Method for classification of the computational problems on the basis of the multifractal division of the complexity classes Artem Potebnia

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Artem Potebnia. Method for classification of the computational problems on the basis of the multi- fractal division of the complexity classes. 3rd International Scientific-Practical Conference ”Problems of Infocommunications Science and Technology” (PIC S&T 2016), IEEE, Oct 2016, Kharkiv, Ukraine. pp.1 - 4, ￿10.1109/INFOCOMMST.2016.7905318￿. ￿hal-01592383￿

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Warning: This is the accepted version of the paper. The final version in the publisher's typesetting could be found here: http://ieeexplore.ieee.org/document/7905318/ Method for Classification of the Computational Problems on the Basis of the Multifractal Division of the Complexity Classes

Artem Potebnia Computer Engineering Department Kyiv National Taras Shevchenko University Kyiv, Ukraine [email protected]

Abstract — This paper proposes the method of the example is the greatest common divisor problem multifractal division of the computational complexity classes, represented in the decision form. which is formalized by introducing the special equivalence relations on these classes. Exposing the self-similarity Therefore, the internal structure of the P class is properties of the complexity classes structure, this method similar to the structure of the wider NP class. This allows performing the accurate classification of the problems observation demonstrates the self-similarity feature of the and demonstrates the capability of adaptation to the new nested complexity classes structure. Based on these advances in the computational complexity theory. considerations, this paper proposes the new method for the proper classification of the computational problems by Keywords — computational problem; P and NP classes; means of the multifractal division of the complexity reduction; equivalence relation; multifractal classes. Within the deflation approach, the fractal division is I. INTRODUCTION constructed by the iterative application of the generator, In order to form the classification of the computational which represents the decomposition rule, to the specified problems, they are grouped into separate classes from the initiator and its smaller copies. The well-known perspective of their complexity. The problems belonging illustrations of the fractal division are the pinwheel, half- to each class have the same type (decision, function, hex, sphinx and many others. For example, in the case of counting, etc.), identical computation model and similar the pinwheel division, the initiator is represented by the requirements to time and space resources [1]. The most right-angled triangle with legs in a ratio of 1:2. The important complexity classes are indicated based on the generator performs the decomposition of such triangle into following concept of reduction: five homothetic triangles that all are smaller copies of the initiator [2]. Definition 1. The reduction ≤ poly BA of the Notice that the multifractal division is more general computational problem A to another problem B is than the regular one, because it allows the application of presented by transformations of the problem instances and multiple generators. The remainder of this paper focuses solutions, respectively denoted by f and h, which have the on the investigation of the division iteration that carries polynomial time complexity. out the detailing of the internal structure of the P class. Let us consider the NP class, which contains the decision problems whose solutions can be verified in II. INTERNAL STRUCTURE OF THE P CLASS polynomial time. Being a preorder on this class, the reduction relation ≤ allows to indicate inside it the As a warm-up observation, note that the problems poly belonging to the P class have different suitability for the NP-complete problems that are the most difficult to solve. development of the parallel solving algorithms. In On the contrary, the class P ⊆ NP covers all problems particular, the problems that are fully deprived of the that can be solved in the polynomial time using the natural parallelism, being inherently sequential, are deterministic Turing machine. Nevertheless, the graph considered as P-complete. On the contrary, the special isomorphism and factoring problems have not been class NC ⊆ P encompasses the problems that allow the included in class P or set of NP-complete problems and efficient parallel implementation of the solving thereby their accurate classification under this approach algorithms. However, the problem of establishing the remains impossible. exact relation between the NC and P classes is still open, ⊂ Moreover, the closer inspection of the P class shows although the assumption NC P is the most common. that it similarly contains the P-complete problems and the The principal requirement for the problems belonging to separate class NC, which respectively encompass the the NC class is formulated as the opportunity to develop hardest and easiest problems according to the criterion of the parallel solving algorithms that achieve the the parallel feasibility. Additionally, the P class also polylogarithmic time complexity k nO )(log using nO c )( contains some problems that are not proven to be either P- parallel processors, where k and c are some constants, complete or included in the NC class. In particular, one while n is the size of input parameters [3].

