CC BY
8
УДК 510.52
Analysis Parameterized Algorithms on the Bases of Elasticity to Functions Complexity
Valentina V. Bykova*
Institute of Mathematics, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 30.10.2010, received in revised form 10.11.2010, accepted 20.12.2010 We give a brief overview of results and problems of parameterized algorithmics as the new direction of computational complexity theory. We offer a new indicator of computational complexity for parameterized algorithm which can be used to measure rate a growth of function complexity from many variables. This indicator is a private elasticity of the function complexity. We offer a two-dimensional classification parameterized algorithms to multiplicative forms a presentation of the functions complexity. We give a mathematical basis to analysis a level impact of parameter for time execution of parameterized algorithm.
Keywords: computation complexity, parameterized algorithms, analysis algorithms, elasticity algorithms.
Introduction
According to the classical complexity theory, many problems that have important real world applications are NP-hard [1]. To deal with NP-hard problems, many approaches have been proposed. Approximation algorithms and parameterized algorithms are two of these approaches. Objective of approximation algorithms is to find in polynomial time solution which is close to the optimal solution. A notable and important class of approximate algorithms for solving NP-hard combinatorial optimization problems is fully polynomial-time approximation schemes [2]. Parameterized algorithms try to give exact solutions for NP-hard problems when its natural parameter k is small even if dimension n of the problem (length input data for algorithm) is big. In parameterized complexity theory measure complexity takes into account not only the n, but the numerical parameter k, whose value may depend on n arbitrary way or at all not depend on n [3]. Roles of parameter take into account the information about structure of input data and identify the main sources nonpolynomial complexity of NP-hard problem. Parameterized complexity theory uses two-dimensional functions of computational complexity of algorithms, so it is also called a two-dimensional complexity theory. Whereas the classical (one-dimensional) complexity theory analyzes and classifies problems and algorithms only in terms of resource (time or memory), which is required for the execution of algorithm depending on size of the input data
[1]. Classical one-dimensionality and orientation in analysis of algorithms on the worst case often makes the problems more complicated than they really are. Parameterized complexity theory provides a basis for detailed analysis of the complexity problems that are intractable in classical sense. Parameterized complexity is a fairly new branch of complexity theory. It was developed by Downey and Fellows in the early 1990s [4]. Over past two decades the ideas from parameterized complexity theory have found their way into various areas of computer science, such as artificial intelligence [5], computational biology [6], database theory [7, 8] and etc.
* bykvalen@mail.ru © Siberian Federal University. All rights reserved
Parameterized Problems and Algorithms
The parameterized complexity theory investigates only of the decision problem, i.e. problem, in which we must answer "yes" or "no" to some question. Generality is not lost. This approach is also used in theory of NP-completeness [1].
We give below basic concepts of parameterized complexity theory [3, 4].
A parameterized problem n is language which is defined as L(n) C E* x N, where E is a finite alphabet, E* is set of all words in E , N is set of all nonnegative integers. If (I, k) e L(n), then I is called the main part and integer number k is called the parameter of parameterized problem n . Each particular pair of values (I, k) is an instance of problem and is also input data for algorithm a that solves this problem. The value |11 = n specifies a length the input data for algorithm a (dimension of parameterized problem n).
We say that an algorithm a solves the parameterized problem n, if for each input (I, k) the algorithm a can determine whether (I, k) is a yes-instance of L(n) (i.e. whether (I, k) is an element of L(n)).The algorithm a is said to be parameterized algorithm, if its computational complexity as the need for resources (further only execution time of algorithm) is measured in terms of both input length |11 = n and value of parameter k. Thus, the function complexity of parameterized algorithm that determines the execution time of algorithm is a function of two variables t(n, k).
A parameterized problem n is fixed-parameter tractable problem (FPT-problem), if it can be solved by some parameterized algorithm a in time
t(n, k) = O(nO(1) • f (k)) (1)
for some function f depending only on k. The growth rate of function f(k) is unlimited. For example, f (k) = 2o(k) or f (k) = 2O(k). It is important that we eliminate the function of form f (n, k), for example, f (n, k) = nk. Denote by FPT the class of all fixed-parameter tractable problems. Appropriate parameterized algorithms, that solve such problems, are called FPT-algorithms.
