CC BY
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Mathematical Structures and Modeling 2015. N. 4(36). PP. 53-66
UDC 510.52
search problems with a promise and graph
isomorphism
G.A. Noskov
Dr.Se.(Phys.-Math.}, S.R., e-mail: g.noskov@googlemail.com Institute of Mathematics, SORAN
Abstract. The Graph isomorphism problem is considered from the point of view of the theory of ’’problems with a promise”, developed by Even, Selman and Yacobi [4]. The ”tau-invariant” of graphs is studied and with its help the search graph isomorphism problem is solved for asymptotically almost all graphs.
Keywords: decision problem, search problem with a promise, graph isomorphism.
Introduction
In this paper we consider the search graph isomorphism problem SGI in the context of ’search problems with a promise” as they defined in [4,6,7].
In existing literature (see for instance [9]) the decision and search variants of the graph isomorphism problem are formulated as follows:
P1: Given two graphs G and H with n vertices each, decide whether they are isomorphic.
P2: Given two graphs G and H, decide whether they are isomorphic, and if so, construct an isomorphism from G to H.
Note that P2 contains P1 as a subproblem. Recently, A.N. Rybalov suggested to isolate the search version from the decision version as follows [12]. By definition, the input set for search graph isomorphism problem SGI is the set of all pairs of isomorphic graphs. Given two isomorphic graphs G and H, one needs to construct the isomorphism between G and H. Thus in Rybalov’s setup of the SGI the input graphs are already assumed to be isomorphic, whereas in version P2 above the input graphs are arbitrary. The seemingly unusual input set in SGI can be elegantly explained in the framework of ’’promise problems” in the sense of [4,6]. Promise problems are a natural generalization of search and decision problems, where one explicitly considers a set of legitimate instances (rather than considering any string as a legitimate instance). Informally, a promise decision problem has the following structure: input x, promise P(x), property R(x), where P and R are unary predicates. An algorithm solves the promise problem if, given an input x, it answers the question whether R(x) given that P(x). The behavior of such an algorithm may be arbitrary on instances x for which the promise P is false [10].
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G.A. Noskov. Search problems with a promise...
In this paper we formulate the search graph isomorphism problems ’’with a promise” and study their reducibility and generic solvability in polynomial time. We make use the ’type-invariant” of a graph introduced in [2,13,14]. Our main result is Theorem 5, which presents the polynomial-time algorithm A, solving the problem SGI (Pisom, R) for asymptotically almost all inputs PiSom П Poblique.
In Section 1 we give the necessary definitions about computation problems. Section 2 is devoted to the formulation of the search graph isomorphism problem (with a promise). Section 3 contains definitions of computational problems in the case of graphs. Section 4 contains the proof of polynomial reducibility of search graph isomorphism problem SGI to the graph isomorphism problem. In Section 5 we discuss the main tool - the graph invariant т and oblique graphs. In Section 6 the efficient solvability of our problems in case of oblique graphs. In Section 7 we solve the SGI problem in generic case. Finally, in the last section we apply our results to probabilistic algorithms.
1. Decision and search with a promise
We start with the standard definitions of decision and search computational problems as they presented in [11] and [7]. Let I = {0,1}* be the set of all words (=binary strings) in the alphabet {0,1}. We consider algorithms as means of computing functions. Specifically, an algorithm A computes the function fA : {0,1}* ^ {0,1}* defined by fA(x) = y if, when invoked on input x, algorithm A halts with output y. We associate the algorithm A with the function fA computed by it; that is, we write A(x) instead of fA(x).
Decision problems. Let’s denote by Ik the k-th direct power I x ■ ■ ■ x I of I for k ^ 1. A decision problem for a subset (=language) L c Ik is to determine for a given tuple w e Ik whether w belongs to L or not. An algorithm A solving this problem is the decision algorithm for L, and in this case the decision problem for L, as well as the language L, is called decidable.
If L is decidable and additionally there are positive constants c,q such that for every instance x e Ik the algorithm A determines the membership of x to L in at most c |x|q steps, then the decision problem for L, as well as L is called polynomialtime decidable (or decidable in polynomial time). Here, for a tuple x = (xb...,xk) we let denote by |x| the maximum max |x,| of lengths of words x,.