2016 Third International Scientific-Practical Conference Problems of Infocommunications. Science and Technology PIC S&T’2016 Notice that the following formal definition of the NC Establishing the relation ≤NC BA imposes the following class uses the concept of a Boolean circuit in order to poly identify the property of the natural parallelism in the simple restrictions on the inclusion of the A and B algorithms. problems in the NC class: If B ∈ NC, then A∈ NC and if A∉ NC, then B ∉ NC. Definition 2. The complexity class NC is specified as a family of subsets NCk , i.e. III. IDENTIFICATION OF THE P-COMPLETE PROBLEMS ∞ BASED ON THE NC-REDUCTION RELATION k NC =  NC . Definition 4. The problem L is considered P-complete k =0 if it belongs to the P class and is linked to all other k NC Each subset NC covers the problems decidable by problems ′∈ PL by the NC-reduction ≤ poly LL .' the Boolean circuits of the depth (logk nO ), which are The process of solving the P-complete problems is composed of the polynomial number of gates nO )1( associated with the presence of the fundamental data having at most two inputs. dependencies (such as the read-after-write (RAW), write- after-read (WAR), write-after-write (WAW) situations), Since the NC class is the subset of the P class, it is which results in an inability to develop the efficient fully composed only of the decision problems. In this parallel algorithms. Most commonly, the P-completeness ⊆ regard, its function extension FPFNC is specified in of the problem ∈ PL is determined by specifying the order to encompass the function problems that have the * NC * high parallel feasibility. Note that the formal criteria for relation ≤ poly LL to the known P-complete problem L . the problem inclusion in the NC and FNC classes are the same. The only difference between the problems In the paper [4], Ladner has proven that the Circuit belonging to these classes occurs only in the form of the Value Problem (CVP) serves as one of the basic P- solution representation. In particular, the circuits complete problems, similarly to the satisfiability problem corresponding to the FNC class problems have multiple (SAT) in the NP-completeness theory. Instances of the outputs in contrast to the circuits for the NC class CVP problem are presented in the form of sequences problems, which are limited to the presence of only one = 0, ..., CCC n composed of the input values Ci = ,0 output. All the problems within the NC class are linked by C = 1 and the basis functions = CC , ∧= CCC , the special NC-reductions in order to determine their i ji kji relative parallel feasibility. ∨= CCC kji , applied to the previous elements of C, i.e. Definition 3. The NC-reduction of the decision , < ikj . At the same time, the element Cn designates the problem A to another decision problem B is denoted by output gate of the circuit C, while its calculated value NC represents the solution of the CVP problem. ≤ poly BA and represents the special case of the general reduction relation where the mapping f could be computed Moreover, the CVP problem has several variations, in the polylogarithmic time using the polynomial number such as TopCVP, MCVP, NANDCVP, NORCVP and of processors, i.e. f ∈ FNC. PCVP, which also are P-complete and differ in the restrictions imposed on the construction of the problem Similar to the general reduction relation, the NC- instances C. Together, these variations provide a powerful reduction is a preorder on the P class, which means that a basis for establishing the reductions to all other problems. NC pair ()P, ≤ poly is a preordered set of problems.