An example, FPT-problem is a problem SATISFIABILITY of formulas propositional logic when parameter k is by the number of variables. In fact, a formula of size n with k variables can be tested by algorithm of brute force in time O(n • 2k). A vertex cover of size k in a graph G with n vertices can also be found by brute force in time O(n • 2k). Therefore, problem VERTEX COVER is FPT-problem where the parameter is the size of the cover. If in problem VERTEX COVER as a parameter to take the treewidth tw(G) of graph G then dynamic programming technique finds a vertex cover of size at most tw(G) in time O(n • 2tw(G)) [9]. Consequently, a parameterization of problem VERTEX COVER with respect tw(G) leads to FPT.
An example, a problem, which does not belong to FPT, is problem GRAPH COLORING (k-coloring of n-vertex graph) with parameter k. It is known that a problem of 2-coloring can be solved in time O(nO(1)). However, 3-coloring is NP-hard problem with respect n [1].
Obviously, FPT contains all polynomial-time computable problems. However, the greatest interest in terms of theory and practice are of FPT-solvable problems that are NP-hard. From a theoretical point of view all these problems can be solved in polynomial time for each fixed parameter value. Meanwhile, really it is possible to implement in most cases only for small values of parameter (it all depends on type of function f (k)). Already there are collections of parameterized problems, including fixed-parameter tractable problems. The most concrete at the moment is a collection Marco Chezati [10].
The parameterized complexity theory is developing on several directions: definition of a class hierarchy of parameterized problems, identification conditions of belonging to the class FPT, identification the relationship between of parameterized complexity and classes of approximate algorithms (in particular, fully polynomial-time approximation schemes), development of parameterized algorithmics (development a methods analysis and design of parameterized algorithms)
and etc. [3]. A parameterized algorithmics is the applied direction of parameterized complexity theory. Here there are a number of open questions.
Some Problems of Parameterized Algorithmics
Analysis of the collection parameterized problems [10] shows that a problem might have different parameterizations with respect to various parameters. An example, in problem VERTEX COVER as a parameter can be size of cover, the number of vertices, the number of edges, treewidth, the maximum degree of vertices to graph, etc. The following questions are natural.
1) What can serve as a parameter for NP-hard problem?
2) How to choose parameter so that a parameterized problem belonged to FPT?
3) Are there special methods for developing FPT-algorithms?
4) How to compare various FPT-algorithms of solving some NP-hard problem?
At some of these questions in parameterized complexity theory not given yet a comprehensive responses. For example, a parameterized complexity theory responds on first two questions follows.
There are internal and external parameters. The parameter is explicit internal parameter, if a parameter appears directly in main part of instance (I, k) for parameterized problem n and its values are bounded by a polynomial respect |11 = n. Such parameters for problem VERTEX COVER are size of cover, number of vertices or number of edges to graph. There are standard parameters. They are most popular. A standard parameter for minimization problems traditionally is a quantity which is necessary to minimize. For problem VERTEX COVER it is size of cover. If k = n then k becomes trivial parameter and parameterized algorithm degenerates into usual algorithm.
Parameter is an implicit internal parameter, if its values are bounded by a polynomial respect |11 = n and information about this parameter is implicitly hidden in I. A typical example: treewidth of graph (values treewidth always no greater than a number of vertices to graph) is an implicit internal parameter. Altogether, all internal parameters characterize in some way a structure of input data for algorithm.
Finally, there are external parameters, which not depend on |11 = n or depend nonpolynomial. Information on these parameters we specify in addition to I, because considered that it no is in I.
