Investigation of Transformations and Landscapes for Combinatorial Optimization Problems

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$approach to the fractal representaton of the complexty classes structure. The nvestgaton of ftness landscapes leads to the establshment of ther key parameters.$

1 Investgaton of Transformatons and Landscapes for Combnatoral Optmzaton Problems Abstract - Ths paper deals wth an analyss of transformatons between combnatoral optmzaton problems and proposes an approach to the fractal representaton of the complexty classes structure. The nvestgaton of ftness landscapes leads to the establshment of ther key parameters. The study results dscover that P and NP class problems can dffer n propertes of ther landscapes. Keywords - Combnatoral optmzaton problem, P and NP classes, Ftness landscape, Reducton. 510 Artem Potebna, Sergy Pogorlyy for whch both A B and B A relatons are specfed, are only the dfferent realzatons of a sngle fundamental problem. I. INTRODUCTION Determnaton of smlartes and dfferences between the combnatoral optmzaton problems s a powerful tool for the effcent algorthms development. In partcular, the reducton method s now wdely used for solvng the new problems, provdng an estmaton of ther complexty and defnng the basc prncples for the formaton of classfcaton [1]. The reducton of the combnatoral optmzaton problem A to another problem B s presented by two transformatons f and h, whch have the nomal tme complexty. Herewth, the algorthm f determnes the mappng of any nstance I for the orgnal problem A to the sample f(i) for the problem B. At the same tme, the algorthm h mplements the transformaton of the global soluton S for the obtaned nstance f(i) to the soluton h(s) for the orgnal sample I. On the bass of these consderatons, any algorthm for solvng the problem B can be appled for calculaton of the problem A solutons by ncludng the specal operatons f and h, as shown n Fg. 1. The reducton descrbed above s denoted by A B. The establshment of such transformaton ndcates that the problem B s at least as complex as the problem A. Therefore, the presence of the nomal tme algorthm for B leads to the possblty of ts development for the orgnal problem A. It should be noted that the convenent property of the reducton relaton s ts transtvty,.e. A B and B C mples that A C. However, ths relaton s not symmetrc. Suppose that there exsts the nomal tme algorthm for solvng the problem A. In ths case, the ndcaton of ts avalablty for the problem B requres the establshment of the reverse reducton B A. Moreover, the problems A and B, Artem Potebna, Sergy Pogorlyy Kyv Natonal Taras Shevchenko Unversty, Acad. Glushkov Ave., 4G, Kyv, 0317, UKRAINE, E-mal: admn@artem.bz.ua Fg.1. Dagram llustratng the reducton of the combnatoral optmzaton problem A to the problem B II. FRACTAL STRUCTURE OF THE COMPLEXITY CLASSES In order to form the classfcaton of the combnatoral optmzaton problems, they are grouped nto separate classes from the perspectve of ther complexty. And the problems belongng to each class have the same computaton model and smlar requrements to tme and space resources []. DTIME f (n), The fundamental famles NTIME f (n), DSPACE f (n) and NSPACE f(n) are organzed as the seres of nested sets, whch total count s about 500. The most mportant among them are the canoncal classes L, NL, P, NP, PSPACE, E, EXP, NEXP and EXPSPACE, whch correspond the logarthmc, exponental and nomal lmtatons of resources nvolved n the calculaton process. Class P s represented by a set of problems whose solutons can be obtaned n a nomal tme usng a determnstc Turng machne. The membershp n ths class has been proven for the computaton of the mnmum spannng trees, convex hulls, shortest dstances n the graphs and many other problems. TCSET'016, February 3 6, 016, Lvv-Slavske, Ukrane