Decision problems with a promise. More general class of partial decision problems was introduced in [4] under the name of ’promise problems”. Formally, a partial decision problem is a pair of decidable subsets (L, P) of Ik, where P is the set of allowed or promised tuples and P = Ik — P is the set of disallowed tuples. The promise problem (L,P) is solved by algorithm A if for every x e P П L it holds that A(x) = 1 and for every x e P — L it holds that A(x) = 0. Shortly,
Thus, the algorithm A is required to distinguish yes-instances P П L from no-
(1)
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instances P — L and A is allowed to have arbitrary behavior on inputs P that are neither yes-instances nor no-instances. A set Li is called a solution to the partial decision problem (L, P) if L1 n P = L. Clearly, when P = Ik we have the notion of a standard decision problem. A partial decision problem (L,P) is polynomial-time decidable if there is a polynomial-time decidable solution L1 of it.
Remark 1. In [7,8] a more restrictive definition is used, namely the promise problems (L,P) are considered with the condition L c P. It is not enough for our purposes, see Section 5.
Search problems. A search computational problem can be described by a binary relation R c Ik x I1 for some fixed k,l ^ 1. The problem is ”given an input x e Ik, find y such that R(x,y) holds, if such y exists”. More precisely, one requires, for a given x e Ik to decide first whether there exists y e I1 such that R(x,y) holds, and only after that to find such y if it exists.
Let us consider R as a multi-valued function R(x) = {y : (x,y) e R}. The associated solution set is Sr = {x : R (x) = 0}. A function f : Ik ^ I1 U {A} is called a branch of R (x) if f (x) e R (x) for all x e Sr and f (x) =L for all x e Sr (thus f (x) =L indicates that x has no solution). We say that the branch function f of R solves the search problem of R. As before, we write A(x) for f (x) for an algorithm A, computing the function f.
Note, that the search problem for R contains the decision problem for a solution set Sr = {x : R (x) = 0} as a subproblem. Indeed, if A computes the branch of R, then x e Sr iff A (x) =L.
A relation R c Ik x I1 is polynomially bounded if there exists a polynomial p such that for every (x,y) e R it holds that |y| A p (|x|). The search problem of a polynomially bounded relation R c Ik x I1 is efficiently solvable if there exists a branch function f of R which is polynomial-time computable. In this case an algorithm A, computing this function, is called a polynomial-time algorithm for the search problem R. We denote by PF the class of polynomially bounded search problems that are efficiently solvable [8].
Search problems with a promise [8, p.143]. A search problem with a promise consists of the input set Ik, a binary relation R c Ik x I1 and a promise set P c Ik. Such a problem is also referred to as the search problem R with promise P and is denoted by (P, R). The search problem (P, R) with promise P is solved by algorithm A if for every x e P it holds that (x, A(x)) e R if x e Sr and A(x) =_L otherwise. Thus, the restriction function A|P is the branch of the relation R n (P, I1). ”We stress that nothing is required of the solver in the case that the input violates the promise (i.e., x e P); in particular, in such a case the algorithm may halt with a wrong output” [8, p.143]. Note that the full search problem Ik, R with a promise Ik is the standard search problem. And at the other extreme there is the so called candid search problem (Sr, R) with promise Sr.
The time complexity of A on inputs in P is defined by
TA|P(n) = max{tA(x) : x e P П {0,1}n},
(2)
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where tA(x) is the running time of A(x). A problem (P,R) is polynomially-time decidable if there is an algorithm A such that function TA|P is bounded by a polynomial in n. ”In this case, it does not matter whether the time complexity of A is defined on inputs in P or on all possible strings. Suppose that A has (polynomial) time complexity T on inputs in P; then we can modify A to halt on any input x after at most T(|x|) steps. This modification may only affect the output of A on inputs not in P (which are inputs that do not matter anyhow). The modification can be implemented in polynomial time by computing t = T(|x|) and emulating the execution of A(x) for t steps.” [7, p.88].