From a theoretical point as a parameter can serve any part input data of algorithm. In practice to get good parameterization (for example, related to FPT) is recommended to select such a part input data of algorithm, which usually takes small values as compared with |11 = n. Ultimately, it all depends on application. For different applications may be acceptable different parameterization of NP-hard problem. Thus, the second question as a whole remains until theoretically open. For some NP-hard problems is proved that individual parameterization of these problems not can be implemented FPT-algorithms. As mentioned earlier, this is a problem GRAPH COLORING (k-coloring of n-vertex graph) with parameter k. The some NP-hard problems have the FPT-algorithms to individual parameters. Example, for many optimization problems on graphs, treewidth proved to be such parameter which usually leads to FPT-algo-rithms [9-11].
As regards the third of above questions, then modern theoretical and practical results provide tools are sufficient for algorithmic practice. Currently a toolbox for construction of parameterized algorithms contains set special methods to creating FPT-algorithms. Above all, this dynamic
programming technique based on tree decomposition for graphs with bounded treewidth [11]. Another known method of construction FPT-algorithms is method of parametric reduction (ker-nelization) [4]. A method of parametric reduction is transformation in polynomial time of each instance (I, k) of parameterized problem n into instance (I', k') of parameterized problem n' so that k' < k. After this for the instance (I', k'), which is called the kernel for I, is used brute force. Example, for problem VERTEX COVER with standard parameter k (size of cover) a method of parametric reduction is reduced to a repeated application of two rules: deletion isolated vertices without changing parameter k, deletion a vertex whose degree is more to k and reduction a parameter k on one. It should be noted, that a method of parametric reduction combined with brute force does not always lead to FPT-algorithm. Meanwhile it is proved, that if a problem belongs to FPT then this method invariably gives FPT-algorithm [4].
Fourth question is theoretically open: in numerous papers on parameterized complexity theory to analysis and compare of parameterized algorithms apply methods of classical (onedimensional) complexity theory. This is predominantly different approaches to defining the functions complexity and study asymptotic behavior of these functions for large values n [12-14]. Meanwhile, classical one-dimensionality limits depth of analysis parameterized algorithms, since for them the computational complexity described by function t(n, k) on two variables.
In this paper we propose a measure of computational complexity of the algorithm which can be used for analysis rate a growth of functions to several variables, for analysis a level impact of structure of input data on the computational complexity parameterized algorithms, for comparison of FPT-algorithms solving the NP-hard problems. This measure is the elasticity to function complexity of algorithm. Presented in this paper results are a generalization and development of author’s results, published in [15-17] for the one-dimensional complexity theory.
Agreements Concerning Functions Complexity
A formal approach to analysis of parameterized algorithms requires assumption of some properties a functions complexity t(n, k) of algorithms. By respect functions t(n, k) we make some natural assumptions, which are feasible for most real algorithms and which are necessary for application of apparatus mathematical analysis:
• we believe that t(n, k) is a monotonous nondecreasing function respect both arguments. The range of values this function is the set of nonnegative real numbers and the domain of definition a function is the set N x N;
• we deviate from discreteness n and k (with formal replacement of n by x, of k by y), i.e. we believe that arguments of function t(n, k) are continuous and necessary values are computed at integer points x = n and y = k. According to this assumption, below instead of t(n, k) we write z(x, y);
• we assume that set under consideration of functions is limited family L of "basically positive" logarithmic-exponential functions. A family L was introduced and studied by G. Hardy [18]. Recall that z(x, y) is "basically positive" function, if there exists x0, y0 such, that z(x, y) > 0 for all x > x0, y > y0. It is known that each such function is continuous and differentiable in domain where it is defined.
These assumptions are the guarantee existence of elasticity for functions complexity of parameterized algorithms. Henceforth, we mean that the L-function always is a monotonous nondecreasing, "basically positive", logarithmic-exponential function of family L.
Elasticity as a Measure Computational Complexity of Algorithm
A elasticity Ex(z) to function z = z(x) is the limit ratio of relative increment this function to relative increment of argument [15]:
, . , (Az Ax\ dz dx dz x ,x w
Ex(z) = lim ------ : --- =-: — = -----------= z'- = x(lnz)'. (2)
z x ) zx dx z z
It is obvious, that for any L-function z = z(x) always exist a elasticity and Ex(z) > 0. In
[16] we showed that when x ^ to the elasticity of L-functions is identically constant, infinitely small or infinitely large logarithmic-exponential function for which the estimates are correct:
Ex(z) = inx^), Ex(czm) = O[Ex(z)], c > 0, m> 0.