2 511 Accordng to the Cobham-Edmonds poston, the necessary condton for effectve solvng of an arbtrary problem A s ts belongng to class P. Per contra, the assgnment of some problems to class P s nonconstructve, and the algorthms of ther nomal tme calculaton reman unknown. The prme example s the recognton of the graph G sutablty for embeddng on a fxed surface L wthout ntersectons of ts edges. Class NP s defned as a set of problems n the decson form, whch solutons can be verfed n nomal tme usng auxlary expressons c called certfcates or wtnesses. In addton, the certfcates dmensonalty s restrcted to nomal space, whle the process of ther formaton can be extremely dffcult. For example, the problem of Hamltonan cycle G V, E corresponds to constructon for the graph the certfcate represented n the form of sequence c v 1, v,..., v V. Therefore, the verfcaton of the problem soluton requres only the edges detecton between specfed pars of vertces. However, applyng of these consderatons to the travelng salesman problem necesstates ts transformaton nto the calculaton of cycles whch length s less compared to some constant. Most mportantly, the queston of class NP closure under complementaton s stll open, and complement problems L for all L NP are attrbuted to the separate set co-np. Ths s assocated wth the hgh complexty of the certfcates formaton for the Fg.. Complexty classes contanng sets NP and co-np problems L. For example, the complement problem for the Hamltonan cycle computaton requres verfyng the absence of the smple cycles contanng V nodes. Certfcate for such problem can be represented as a set of all possble cycles, but t has the exponental dmensonalty that doesn t meet the elgblty crtera of class NP problems. Class co-np contans a number of mportant problems, ncludng determnaton, whether a gven Boolean expresson s a tautology, and dentfcaton of the expander propertes n graphs. The qualty of expanders s defned by the condton of the strong connectedness, and the area of ther applcaton covers the creaton of correctng codes, extractors, generators of pseudo-random numbers and hash functons. Fg. shows that NP and co-np classes are ncluded n a number of much wder tme and space sets wth the formaton of the correspondng fractal generatons. Due to the self-smlarty propertes, the establshment of the relatonshp P = NP would lead to the collapse of the numerous levels of ths herarchy, for example, the nfnte sets of classes and would be curtaled nto the common class P. In other words, the P versus NP problem s assocated wth the fundamental propertes of ths fractal and s repeatedly reproduced for ts dfferent generatons. However, the ntersecton NP co - NP forms an Edmonds set, whch ncludes class P and, f NP co - NP, doesn t contan any NP-complete TCSET'016, February 3 6, 016, Lvv-Slavske, Ukrane

51 problem. The most modern cryptosystems are based on the factorng problem, whch has a sgnfcant computatonal complexty and assumes the decomposton of the number N nto the set of prme factors p 1, p,.

However, due to the presence of only one possble decomposton, there exst also certfcates for the complement problem, represented by expressons c p, p,.

3 51 problem. The most modern cryptosystems are based on the factorng problem, whch has a sgnfcant computatonal complexty and assumes the decomposton of the number N nto the set of prme factors p 1, p,..., p k. The decson verson of ths problem requres fndng of the factor p b, where b s the specfed threshold. The value p s enough to form the certfcate that mples the problem ncluson n class NP. However, due to the presence of only one possble decomposton, there exst also certfcates for the complement problem, represented by expressons c p, p,..., p, whle the procedure of ther 1 k processng requres fndng factor p b. Therefore, the factorng problem s also ncluded n class co-np and ntersecton NP co - NP. These consderatons lead to the exstence of the problem hdden propertes that can greatly smplfy the process of the soluton formaton. In partcular, the factorng problem can be transformed to the determnaton of the perod r for functon f x a mod N, where a and N are the relatvely N x prme numbers. For example, n the case of N 15 and a ths functon s represented by the sequence f, 4, 8, 1,, 4, 8, 1,..., for whch r 4. The 15 factors p and q are defned as the greatest common r/ dvsors of numbers a 1 and N. In the gven example, the soluton s represented by factors p 5 and q 3. But ths approach s not suffcent for effectve solvng of the problem due to the large dmensonalty of the sequences f x and hgh N complexty of ther processng. However, determnaton of the perod doesn t requre selecton of only one tem and s assocated wth the dentfcaton of the global N propertes nherent to all members of f x. These features provde the possblty of the effectve computaton n a nomal tme by usng a quantum computer archtecture. Specal class BQP covers all problems whch have the smlar propertes. But the relatonshp between classes NP and BQP s stll not establshed. The known relatons are only P BQP and BQP PSPACE due to the nomal lmtaton of memory. The detaled nternal fractal structure of classes NP and co-np wth an estmated poston of BQP set s gven n Fg. 3. It should be noted that the graph somorphsm problem occupes the specal place n the structure of class NP. Despte the belongng of ths problem to class NP, t has not been ncluded n class P or set of NPcomplete problems. In addton, there s no proof of ts accesson to class co-np, whch s assocated wth the hgh complexty of the certfcates formaton. Fg.3. Internal fractal structure of classes NP and co-np TCSET'016, February 3 6, 016, Lvv-Slavske, Ukrane