NC d Fig. 1. Example of the NC-reduction MCVP ≤ poly MFP

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As an illustration, let us show the P-completeness of total capacity, because only completely filled flows pass the maximum flow problem MFPd (formulated in the through the specified partition. This means that the decision form as the determination of the flow value standard flow is the maximum in the network G′. NC d Excluding the arcs used for balancing the network, only parity) by specifying the reduction MCVP ≤ poly .MFP flow passing through the edge n ,( tv ) can have even or Instances of the MCVP problem are represented by the odd value depending on the output of the element C . monotone circuits C constructed over the incomplete basis n {}, ∨∧ without application of the negation elements This observation provides the basis for establishing the mapping h between the solutions of the CVP and MFPd = CC ji . The transformation of instances f converts such problems. Fig. 1 demonstrates the example of the monotone circuits C into networks ′ = ,( EVG ′′ ) NC d reduction MCVP ≤ poly ,MFP which clearly illustrates composed of the vertices ′ = {}{}, 10 , ..., n  , tsvvvV , all above considerations. where s and t designate, respectively, the source and sink nodes of the network. The edges of the network ′∈ Ee ′ IV. DIVISION OF THE COMPLEXITY CLASSES INTO THE and their capacity values ec ′)( are formed by EQUIVALENCE CLASSES implementing the following operations: NC NC The specification of both ≤ poly BA and ≤ poly AB 1. The input gates Ci = 1 of the circuit instance C are reductions means that the A and B problems are equivalent ′ NC reflected in the structure of G by attaching the under the ≤ poly relation and have the same parallel − jn arcs ,( vs i ) with capacities ,( vsc i = .2) feasibility in terms of the NC class theory. In order to 2. The network is expanded by adding the edge describe such case in detail, let us introduce the special − jn subrelation ~NC ≤⊆ NC . ) ,( vv ij ) with ,( vvc ij = 2) for every argument poly poly

C j of the function implemented by the Ci NC Definition 5. The binary relation ~ poly over the P element. class is defined as the intersection of the original ≤NC 3. The network vertex vn that corresponds to the poly output gate of the circuit is linked to the sink node NC −1 relation with the inverse relation ≤ poly , i.e. by attaching the arc n ,( tv ) with n ,( tvc = .1) − 4. For each gate ∧= CCC kji , the network structure NC NC NC 1 . ~ poly poly  ≤≤= poly . is modified by forming edges ,( tv ) , whose i In this expression, the relation resulting from the capacity values are limited only by the condition of operation of the relations intersection is composed of all the flows conservation. pairs of problems belonging to both relations that are the 5. In a similar manner, the elements ∨= CCC kji are arguments of the operation, i.e.

reflected in the network structure by adding the NC  NC NC −1 poly =  BABA ),(|),(~ ≤∈ poly and BA ),( ≤∈ poly . backward connections to the source node i ,( sv ).   Notice that the edges ,( tv ) and ,( sv ) excepting The inverse relation, in turn, is obtained from the i i original one by switching the order of problems in each n ,( tv ) serve for diverting the redundant flows and pair, i.e. balancing the overall network. Moreover, the described algorithm f performs the construction of the network G′ NC −1 NC poly =≤ { ABBA ),(|),( ≤∈ poly } . by processing the circuit elements in the independent NC manner, which means that f ∈ FNC. Let us combine the Theorem 1. The relation ~ poly is an equivalence intermediate vertices of the network = ′ \ {}, tsVV into the relation on the P class. ∈ T and F subsets such that the nodes Tv correspond to ▲ In order to prove this theorem, we need to show that the the true gates of the circuit, while the vertices ∈ Fv NC correspond to the false ones. The network G′ holds the relation ~ poly holds the properties of the reflexivity, standard flow formed by filling the flows passing through symmetry, and transitivity. Above all, notice that the the edges t ,( vv ) up to their capacity and assigning zero properties of the reflexivity and transitivity are stable under the operation of the inverse relation formation. This values to the flows that correspond to the edges f ,( vv ), NC NC −1 implies that both ≤ poly and ≤ poly relations are where t ∈Tv , f ∈ Fv and ∈Vv . In addition, the specification of the standard flow requires setting reflexive and transitive and, consequently, are preorders 1) ,( tvf = 1) if C = .1 on the P class. The following part of the proof is organized n n NC into three stages, which show that the intersection ~ poly In order to explore in detail the standard flow, let us preserves the reflexivity and transitivity properties of both consider the (s, t)-cut { },  tFsT }{ of the network relations and additionally is symmetric. G′. Note that the overall flow in such cut is equal to its