Last estimate means that when x ^ to behavior of elasticity is invariant with respect to a polynomial transformation, which consists of multiplying function by constant c > 0 and exponentiation m > 0.
It is known that two arbitrary L-functions zi(x) and z2(x) is always comparable in order of growth [18]: if z1(x), z2(x) G L then for x ^ to surely one of three relations is true:
zi(x) -< z2(x), z2(x) -< zi(x), zi(x) = O [z2(x)] .
In addition it is proved, that a hierarchy of elasticities generates an identical hierarchy of L-functions [16], i.e. if Ex(z1) -< Ex(z2) then z1(x) -< z2(x). Note that here the relation -< is interpreted as follows: z1(x) -< z2 (x) if and only if z1(x) = o[z2(x)]. The O-symbol here denotes the asymptotic proportionality of functions: z1(x) = O[z2(x)] if and only if z1(x) ~ cz2(x), c > 0. Further we shall stick to this interpretation of asymptotic notation.
We note additionally several important features of elasticity L-functions. By (2) for sufficiently small Ax the following approximation is correct:
Az Ax
— « Ex(z)------,
zx
which means that Ex(z) is the coefficient of proportionality between a growth rate variables z and x. Elasticity is a dimensionless quantity, since Ex(z) = Eax(bz) for all constants a > 0, b > 0. Of the elasticity inherent properties similar to the properties of logarithm and differentiation operations [17], so the elasticity to logarithmic-exponential functions can be easily calculated.
All these properties allow the use elasticity as a measure computational complexity of algorithm, because elasticity satisfies essential requirements such as comparability, interpretability and computability. Dimensionless of elasticity and invariance with respect to calculations model is the guarantee comparability: because transition from one model to another is only a polynomial transformation of computational complexity algorithm. Elasticity has a clear interpretation: if z = z(x) is a function computational complexity of algorithm, this means that when value of x (length input data of algorithm) increases by one percent a value of z (execution time of algorithm) will increase by about Ex(z) per cent. Computability also holds: elasticity exists for every L-function, elasticity of L-function z = z(x) can always be found directly from (2) or from properties of elasticity. Ease of computation is the main advantage of elasticity in comparison with other measures computational complexity of algorithms.
Elasticity is a local characteristic of function: in general its values change at transition from one value to another value of argument. Meanwhile, for large values of argument in the behavior of elasticity observed certain regularity: different classes of L-functions (different in growth rate) have fundamentally different behavior of elasticity and on basis asymptotic of elasticity can define a class to which belongs L-function. Following theorem states that this is true.
Theorem 1 (on classification of L-functions [16]). Splitting family of monotonous nondecreasing, "basically positive" L-functions on the classes Subpoly, Poly, Subexp, Exp, Hyperexp in order of their growth is equivalent a proper splitting according to asymptotic behavior of elasticity
these functions at infinity:
Subpoly = {z(x) | z(x) -< eO(lnx)} = {z(x) | Ex(z) = o(1)}; (3)
Poly = {z(x) | z(x) = O[eO(lnx)]} = {z(x) | Ex(z) = O(1)}; (4)
Subexp = {z(x) | eO(lnx) -< z(x) -< eO(x)} = {z(x) | 1 -< Ex(z) -< x}; (5)
Exp = {z(x) | z(x) = O[eO(x)]} = {z(x) | Ex(z) = O(x)}; (6)
Hyperexp = {z(x) | eO(x) -< z(x)} = {z(x) | x -< Ex(z)}. (7)
A theorem on classification of L-functions generates five classes complexity of algorithms (subpolinomicial, polynomial, subexponential, exponential and hyperexponential respectively), that fully meet modern classification of algorithms. Properties of elasticity and equivalence (3)-
(7) make "transparent" the classes complexity of algorithms, which is extremely important for algorithmic practice.