513 These unque propertes have caused the formaton of the separate class GI and the correspondng set of GIcomplete problems, whch complexty s equvalent to the problem of somorphsm establshng [].

4 513 These unque propertes have caused the formaton of the separate class GI and the correspondng set of GIcomplete problems, whch complexty s equvalent to the problem of somorphsm establshng []. The examples of GI-complete problems are the recognton of self-complementary graphs and determnaton of automorphsm wth dscoverng of the complex symmetry forms. III. SYSTEM OF TRANSFORMATIONS AND THE FUNDAMENTAL PARENT PROBLEM The condton of NP-completeness of the problem L NP s the presence of reductons system L L for all problems L NP. Accordng to the Cook-Levn theorem, the Boolean satsfablty problem (SAT) s a fundamental n NP-complete class. The nstances of SAT problem are represented as formulas n the conjunctve normal form whle the solutons are the sets the varable values that ensure satsfyng of these formulas. k -SAT problem s a partcular case of SAT, n whch each elementary dsjuncton D 1,..., D m contans no more than k lterals. It s mportant that k -SAT belongs to the set of NP-complete problems f k 3. Relatons of the transformatons system cover problems wth dfferent representatons of the nstances and solutons n the form of Boolean expressons, graphs, cyclc structures, subsets, etc. They all are realzatons of a sngle fundamental NP-complete parent problem, adapted for modelng of the real complex systems. Fg. 4 demonstrates the example of transformaton 3-SAT 3- GC, where GC denotes the graph colorng problem. The functon f provdes the formaton of graph G by the ntroducton of the ndvdual vertces for the lterals of the nput expresson. In addton, the structure of the receved nstance s expanded by attachng of the specal basc subgraph consstng of nterconnected C (control), F (false) and T (true) nodes, whch correspond to dfferent color values. The edges set E of the obtaned graph contans the connectons requred for the formaton of the clques between the control node C and vertces, correspondng to expresson varables and ther negatons. The next operatons provde the creaton of the addtonal basc subgraphs for each elementary dsjuncton and ther lnkng wth the node T. Under ths approach, the color assgnment s possble f there s an nput node labeled by color T for each dsjuncton D. The transformaton of solutons h s mplemented on the bass of the vertces dstrbuton by classes T and F. The formaton of reductons 3-SAT 3- GC and 3-SAT HC, where HC denotes the Hamltonan cycle problem, leads to a sgnfcant ncrease n the dmensonalty of nstances after applcaton of mappng f. Therefore, the most evdent presentatons of the sngle fundamental problem at the same tme are the least effectve. Fg.4. Example of the reducton 3-SAT 3- GC IV. LANDSCAPES AND THEIR PARAMETERS The hgh complexty of the optmzaton problems s resultng from the sgnfcant dmensonalty of problem solutons space S, whch makes ts straghtforward enumeraton mpossble, and s known as a combnatoral exploson. Therefore, the local search approach s rendered one of the most effcent methods for solvng them. These methods assume the search space constrcton to the neghborhood of the prescrbed soluton N(s 0) S for copng wth the curse of dmensonalty. In ths case, problem solvng s constraned to fndng local mnmum s 0, for whch F(s 0 ) F(s) wth any s N(s 0). The hgh speed of these methods s accompaned by losng the best global solutons located outsde N(s 0), as shown n Fg. 5. Specfc neghborhood graphs G (S, E) are wdely used for local optmzaton algorthms research [3]. Ther nodes represent possble problem solutons, and connecton system s composed of ordered couples (s, s ), where s N(s). Each node corresponds to the TCSET'016, February 3 6, 016, Lvv-Slavske, Ukrane

