Lower bounds in algebraic computational complexity - Laboratory of

LOWER BOUNDS IN ALGEBRAIC COMPUTATIONAL COMPLEXITY D. Yu. Grigor'ev

UDC 519.5

The present article is a survey of selected methods for obtaining lower bounds in algebraic complexity. Introduction.

i.

We present the contents.

Basic concepts.

Chapter I.

Algebraic-geometric approach to

obtaining lower bounds of computational complexity of polynomials. a polynomial with "general" coefficients. vidual polynomials.

4.

6.

Computational complexity of indi-

5.

The degree method (the case of a finite ground

Additive complexity and real roots.

Chapter II.

multiplicative complexity for problems of linear algebra. plexity and rank.

8.

Rank of a pair of bilinear forms.

plexity of a bilinear form over a commutative ring. algebras,

ii.

Linearized multiplicative complexity.

for straight-line programs of nonstandard types. complexity of algebraic functions. tradeoffs.

15.

Evaluating

The degree method and its generalizations (the case of

an infinite ground field). field).

3.

2.

13.

12.

i0.

7. 9.

Lower bounds on Multiplicative comMultiplicative com-

Bounds on the rank of

Chapter III.

Complexity

Irrational computational

Monotone programs.

14.

Graph-theoretic methods in algebraic complexity.

Time-space 16.

Additive

complexity in triangular and directed computations and Bruhat decomposition.

INTRODUCTION The problem of lower bounds is one of the most difficult ones in computational complexity theory, and it can be said without exaggeration that their obtaining constitutes the naturally fundamental topic of complexity theory, since the establishment of lower bounds, i.e., the construction of sufficiently fast algorithms, is, rather, the prerogative of the other mathematical sciences from which the concrete computational problems originate.

In spite of the

fact that the problem of obtaining nontrivial lower bounds (i.e., of proving the impossibility of sufficiently fast algorithms for given computational problems, and, by the same token, the penetration of the secrets of fast algorithms) is far from completely solved, in it there are certain interesting advances, particularly in that part of complexity theory which relates to the problems traceable to algebra, called algebraic complexity (see Bel'tyukov's survey in the present issue on lower bounds in some other sections of complexity theory). Algebraic complexity is one of the oldest branches of complexity theory (but it is one that is being most intensively worked on at the present time); it has been around for nearly 25 years, but a sufficiently complete survey devoted to it has not yet appeared in Russian. Among the foreign publications we should note, in the first place, the book [27], as well as [I, 13], but lower bounds are in fact absent in the latter, while [27] does not go into the achievements of recent years. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 118, pp. 25-82, 1982.

1388

0090-4104/85/2904-1388509.50

9 1985 Plenum Publishing Corporation

To no extent does the author pretend at completeness of the exposition of all the resuits in the area of obtaining lower bounds in algebraic computational complexity; rather, the present text is a survey of selected methods and achievements, whose aim is to fill in the gaps existing in the Russian literature. propagated,

The methods that have already been widely

as well as those that do not as yet have sufficiently strong applications,

presented in lesser detail.

are

The number of proofs given in the present survey is compara-

tively small; a sufficiently complete list of references permits us to refer to appropriate literature when necessary. The author tried to pay most attention to those methods of establishing lower bounds which are connected with nontrivial algebraic methods.

The profound connections with clas-

sical algebra, as also the problem statements, being atypical of traditional algebra, are, in general, a characteristic

feature of algebraic complexity, which can make it attractive

for algebraists. A survey of methods in algebraic complexity leaves a somewhat mosaiclike impression. This is due, it seems, to the fact this branch of mathematics

is still quite young and as yet

no unified ideas have been formulated in it, the problems are difficult and have to be approached individually. each other

Therefore,

the different chapters are formally little connected with

(except Sec. I which gives the definitions needed for understanding what follows).

Essentially,

each section contains a description of an individual method; at the same time,

the ordering of the material is not random and has definite historical and methodical reasons

(if the ontogeny

and

phylogeny of algebraic complexity is desired).

We note that

the contents of Secs. 6, ii, 16, and a part of Sec. 15 are being published for the first time. The numbering of the sections and of the formulas is consecutive; and corollaries are two-numbered,

the theorems, lemmas,

the first of which is the number of the corresponding sec-

tion. i.