The classes of complexity are needed primarily to compare algorithms. In one-dimensional theory of complexity for recognition of class to which belongs to an algorithm with a function of z(x) Є L, we perform following steps: we calculate Ex(z), we find an asymptotic estimate for Ex(z) when x —— to and we define class complexity using (3)-(7). It should be noted that hierarchy classes of L-functions generates the hierarchy classes of complexity algorithms:
Subpoly -< Poly -< Subexp -< Exp -< Hyperexp.
Here Ki -< K2, where Ki = and Ki,K2 Є {Subpoly, Poly, Subexp, Exp, Hyperexp}, means that for any z1(x) Є K1,z2(x) Є K2 always true z1(x) ^ z2(x).
Suppose you want to compare algorithms a1 and a2 with functions z1(x) and z2(x) respectively. First, we need to establish of classes complexity for a1 and a2. If these algorithms belong to different classes then hierarchy of these classes defines a relation between a1 and a2 in terms the execution time of algorithms. When algorithms a1 and a2 belong to same class then much depends on this class:
• if z1(x),z2(x) Є Subpoly, Subexp, Hyperexp then for Ex(z1) Ex(z2) is always true z1(x) ^ z2(x), i.e. algorithm a1 is asymptotically time-faster than algorithm a2;
• if z1(x), z2(x) Є Poly, Exp then Ex(z1) ~ cEx(z2), c > 0. When great length input data of algorithm and 0 < c < 1 the execution time of algorithm a1 less approximately 1/c times than the execution time of algorithm a2. When c > 1 on the contrary, algorithm a1 is asymptotically time-slower approximately c times than algorithm a2. For c =1 is required the analysis in Ex(z1), Ex(z2) members of a lower order than constant.
Thus, the use of elasticity can significantly simplify process of comparing algorithms. Because elasticity has simpler form than of the appropriate L-function, see property 4 of elasticity and
equivalence (3)-(7). The concept of elasticity also extends to functions of many variables. This makes it possible to expand the range of applications of elasticity, for example, use this indicator in analysis of parameterized algorithms.
A private elasticity Ex(z) to function z = z(x,y) respect argument x is elasticity variable z, which is considered a function only of x and at constant y. A private elasticity and private derivative respect argument x associated relation:
x
Ex(z) = zX z = x(ln z)X- (8)
A private elasticity Ey(z) to function z = z(x, y) respect argument y defined as follows:
Ey (z) = zy y = y(ln z)y • (9)
Since every L-function z = z(x,y) is continuous and differentiable in domain where it is defined, then for it in this domain always exist a partial elasticity. Expressions (8), (9) are similar to (2).
Consequently, Ex(z) and Ey (z) have all basic properties of elasticity one variable. We mention
the most important of these properties and we write them for Ex(z).
1) If z = z(x,y) does not depend on x then Ex(z) = 0.
2) A private elasticity is dimensionless quantity: Ex(z) = Eax(bz).
3) A private elasticity of product (of relation) functions zi = zi(x, y) and z2 = z2(x, y) is equal
to sum (difference) of their private elasticity:
Ex(zi • z2) = Ex(zi) + Ex(z2), Ex(zi/z2) = Ex(zi) - Ex(z2).
4) A private elasticity for L-functions of general form z(x,y) = ew(x,y), w(x,y) G L, given by:
Ex[ew(x’y)]= w(x,y)Ex[w(x,y)].
5) A elasticity to function z = z(x,y) respect argument t, where x = x(t) and y = y(t), can express in terms of partial elasticity by formula:
Et(z) = Ex (z)Et(x) + Ey (z)Et(y).
Let z = z(n, k) is a function complexity for parameterized algorithm, where n is length input data of algorithm and k is parameter of algorithm. After the formal replacement of n by x, of k by y we have z = z(x,y). We assume that z = z(x, y) G L. Then Ex(z) is coefficient of proportionality between a growth rate the execution time and the length input data of parameterized algorithm. Similarly, Ey(z) is coefficient of proportionality between a growth rate the execution time and the parameter y.