Algorithms which are polynomial-time on asymptotically almost all inputs. By the very definition of the search problem (P,R), it becomes easier when decreasing the promise P. If the given problem (P,R) is hard, then it makes sense to consider a restriction problem (P',R) with a promise set P' c P and ask whether (P',R) is polynomial-time decidable. This may happen when P' is small enough, for instance when P' is finite. Thus, it is highly desirable to find out a polynomial-time decidable restriction (P',R) with P' being maximally close to
P. For a rigorous definition of closedness we need a size function on the promise set P, i.e. any computable function x m ||x|| e N such that for every n e N the set Pn = {x e P : ||x|| = n} is finite. Furthermore, we need an ensemble of distributions D = (Dn) with Dn a distribution on the set Pn [1]. The closedness of P' to P can be measured by the asymptotic density (=asymptotic probability) (if exists)
D (P') = lim Dn (P' П Pn). (3)
n—
A subset P' c P is said to be asymptotically almost certain (=generic) if
D (P') = 1. In the theory of random graphs another terminology is accepted: the event P' c P as above happens asymptotically almost surely (=a.a.s) or with high probability (=w.h.p.). The complement of a generic set is said to be negligible.
A search problem (P, R) is polynomial-time decidable with high probability (=for asymptotically almost all inputs) if it admits a polynomial-time decidable restriction (P',R) such that P' is asymptotically almost certain in P (relative to a fixed size function and fixed distribution ensemble. Similar definitions can be given in case of promise decision problems.
2. Graph problems
Let Gn denote the set of all graphs on the set of vertices [1; n] = {1,..., n}. The set Gn is naturally acted upon by the symmetric group Sn on the set [1; n]. The orbits of this action are precisely the isomorphism classes of graphs from Gn.
We encode graphs as binary strings as usual. Namely, let (g^) be the adjacency matrix of G then we encode G by the string (g12g13 ••• g1ng23 ••• g2n ••• gn-i,n).
This encodes Gn bijectively onto In(n-1). The permutations ф e Sn are in one-one
2
correspondence with n x n monomial matrices and the last ones can be encoded row by row by binary strings in In2. We consider the graph isomorphism problem
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GI(L,P) as a partial decision problem (L,P) whose input set is I x I , whose promise set is a decidable subset P c PaU, where
Pall Un (Qn x Qn) Un ^In(n-l) x 1 n(n-l^ (4)
and whose language L of yes-instances is defined by
L - Pisom = {(G, H) e Pall : G - H} . (5)
The standard GI-problem is (Pisom,Pall). It is to determine for a given tuple (G, H) e 12 whether G — H or not.
In view of exceptional hardness of (Pisom,Pall) the problems (Pisom,P) have been considered with P being the pairs of graphs from some interesting classes of graphs, such as trees, planar graphs, graphs of bounded valence, etc. For instance, it is well known, that promise decision problem (Pisom, Ptrees), where Ptrees is the class of all finite trees, is polynomial-time decidable.
Accompanying to GI is SGI - the search graph isomorphism problem with a promise. This problem is denoted by SGI (P,R) and consists of the input set I2, relation R c I2 x I and a promise set P c 12. The relation R is defined by
R - {(G,H,4>) : 3n such that G, H eQu^ e Sn C In2 and ф : G — H} . (6)
The promise set P is an arbitrary decidable subset of Pall. The problem requires, for a given pair of graphs (G, H) e P to decide first whether they are isomorphic and then to find an isomorphism ф : G — H.
The problem SGI (P, R) with promise P is solved by algorithm A if for every (G, H) e P it holds that ((G, H),A(G,H)) e R if G — H and A(G, H) -A otherwise. We see, that the full search problem (Pall,R) with a promise Pall is the standard SGI-problem as it is defined for instance in [9]. This problem contains the standard decision problem GI (Pisom,Pall) as a subproblem. Indeed, if an algorithm A solves SGI (Pall, R) then (G, H) e Pisom if and only if A(G, H) -A, thus A recognizes whether an isomorphism between G and H exists or not.
At the other extreme there is the candid search problem (SR, R) with promise SR, where SR - {x : R (x) - 0} is the solution set of R . In our case it is clear
that { }
Sr - Pisom - {(G, H) e I2 : G — H} . (7)
And again, the full search problem SGI (Pall,R) contains the candid search problem SGI (Sr,R) as a subproblem. We consider SGI (Sr,R) as an adequate formulation (in terms of promise problems) for the problem SGI from [12]. We find the following problem rather intriguing:
Problem 1. Is the standard isomorphism problem GI (Pisom, Pall) polynomialtime reducible to SGI (SR, R)?
