of PQGA1 and PQGA2 were equal to 12 and 3, resectively. The migration eriod of PQGA1 is 2 generations. In PQGA2, there are 4 sub-grous in the overall structure, and each sub-grou includes 4 rocessors. There are 3 suboulations in each rocessor. The migration between the rocessors occurs every 2 generations, and the w i

6 ] of CGAs QGAs PQGAs items Pen1 Pen2 Re1 Re2 Dec1 P 2R1 QGA1 QGA2 PQGA1 PQGA2 b rots m w t(sec=run) b rots m w t(sec=run) b rots m w t(sec=run) Table 2: Exerimental results of the knasack roblem: the maximum number of generations 1, the number of runs 3. P 2R1 means the algorithm imlemented by P en2 and Re1, and b:, m:, and w: means best, mean, and worst, resectively. t(sec=run) reresents the elased time er one run. migration between the sub-grous takes lace every 5 generations. As a erformance measure of the algorithm we collected the best solution found within 1 generations over 3 runs, and we checked the elased time er one run. A Pentium-III 8MHz was used, running Visual C Table 2 shows the exerimental results of the knasack roblems with 1, 25, and 5 items. In the case of 1 items, PQGA yielded suerior results as comared to all the other CGAs and QGAs. The CGA designed by using a linear enalty function and random reair algorithm outerformed all other CGAs, but is behind QGAs as well as PQGAs in erformance. The results show that PQGAs and QGAs erform well in site of small size of oulation. Judging from the results, PQGAs and QGAs can search solutions near the otimum within a short time as comared to CGAs, and although the total oulation number of PQGAs is equal to that of QGA2 (oulation size 48), PQGAs outerformed QGA2 in best solution and comutation time. In the cases of 25 and 5 items, the exerimental results again demonstrate the sueriority of PQGAs. Figure 3 shows the rogress of the mean of best rots and the mean of average rots of oulation found by PQGA1, PQGA2, QGA1, QGA2 and CGA over 3 runs for 1, 25, and 5 items. PQGAs erforms better than QGAs and CGAs in terms of convergence rate and nal results. Initially, QGA2 including 48 oulations shows the fastest convergence rate. PQGAs shows a slower convergence rate than QGAs due to its small oulation number in one rocessor. But in 2 generations, PQGAs outace QGA2. It is caused by the best migration between rocessors. Esecillay, in the cases of 25 items and 5 items, PQGA2 outerforms PQGA1 in best solution. The reason is that the structure of PQGA2 can increase the oulation diversity. PQGAs nal results are better than QGA s and CGA s in 1 generations. The tency of convergence rate can be shown clearly in the results of the mean of average rots of oulation. In the ning, convergence rates of all the algorithms increase. But CGA maintains a nearly constant rot due to its remature convergence, while QGA aroaches towards the neighborhood of global otima with a constant convergence rate. Esecially, after migrations, PQGAs have a faster convergence rate to nd out global otima. The exerimental results demonstrate the effectiveness and the alicability of PQGA. Esecially, gure 3 shows excellent global search ability and sueriority of convergence ability of PQGA. 6 Conclusions This aer roosed a new arallel evolutionary algorithm, PQGA with a quantum reresentation and a coarse-grained arallel scheme. Since QGA is based on the rinciles of quantum comuting such as concets of qubits and suerosition of states, it can reresent a linear suerosition of states, and there is no need to include many individuals. QGA has excellent ability of global search due to its diversity caused by the robabilistic reresentation, and it can aroach better solutions than CGA s in a short time. These characteristics of QGA are suitable for arallel structure. Also, the migration in PQGA can imrove the caability of exloitation and exloration. The knasack roblem, a kind of combinatorial otimization roblem, is used to discuss the erformance of PQGA. It was shown that PQGA s convergence and global search ability are suerior to QGA s and CGA s. The exerimental results demonstrate the effectiveness and the alicability of PQGA.

7 Best rots Average rots Best of PQGA2 Best of PQGA1 Best of QGA2 Best of QGA1 Average of PQGA2 Average of PQGA1 Average of QGA2 Average of QGA Best of CGA Average of CGA (a) best rots (1 items) (b) average rots (1 items) Best rots Average rots Best of PQGA2 Best of PQGA Best of QGA2 146 Best of QGA1 Average of PQGA2 15 Average of PQGA1 Average of QGA2 145 Average of QGA Best of CGA Average of CGA (c) best rots (25 items) (d) average rots (25 items) Best rots Average rots 3 3 Best of PQGA2 Average of PQGA Best of PQGA1 29 Average of PQGA1 285 Best of QGA2 285 Average of QGA2 28 Best of QGA1 28 Average of QGA1 275 Best of CGA Average of CGA (e) best rots (5 items) (f) average rots (5 items) Figure 3: Comarison of PQGAs, QGAs and CGAs on the knasack roblem. The vertical axis is the rot value of knasack, and the horizontal axis is the number of generations. (a), (c), (e) show the best rots, and (b), (d), (f) show the average rots. Both were averaged over 3 runs

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