2016 Third International Scientific-Practical Conference Problems of Infocommunications. Science and Technology PIC S&T’2016 −1 NC NC NP poly .~/ Therefore, we can conclude that based on the Since the relations ≤ poly and ≤ poly are reflexive, it set NP poly ,~/ the generator performs the formation of NC NC −1 follows that AA ),( ≤∈ poly and AA ),( ≤∈ poly for any the subsequent iteration of the fractal division by replacing problem ∈ PA . Then, by the definition of the intersection, the equivalence class []E that contains the easiest ~ ~ poly NC we have that AA ),( ∈ ~ poly , which means that the relation NC problems with the quotient set P poly .~/ NC ~ poly is reflexive. Assume there exist problems A, B and C such that NC NC BA ),( ∈ ~ poly and CB ),( ∈ poly .~ From the definition of NC the intersection, we receive that BA ),( ≤∈ poly , NC NC −1 NC −1 CB ),( ≤∈ poly , BA ),( ≤∈ poly and CB ),( ≤∈ poly . NC NC −1 Moreover, CA ),( ≤∈ poly and CA ),( ≤∈ poly , because both relations are transitive. Finally, applying the definition of the intersection again, we obtain that NC CA ),( ∈ poly .~ This clearly shows the transitivity of the NC ~ poly relation.

NC Suppose that BA ),( ∈ ~ poly for some A and B Fig. 2. Simplified example of specifying the collection of the problems. As in the previous stage, considering the equivalence classes inside the sample class P NC definition of the intersection, we have that BA ),( ≤∈ poly −1 V. CONCLUSIONS and BA ),( ≤∈ NC . Applying the operation of the poly The method for classification of the computational inverse relation formation, we receive that problems based on the equivalence classes of the relations −1 −1 −1 NC NC  NC  ~ poly and ~ poly allows for a much more detailed AB ),( ≤∈ poly and AB ),( ≤∈ poly  . Obviously, the   description of the internal structure of the complexity NC classes. In particular, such approach is sufficiently flexible last expression is equivalent to AB ),( ≤∈ poly . Therefore, NC from the definition of the intersection, we obtain that to address the situations where NC = P ( P ~/ poly NC NC ⊂ AB ),( ∈ poly ,~ which shows that the relation ~ poly is contains just one equivalence class) or NC P (in this case, the P class can be divided into any number of subsets symmetric and finally completes the proof. ▼ by introducing the corresponding equivalence classes). In the light of the above theorem, we can conclude that NC the relation ~ poly splits the P class into a collection of REFERENCES disjoint equivalence classes, forming a quotient set [1] A. Potebnia and S. Pogorilyy. “Investigation of transformations and landscapes for combinatorial optimization problems,” in 2016 13th International Conference on Modern Problems of Radio NC   P ~/ poly = [] ~ NC ∈ PEE .| Engineering, Telecommunications and Computer Science (TCSET),  poly  pp.510-514, 23-26 Feb. 2016. Here each equivalence class []E ~ NC is denoted by [2] M. Senechal. Quasicrystals and Geometry. New York: Cambridge poly University Press, 1996. some representative problem E and covers all problems [3] A. Potebnia and S. Pogorilyy. “Fast output-sensitive approach for NC minimum convex hulls formation,” in Recent Advances in that are related to E by ~ poly . Therefore, the problems Computational Optimization: Results of the Workshop on that have the same parallel feasibility in terms of the NC Computational Optimization WCO 2015, Vol. 655. Cham: Springer class theory are grouped in one equivalence class. Fig. 2 International Publishing, 2016, pp.1-20. presents the simplified example of splitting the sample [4] R. Ladner. "The circuit value problem is log space complete for P," class P containing five problems into the equivalence SIGACT News, vol. 7, pp.18-20, 1975. NC classes of the subrelation ~ poly on the basis of the pre- NC defined relation ≤ poly . In a similar manner, we can specify the equivalence −1 relation ~ poly poly  ≤≤= poly over the wider NP class, thereby forming the corresponding quotient set

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