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3.
The problem SGI (Pall , R) is polynomial-time reducible to
GI (Pisom, Pall)
Theorem 1. The full search isomorphism problem SGI (Pan, R) Is polynomialtime reducible to the standard isomorphism problem GI (Pisom,Paii)■
Proof. (Cf. [9, Thm,6, p.29]) It is easy to see that SGI can be reduced to the case of graphs without isolated vertices.
Ascent. Suppose we have a polynomial-time algorithm A recognizing the graph isomorphism. Given two isomorphic graphs G,H on n vertices and the list of vertices VG = [gi,..., gn}, we will construct inductively and efficiently (=in polynomial time) the sets of graphs G = G0 < Gi < ■ ■ ■ < Gn, H = H0 < Hi <
■ ■ ■ < Hn and the vertices hi,..., hn e VH such that:
1) Gi ~ К, for all i,
2) Every isomorphism between G, and H, takes Gi-i to Hi-i for all i = 1,..., n,
3) Every isomorphism between G, and H, takes g, to h, for all i = 1,..., n.
The construction is defined as follows. Let Kn+i denote the complete graph
on n + 1 vertices. Fix ki e Kn+i and consider
Gi G Ug1=kl Kn+b Hi (h) H Uh=kl Kn+b
where Ux=y denote the gluing operation on graphs with identified vertices x = y. Since G ~ H, there exists h such that Gi ~ Hi = Hi (h), namely h is the image of gi under isomorphism G ~ H. We find out h by applying A to all n pairs (Gi,Hi(h)). Set hi = h. By assumption, G,H have no isolated vertices, so gi,hi are unique vertices of highest degree in Gi,Hi. Hence every isomorphism between Gi and Hi takes gi to hi. Moreover, all vertices of G distinct from gi have degree
(in Gi) no greater than n — 1, whereas all the vertices of Kn+i have degree (in Gi) at least n. Therefore, every isomorphism between Gi and Hi takes G0 to H0. Thus
1)-3) hold for i = 1.
Fix k2 e Kn+2. For all h e VH construct the graphs
G2 = Gi Ug2=k2 Kn+2, H2 (h) = Hi Uh=k2 Kn+2. (8)
Since Gi,Hi are isomorphic, there exists h such that G2 ~ H2 and we can find h applying A to all n pairs of graphs G2, H2 (h). Fix any such h and denote it by h2. Continuing, we find out the desired hi,..., hn e H and G = G0 < Gi < ■ ■ ■ < Gn,
H = H0 < Hi < ■■ ■ < Hn.
Descent. The map g, m h, is an isomorphism between G and H! Indeed, by 1) there exists an isomorphism ф : Gn m Hn. By 2) ф takes gn to hn. The restriction ф|Gn-1 takes Gn-i to Hn-i by 2) and takes gn-i to hn-i by 3). Then ф^п-1 takes Gn-2 to Hn-2 and we can continue this descent process until obtaining that ф takes every g-i to h,. Hence the map g, m h, is an isomorphism in question. ■
4. Graph invariant т and oblique graphs
Graph invariant т [2,13,14].
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Let N denote the set of all natural numbers (taken as 1,2,3,4,...A string over N is a finite sequence of natural numbers, that is, an integer n > 0 and a mapping {1,...., n} m N. If n = 0, the domain is the nullset and there is a unique such mapping, called the nullstring and generally denoted e [3]. Let N* denote the set of all finite strings of natural numbers (including the nullstring e). The lexicographic order ranks strings of the same length in N*, by comparing the letters in the first position where the strings differ. We define the ShortLex order on N*: v < w if and only if v is shorter than w, or they have the same length and v comes before w in lexicographical order. ShortLex order is a well-ordering.
Let G = (V, E) be a simple graph. By d(x) we denote the degree of the vertex x. The set of all vertices adjacent to v is denoted by N (v).
The type-string tg(v) e N* of a vertex v is the string of degrees of vertices of N (v) in non-decreasing order. We set tg (v) = e for isolated vertex v. In detail, tg (v) = (db..., dd(v)) is the degree sequence of the vertices adjacent to v, arranged in non-decreasing order: di ^ d2 ^ ... ^ dd(v). Clearly a type-string is preserved under graph isomorphisms: if ф : G m H is a graph isomorphism then th (ф (v)) = tg (v) for all v e V (G), i.e. тн о ф = tg.