Two-Dimensional Classification FPT-algorithms Respect to the Complexity
In general, the partial elasticity of Ex(z),Ey(z) are functions that depend on two variables x and y. However, a situation is considerably simplified if we consider the fact that for most of parameterized algorithms the execution time of algorithm is described by L-function:
z(x, y) = q(x) • f (y), (10)
where q(x) G L is a quantitative component, and f(y) G L is a parametric component for function z(x,y). The multiplicative form of (10) is typical for algorithms with parametrically dependent cycle that performs an exhaustive search of all possible options with different values of parameter, whereas in body of this cycle performed test the current version and time test depends only on the length of input data algorithm. Method of parametric reduction generates it such algorithms. According to (1), a multiplicative form of (10) for the FPT-algorithms is represented formula:
z(x, y) = O(xO(i) • f (y)). (11)
Let z(x,y) = q(x) • f(y) G L. By properties of elasticity of 1 and 3, partial elasticity of
Ex(z), Ey(z) degenerate into conventional elasticity of function one variable:
Ex(z) = Ex[q(x) • f(y)] = Ex[q(x)] + Ex[f(y)] = Ex[q(x)L (12)
Ey(z) = Ey[q(x) • f(y)] = Ey[q(x)] + Ey[f(y)] = Ey[f(y)]. (13)
Now Ex(z) depends only on x, and Ey(z) depends only on y. Since q(x),f(y) G L then each of these functions belongs to only one class a Subpoly, Poly, Subexp, Exp, Hyperexp respect corresponding argument.
We denote by Kx the class complexity for L-functions z(x, y) = q(x) • f (y) respect argument x and denote by Ky the class complexity for z(x,y) = q(x) • f (y) respect argument y. Then for
every parameterized algorithm with a function complexity of z(x, y) = q(x) • f (y) G L there exist
a pair:
(Kx, Ky) G {Subpoly, Poly, Subexp, Exp, Hyperexp}x{Subpoly, Poly, Subexp, Exp, Hyperexp},
which characterizes the complexity of this algorithm respect a length of input data x and respect value of parameter y. Thus, we arrive at two-dimensional classification parameterized algorithms by respect complexity.
When we rely on two-dimensional approach, then more accurately and a general statement gets the definition of FPT-algorithm and a condition of (11): parameterized algorithm called FPT-algorithm, if the time of its execution represent by function z(x,y) = q(x) • f (y) G L, for which is true formula:
(Kx,Ky) G {Subpoly, Poly} x {Subpoly, Poly, Subexp, Exp, Hyperexp}.
For example, FPT-algorithm for which (Kx, Ky) = (Poly, Subpoly) may has z = z(x,y) of the form:
z(x, y) = x3 • yA(ln y)m-1 = e3ln x • eA(ln y)m, A > 0, 0 < m < 1,
because Ex(z) = 3 = O(1) for x ^ to , Ey(z) = Am(lny)m-i = o(1) for A > 0, 0 < m < 1 and
y ^ to. The FPT-algorithm with function
z(x, y) = (ln x)x2 • yA(ln y)m-1 = eln ln xe2 ln x • eA(ln y)m, A > 0, m > 1
characterized by complexity (Kx,Ky) = (Poly, Subexp), as Ex(z) = 1/ lnx + 2 = O(1) for
x ^ to, Ey(z) = Am(lny)m-i = o(y) for A > 0, m > 1 and y ^ to. If
z(x, y) = x5 • eAy = e5ln x • eAy, A > 0, (14)
then FPT-algorithm has complexity (Kx, Ky) = (Poly, Exp), since Ex(z) = 5 = O(1) for
x ^ to, Ey(z) = Ay = O(y) for A > 0 and y ^ to. The FPT-algorithm with function
z(x, y) = ln x • yy = elnln x • ey ln y
has complexity (Kx,Ky) = (Subpoly, Hyperexp), because Ex(z) = 1/lnx = o(1) for x ^ to and Ey(z) = y lny(1 + 1/lny) = O(y lny) for y ^ to.