Basic Concepts The basic computing model used in algebraic complexity is the straight-line program

(see [i, 27], for example) which we now describe. i) a collection of input variables 2) a ring

K

Let there be given:

~4,...,~

(usually this is a field which will be denoted by

P

) which is subse-

quently called the ground ring; 3) a set

~

of base operations

are binary arithmetic operations, The variables

~...~

x~

(usually

~-

[+,X,/} U

~X~e K

,

where

+,X,/

is a unary operation, viz., multiplication by ~

can be assumed or not to be pairwise commuting;

).

often this is

clear from the substance of the problem being analyzed. 4) a straight-line program (SLP) proper is a sequence of rows (instructions)

the

~ -th

of which has the following form:

9~ = ~{7.~,...,Z~,~I,,...,~), where ~~

~ -th degree polynomial from values at (~§

), its multiplicative complexity also equals

in order (we recall that we would have the same bound for these two problems in

the case of an infinite field -- see the applications of Theorem 4.1).

The matter is different

for elementary symmetric functions: in [17] it is shown that even the total complexity %(~,...,~)

is linear in

~

over a finite field

~

(cf. Sec. 4).

It is interesting to note that the reverse situation occurs in certain natural cases, i.e., the complexity of evaluation of a family of polynomials over a finite field can be greater than the complexity of evaluation of this family over an infinite field. the multiplicative complexity of the multiplication of two ~ infinite field

~

equals

~I

two elements a lower bound of

For example,

-th degree polynomials over an

(see [31], for instance); in the case of a field 8,~W

~

of

was proved in [28] for the multiplicative complexity in

this problem (the best known upper bound for it to-date is W.~(W),

where

~

is some function

growing more slowly than any fixed iteration of the logarithm; see [8, 9, 32, 33]). 6.

Additive Complexity and Real Roots In the preceding two sections we established lower bounds for the multiplicative com-

plexity

C~(~1,:..,~

(~_~,...%~x)= ~). the system

of a family of polynomials in terms of a power of the graph W = ~ Since

~ .... = ~ K = 0

~

~

is not less than the number

over field

~

,

we have

~

of discrete roots of

Cm(~,..,,~)~N~

(cf. Theorem 4.1).

In the present section, on the basis of Khovanskii's work in [21], we shall find a lower bound for the additive complexity ....J ~ ' = A~-A+

0

9

~§

in terms of the number of roots of the system

In this section, in the notation of Sec. i, ~ = ~ ;

(see the beginning of Sec. 2); ~)...,~ W ~ [ ~ , . . . , ~ ]

ulated, all the polynomials are assumed real.

.

~+,X,/}U{X~}~m ~

;

Below, unless otherwise stip-

The existence of the bound mentioned was

assumed a long time ago and this assumption was based on the Descartes principle: the number of nonnegative roots of a polynomial in one variable does not exceed the number of its monomials.

For one polynomial

~

in one variable a bound weaker than the one established below

was obtained in [26] (it appears that the method of proof of the main theorem in this paper is of independent interest). 1402

By

~

we denote the set of nonzero real roots.

THEOREM 6.1 [21].

~(t4,+~,)K~ K(K+'I)/~'

than

The system of equations ~1--...--~--0{~r.,~e~[~...,~)

mials in all the polynomials COROLLARY 6.29

~,;..,g~

IZ*)", then C~(~,

Proof of the Corollary.

Let

K

where

(~)~

0

has

is the total number of mono-

N

simple (i.e., of multiplicity

~[-~"

,~)~-

~I,..9

(the roots from

,

.

If the system 9'..... ~ -

one) discrete roots in ~ ....=~*----- 0

('~)Hi,

discrete roots in

has no more

be the simple roots mentioned of the system will be called nontrivlal).

To derive the corol-

lary from the theoremwe make use of the well-known canonic form for the SLP (see [26], for example)

containing no more than

Ir----~,(~l,...p~) multiplications:

e

:

:.(r~t) _;.~,o

'T' r o T "

'1"~

~----- 'I .... ~

c~,~)

~

"

c~.o :($+4)

~,},~,W,p,~

w(p)

(4)

"''~

-(~) -t~) -(~) ~ ' - - '1 . . . . ~" ~1 " ' '

where the

;(c~o ~

,r.t~ .'p~4 ...

4~~)

with subscripts are integers;

T~

is the SLP's working variabZe (see

Sec. i) in the instruction in which the ~ -th multiplication operation, by count, takes place; the value of the working variable

~;

equals

Let us show that we can so modify system (4) numbers (~r, 1~)

~,..., ~

ping ~

(If+1~) unknowns

Since ~I,"',~N

;'~ ....~ .)>~

~

&(~)=O

~

by replacing

~I,...,~,TI,...,T~

~"

is bijective for each

coincide and

yA=(~) ~

~ H

Y~ . for all

in some neighborhood

~-