The type-vector tg of G is the ShortLex ordered sequence (tg (vi),... ,tg (vn)), n = |V| of all type-strings of all vertices (thus tg is used to denote a vector as well as a map). Clearly the type-vector function G m tg is a graph invariant, i.e. tg = th for isomorphic graphs G, H. However, as the next example will show, the type-vector is not a complete graph invariant, i.e. there exist non-isomorphic graphs with the same type-vector. The examples will be found among regular graphs. A graph G is said to be k-regular if all the vertices of G have the same degree k. In this case tg = (k*k,...,k*k) has n coordinates, n = |VG|, where k*k = (k,..., k) e N* is a string with k coordinates. Thus all k-regular graphs on n vertices have the same т-invariant.
Proposition 1. There exist two non-isomorphic graphs Gi,G2 on 8 vertices all of whose degrees are equal 5. In particular Gi ,G2 have the same type-vector
(5*5,5*5,5*5,5*5,5*5,5*5,5*5,5*5)^ where 5*5 = (5,5,5,5,5) e N*.
Proof. Define Gi,G2 as on Figure 1.
Degree conditions are clearly fulfilled. It is left to ensure that the graphs are non-isomorphic. For this it is enough to ensure that the complement graphs are non-isomorphic. The complement to the first graph is two disjoint 4-cycles ACHF and BDGE (so it is disconnected), whereas the complement to the second graph is an 8-cycle AFEBHCDG (hence it is connected). The non-isomorphism is clear now.
Despite the fact that т is not complete, it turns out to be asymptotically complete, i.e. for asymptotically almost all graphs G,H the equality tg = th implies that G, H are isomorphic (see [2,13] and Section 7).
A graph G is said to be oblique if the map tg : V m N* is injective or, in other words, there are no distinct vertices u,v e V(G) such that tg (u) = tg (v) [14]. Figure 3 shows an example of oblique graph on 8 vertices [5].
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G.A. Noskov, Search problems with a promise,,.
Figure 1. Graphs G1 and G2.
Figure 2. Complement graphs Gi and G2.
Lemma 1 (uniqueness of the type-vector). If a graph G is oblique then its type vector tg = (tg (vi),...,tg (vn)) is unique in the sense that (tg (vi),..., TG (vn)) = (tg (w 1),... ,TG (Wn)) implies that v = Wi for all i.
Proof. Indeed, by assumption tg (vi) = tg (wi) for all i, hence by obliqueness vi = wi for all i. ■
Lemma 2 (retrieving an isomorphism). Let tg = (tg (g1),... ,tg (gn)) ,th = = (th (h1),... ,th (hn)) be the type-vectors of graphs G,H respectively. If G is oblique and G ~ H then the map gi ^ hi is the isomorphism of G onto H and there is no other isomorphism between G and H can be drawn. In particular, every oblique graph has a trivial automorphism group.
Proof. Let tg = (tg (gi),..., tg (gn)) ,th = (th (hi) ,...,th (hn)) be the type-vectors of graphs G,H respectively, let G be oblique and let ф : G ^ H be an
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(3.5)
(4.5) (4, 5,5) (2, 4,5)
5
5
4
4
(2, 3,4, 4, 5) (2, 3, 3, 4, 5) (2, 3,4, 5) (3, 4,5, 5)
Figure 3.
A vertex-oblique graph on 8 vertices. The degrees of vertices and their neighbors are
written in columns
arbitrary isomorphism. By assumption tg : V (G) m N* is injective, so th = = tg о ф-1 : V (H) m N* is injective also and thus H is oblique. The vector of strings
(th (ф91),..., th (ф9и)) = (tg (9i),..., tg (9n)) (9)
is the type-vector of H because its coordinates are all type-strings of all vertices of H (looking at the left-hand side) and they are ShortLex ordered (looking at the right-hand side). Thus we have two type-vectors for H:
(th (ф91) ,...,TH (ф9п)) = TH = (th (hi) ,...,TH (hn)). (10)
By the uniqueness Lemma фд-i = hi for all i. Hence the map д» m hi = фд-i is an isomorphism of G onto H ; moreover, there is no other isomorphism between G and H. ■
5. Efficient solvability of GI (Pisom, PobUque)
End SGI (Pisom ^ Poblique, R)
Consider the following sets of pairs of graphs:
Pall = Un (Qn x Qn) , (11)
Pisom = {(G, H) E Pall : G ~ H} , (12)
Pobiique = {(G, H) e Pall : G and H are oblique} . (13)
It is easy to see that all these sets are decidable.