When a parameterized problem is not only FPT, but polynomial-time computable problem, then it has FPT-algorithms whose complexity corresponds to pairs
(Kx,Ky) G {Subpoly, Poly} x {Subpoly, Poly}. (15)
Such parameterized algorithms are naturally called polynomial FPT-algorithms. Parameterized algorithms whose complexity corresponds to pairs
(Kx,Ky) G {Subpoly, Poly} x {Subexp}, (16)
(Kx,Ky) G {Subpoly, Poly} x {Exp}, (17)
(Kx,Ky) G {Subpoly, Poly} x {Hyperexp}, (18)
it is expedient to named as subexponential, exponential and hyperexponential FPT-algorithms respectively. Such FPT-algorithms are specific to NP-hard problems.
Two-dimensional classification FPT-algorithms does not contradict concepts that are currently used in parameterized complexity theory as applied to FPT-algorithms, but merely formally clarifies them. For example, today there is a demand for subexponential FPT-algorithms for solving of parameterized versions NP-hard problems [19, 20]. It is considered that for such algorithms of time execution equal to
z(x, y) = xO(i) • 2o(y)
and satisfies (17). It is obvious, that there always the Cartesian product
{Subpoly, Poly, Subexp, Exp, Hyperexp} x {Subpoly, Poly, Subexp, Exp, Hyperexp},
can be divided into smaller subsets of pairs (Kx, Ky) and, as a consequence, define more detailed classification of parameterized algorithms than (15)-(18).
For trivial parameterization is true y = x and expession (12), (13) becomes:
Ex(z) = Ex[q(x) • f(x)] = Ex(q) + Ex(f).
In this particular case we arrive at one-dimensional classification of algorithms. A similar case arises when f (y) is identically constant, i.e. f (y) = c > 0.
Classification Parameterized Algorithms in Terms Impact of Parameter
For parameterized algorithms of practical interest is the classification, which aims to clarification the level impact parametric component f (y) the multiplicative form (10) on time execution of algorithm. Traditionally, we distinguish between [4, 9]:
• parameterized algorithms with a weak level of impact parameter. The time execution such algorithms depends weakly on parametric component f(y). It establishes a predominantly coefficient for q(x);
• parameterized algorithms for which parametric component f (y) and quantitative component q(x) have a comparable effect;
• parameterized algorithms with a dominant level of impact parametric component. For them, the parameter identifies the main source of complexity. It is these algorithms interesting for parameterized complexity theory.
Unfortunately, in literature on parameterized complexity until are no formal methods of identifying parameterized algorithms with a dominant dependence on parameter. Application for these purposes elasticity gives the positive results for multiplicative forms of presentation functions of complexity and for the family L-functions.
In fact, a comparison of components q(x) and f (y) is only possible when there is a dependence y = y(x), which describes the relationship between values of parameter and length input data of algorithm. In these circumstances by properties of elasticity 3 and 5, the expression (12) becomes:
Ex(z) = Ex[q(x) •f(y)] = Ex[q(x)] + Ex[f(y)] = Ex(q) + Ey(f) • Ex(y). (19)
Note that, in (19) the terms Ex(q) and Ey (f) • Ex(y) are identically constant or infinitely small or infinitely large logarithmic-exponential functions, which depend only on x, and because they always are comparable with each other in order of growth rate. We assume that z(x, y) = q(x) • f (y) is not identically constant (case z(x, y) = c > 0 is not interesting). We have the following situations.
If when x ^ to relation
Ey(f) • Ex(y) -< Ex(q) (20)
holds then
Ex(z) = O[Ex(q) + Ey(f) • Ex(y)] = O[Ex(q)].
This means that parameterized component has weak effect on value of z(x,y) = q(x) • f(y). If when x ^ to relation
Ex(q)= O[Ey(f) • Ex(y)] (21)
is correct then
Ex(z) = O[Ex(q) + Ey(f) • Ex(y)] = O[Ex(q)] = O[Ey(f) • Ex(y)],
which indicates a comparable effect of parametric and quantitative components. Finally, if for x ^ to relation
Ex(q) -< Ey(f) • Ex(y) (22)
is true then
Ex(z) = O[Ex (q) + Ey (f) • Ex(y)] = O[Ey (f) • Ex(y)].