5 514 weght coeffcent evaluated by takng nto consderaton the problem s objectve functon. Many studes menton such graphs as ftness landscapes because ther structure has the sgnfcant nfluence on the search effcency. However, assgnment of solutons S dstrbuton by neghborhoods N:S requres consderaton of crtcal parameters for each specfc problem. Fg.5. Search space restrcton appled by local optmzaton methods In partcular, analyss of the accurate neghborhoods that have exponental dmensonalty s one of the most promsng research drectons. Polynomal-tme algorthms, sutable for processng such landscapes and fndng the best soluton wthn the neghborhood, have substantal advantages over other methods. A prme example s a known approach to the constructon of the exponental neghborhood for the travellng salesman problem, whch envsages the lmtaton of the maxmum dstance between any two routes to 4 nodes [4]. Ths value does not depend on the problem dmensonalty, whch allows the optmum soluton fndng just n several teratons. However, attanng these advantages requres the formaton of nomal algorthms for choosng nodes s N(s), whch remans an mpossble task for exponental neghborhoods. Hence, the most effcent local optmzaton methods are lmted nowadays to the usage of nomal search space. Landscape structure complexty s an mportant parameter because a great number of local extremes can deprecate the algorthm s effcency. Thereby, a specal autocorrelaton functon s offered for the symmetrcal landscapes qualty evaluaton: 1 d(s 1, s ) f (s 1) f (s ) f (s ) f (s ) p(d) 1, where d s the dstance between solutons s 1 and s, 1 f (s ) f (s ) denotes the average value of f (s ) f (s ) 1 d(s 1, s ) for all (s 1, s ) couples located n dstance d wthn the graph. Partcularly, f p(d) 1, then the dfferences between neghbor solutons are not sgnfcant that makes a landscape sutable for the local search procedure [3]. Specal autocorrelaton coeffcent values 1/ (1 p(1)) are also used for estmatng the qualty of landscapes. Elementary landscapes represent an exclusve case assocated wth the possblty of establshng wave equatons for the neghborhoods of correspondng problems. Such equatons determne a relaton between f (s) and the average value of the objectve functon for neghbor nodes s N(s). Hgh values receved for the travellng salesman problem ndcate the hgh effcency of usng local search methods for ts solvng. Moreover, the -opt neghborhood has more flat landscape compared to - exchange and provdes better solutons n conformty wth the expermental research. For the graph partton problem, -Flp neghborhood assumes extendng of the search space by two tmes n comparson to Swap. Ths leads to decreasng n the ruggedness of the landscape, whch have a postve nfluence on the search results. V. CONCLUSIONS The foregong results demonstrate the possblty of settng a relaton between the landscape structure and NP-complexty of the correspondng problems. Partcularly, the problem of matchng formaton wth known nomal algorthms dffers n much greater landscape flatness at relatvely low neghborhood graph dameter. Therefore, the dfference between the problems of P and NP classes can be assocated wth the propertes of ther landscape. REFERENCES [1] A. Potebna, S. Pogorlyy, "Innovatve GPU accelerated algorthm for fast mnmum convex hulls computaton," n Computer Scence and Informaton Systems (FedCSIS), 015 Federated Conference on, pp , Sept [] S. Arora and B. Barak, Computatonal complexty: a modern approach. New York: Cambrdge Unversty Press, 009. [3] C. Redys, P. Stadler, "Combnatoral Landscapes," SIAM REVIEW, vol. 44, pp.3-54, 00. [4] G. Gutn, A. Yeo, "Small dameter neghborhood graphs for the travelng salesman problem: four moves from tour to tour," Computers and Operatons Research, vol. 6, pp.31-37, TCSET'016, February 3 6, 016, Lvv-Slavske, Ukrane

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