Theorem 2. The promise decision problem GI (Pisom, Poblique) is polynomialtime solvable.
Proof, (Cf. [2,13] ) This time the promise set Poblique does not contain the language Pisom, so we need the formalism of promise problems in full generality. Precisely, an algorithm A solves instance (G, H) of Poblique iff
A (G,H)
1 if H) E Poblique G Pisom
0 if (G, H) E Poblique Pisom
(14)
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On instances out of P0biique the behavior of A may be rather arbitrary. Since Pobiique is decidable, we can construct the required algorithm A by splitting into cases depending on whether a given instance (G, H) belongs to Poblique or not. Precisely, we put A (G, H) = 0 on instances (G, H) e I2 — Poblique. The idea of the restriction algorithm A|Poblique is to distinguish all vertices of a graph using the invariant т. Given a pair (G, H) e Poblique the required algorithm A first computes the type-vectors
TG = (tG (gi),..., TG (9n)) , TH = (tH (h1) ,..., TH (hn)). (15)
If тG = тH then A checks whether the map ф : gi m hi is an isomorphism from G to H or not. If it is an isomorphism then we put A (G,H) = 1. Otherwise Lemma 2 asserts that G, H are not isomorphic, so A (G, H) = O.Thus A is well defined on all inputs I2 and it solves the given problem. It is easy to see that the running time of A is linear in |V| + |E|. ■
Theorem 3. The problem SGI (Pisom П Poblique,R) with relation
R = {((G, H), ф) e 12 x 1: 3n such that G, H e Gn, ф e Sn c /„2 and ф : G ~ H}
(16)
is polynomial-time solvable.
Proof. By definition the given problem is solved by algorithm A if for every (G, H) e Pisom П Poblique the inclusion ((G, H), A(G, H)) e R holds in case G ~ H and A(G, H) =± otherwise. Given the n—vertex graphs G, H the required algorithm A first computes the type-vectors
TG = (tG (91) , . . . , TG (9n)) , TH = (tH (h1) , . . . , TH (hn)) . (17)
If the coordinates either of tg or th are not distinct then at least one of graphs is not oblique i.e. (G, H) e Pisom П Poblique. In this case we let A (G, H) = id e e Sn. Otherwise both of graphs are oblique and then algorithm compares the type-vectors. If tg = th, then G,H are not isomorphic and thus A (G, H) is already defined. If tg = th then the algorithm checks whether the map ф : gi m hi is an isomorphism or not. If this map turns out to be an isomorphism then the algorithm gives ф as the answer to the problem. Otherwise, if the map is not an isomorphism, Lemma 2 asserts that there is no isomorphism between the given graphs at all. That is this case is impossible by the very setup of the problem. Thus the algorithm A gives the right answer for the set of inputs Pisom П Poblique and A (G, H) = id on the complement set Pisom П Poblique, which may be the wrong answer (but also may be right answer occasionally). ■
6. Generic algorithms for GI and SGI and their failure probability
Theorem 4. Endow the set Pall with the ensemble of distributions (Dn), where Dn is the uniform distribution on the set P^ of pairs of n-vertex graphs. Then Pisom П Poblique is asymptotically almost certain in Pisom.
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Proof, Let Gn denote the set of oblique graphs on vertices 1,2,...,n. We rely on the result from [2,13] which says that the property of a graph G e Gn to be oblique holds asymptotically almost surely, i.e.
no lim -[Gni = 1, n—~ \Gn \
or, equivalently, 1' \Sn\ П lim 1Г 1 = 0 n—~ \Gn \
where Sn = Gn - Gn. We have to prove that | pn n Pn 1 lim 1 isom 1 1 obliqv£\ 1 n—^ I Pn I \ isom \
(18)
(19)
(20)
or, equivalently,
lim
П—
\РГ8от П (Sn X Sn)|
\PSom\
0.