This indicates that parametric component is dominant.
Consequently, we can mathematically rigorously identify a level of impact parametric component f (y) the multiplicative form (10) on time execution of algorithm by examining relations (20)-(22) and using properties of elasticity. The verification process is simplified if we consider
the fact that the dependence y = y(x) usually holds for internal parameter of parameterized algorithm. In this case y = y(x) is a polynomial or subexponential function (in growth rate) and hence Ex(y) = O(1) or Ex(y) = o(1) when x ^ to. If Ex(y) = O(1) then relations (21)-(23) simplifies and takes form:
Ey (f) ^ Ex(q)
Ex (q) = O[Ey (f)],
Ex(q) -< Ey (f).
Thus, if we have a polynomial dependence of parameter on length of input data, then it all boils down to a comparison of two L-functions (more precisely to a comparison of their elasticity). When Ex(y) = o(1) we should take into account in (20)-(22) both factors of expression Ey (f) • Ex(y). For the trivial parameterization of relation between complexity classes in (Kx, Ky) identifies the level of impact parameter on time execution of algorithm. If Ky -< Kx then time execution of parameterized algorithm depends only weakly on parameter. If Ky = Kx then the parametric component and quantitative component have a comparable effect. If Kx -< Ky then we have a parameterized algorithm with a dominant level of impact of the parameter.
For example, consider the function (14), which is characterized by two-dimensional complexity (Kx, Ky) = (Poly, Exp) and Ex(q) = 5, Ey(f) = Ay. Let y = lnx. Then Ex(y) = 1/ lnx = o(1). Since at x ^ to holds estimate
Ey(f) • Ex(y) = (Alnx • (1/lnx)) = A = O(1)
then relation (21) is true, i.e. the parametric component and quantitative component have a comparable effect, although there Kx -< Ky. Now let y = x. We have a trivial parameterization and Ex(y) = 1. The estimate
Ey(f) • Ex(y) = Ax = O(x)
satisfies (22), satisfies relation Kx ^ Ky and shows dominant a level of impact parametric component.
Conclusions
Parameterized complexity theory has received in recent years widespread and is an extremely suitable tool for solving NP-hard problems associated with a number of important applications. It draws our attention to how represent the input data, what data are of interest to problem being solved and what data are sources of the nonpolynomial-time for NP-hard problems. Parameterized complexity theory has given impetus to development of new approaches into designing algorithms (algorithms solving the FPT-problems). At the same time, this theory has generated a wide range of problems. One problem is the lack of simple and convenient tools for analysis of parameterized algorithm where function of complexity depends on many variables.
In this paper we proposed a new indicator of computational complexity for parameterized algorithm, which can be used to measure rate of growth function of complexity from many variables, by which we can to compare the FPT-algorithms for NP-hard problems and to perform analysis level of impact parameter for time execution of the parameterized algorithm. This indicator is the elasticity of function complexity algorithm. In this paper we proposed a twodimensional classification of parameterized algorithms to multiplicative forms of presentation the functions complexity, we accurately identify the class of FPT-algorithms and we proposed the mathematical method of analysis level of impact parameter for time execution of parameterized
algorithm. The proposed approach to analysis of parameterized algorithms admits further development in terms of increasing a number of parameters, the study of other forms for representing the function complexity.
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Анализ параметризированных алгоритмов на основе эластичности функций сложности
Валентина В. Быкова
Дан краткий обзор результатов и проблем параметризированной алгоритмики — нового направления теории сложности вычислений. Предложен новый показатель вычислительной сложности параметризированного алгоритма, с помощью которого можно измерять темп роста функции сложности многих переменных. Этим показателем является частная эластичность функции сложности. Предложена двумерная классификация параметризированных алгоритмов для мультипликативной формы представления функций сложности. Математически обоснован метод анализа уровня влияния параметра на время работы параметризированного алгоритма.
Ключевые слова: сложность вычислений, параметризированные алгоритмы, анализ алгоритмов, эластичность алгоритмов.