(21)
Consider the natural projection map onto the first coordinate pr1 : P™om ^ Gn. What is the cardinality of the inverse image (=fiber over G) of a graph G under pr1? Each isomorphism class of G e Gn consists of at most n! elements, so |pr-1G| ^ n! for every G e Gn, hence
\PZom\ > n! \Gn\ . (22)
In case G is oblique, we can say more: |pr-1G| = n!. Indeed, by Lemma 2, Aut (G) = 1 and thus the orbit SnG consists of n! graphs and the set (G, SnG) is the inverse image of G under pr1. Thus each fiber of the map pr1 : P™om П (G°n x G°n) ^ ^ Gn has cardinality n!. Hence \P™om П (Gn x Gn)\ = n! \Gn\ and so
\PnSoJ > n! \Gn\.
(23)
Considering the fibers of pr1 over the set Sn , we conclude that
\PSom П (Sn xSn)\ ^ n! \Sn\ .
(24)
Finally, combining inequalities 23 and 24 and the result 19, we obtain
n
isom
I pn I
\ isom \
\Sn
n! \Sn\ \Sn\ \Sn\ (25)
n! \Gn \ = \Gn \ = \Gn \ — \Sn\
1 ) |Sn| ' |Gn| / ^ 0, n ^ TO. (26)
■
n
n
n
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G.A. Noskov. Search problems with a promise...
Remark 2. It is shown in [2] a linear time, high probability canonical labeling algorithm for G(n,p) graphs for p = w(ln4n/nlnlnn) and p ^ 1/2. Here, high probability means probability at least 1 — O(n-c) for every c > 0. It follows that Pisom П Pobiique is asymptotically almost certain with the rate of convergence at least 1 — O(n-c) for every c > 0.
Here we prove the main theorem in the following more precise formulation.
Theorem 5. Any algorithm A, constructed in Section 5. to solve the problem SGI (Pisom П Poblique,R) , solves also the problem SGI (Pisom,R) for asymptotically almost all inputs Pisom П Poblique. In other words, the failure probability of the algorithm A tends to zero as n tends to infinity.
7. Applications to probabilistic algorithms
Recall that the input set of the problem SGI (Pisom,R) is I2 x I, the promise set is
Pisom = {(G, H) : 3n such that G,H e Gn, G - H} , (27)
and the relation is
R = {(G,H,4>) : 3n such that G, H eGnA e Sn С b and ф : G - H} . (28)
A.N. Rybalov considered a search graph isomorphism problem with particularly small promise set PY с Pisom [12]. Namely, fix an infinite sequence of graphs Y = (Gn)neN , such that Gn e Gn for all n and define
Pj = {(G, Gn) : n e N, and G, Gn e Gn, G — Gn} . (29)
A.N. Rybalov studied the problems of the type SGI (PY,R) which clearly are all the restrictions of the candid search problem SGI (Pisom,R).
The main result of A.N.Rybalov is the following.
Theorem 6. [12] If there exists a polynomial generic algorithm for
SGI (PY ,R), then there exists a polynomial probabilistic algorithm computing SGI (PY, R) for all inputs.
From this and from our Theorem 5 we derive the following
Theorem 7. If the sequence of graphs у = (Gn)neN consists of oblique graphs, then the problem SGI (PY,R) is polynomial-time solvable and there exists a polynomial probabilistic algorithm computing SGI (PY,R) for all inputs.
Acknowledgements
The author is thankful to A.N.Rybalov for stating the SGI.
Mathematical Structures and Modeling, 2015. N.4(36)
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G.A. Noskov. Search problems with a promise...
проблемы поиска с посулом и изоморфизм графов
Г.А. Носков
д.Ф.-м.н., с.н.с., e-mail: g.noskov@googlemail.com Институт Математики им. С.Л. Соболева СОРАН
Аннотация. Проблема изоморфизма графов рассматривается с точки зрения теории «проблем с посулом», развитой Ивеном, Зелманом и Якоби [4]. Изучается «тау-инвариант» графов и с его помощью решается проблема поиска изоморфизма для асимптотически почти всех графов.
Ключевые слова: проблема разрешимости, проблема поиска с посулом, изоморфизм графов.
