The -Simultaneous Conjugacy problem in the symmetric group asks whether there exists a permutation in which simultaneously conjugates two given -tuples of permutations from . More formally, given two ordered -tuples and of permutations from , is there a permutation such that holds for all indices ? To save words, we shall refer to this problem as -SCP in or even just as SCP.
This problem arises in many forms in various fields of mathematics and computer science, in particular, when deciding whether two objects from a given class are structurally equivalent. A brief list includes the following: in the theory of covering graphs, the problem of equivalence of covering projections , and moreover, the construction of all regular covering projections along which a given group of automorphisms lifts [8, 9]; in the theory of maps on surfaces, the question whether two oriented maps on a closed surface are combinatorially isomorphic ; in computational group theory, the problem whether the centralizer in the symmetric group of a given group is non-trivial . Last but not least, in the design of efficient and fast interconnection networks for computer systems, the question of equivalence of permutation networks also reduces to the SCP [11, 12].
Because of its fundamental importance, the complexity of the -SCP in has been studied since mid seventies . The problem can be viewed as a special case of the graph isomorphism problem. More precisely, let be an arc-colored (multi)digraph on the vertex set such that there is an arc from to colored if and only if maps to , for . The permutation digraph is defined in a similar fashion. (See Section 2 for a more formal definition.) The permutation that simultaneously conjugates the two tuples is precisely a color and direction preserving isomorphism from onto (assuming that permutations in are multiplied from left to right). The graph isomorphism problem is hard in general: no polynomial time algorithm is known, nor is the problem known to be NP-complete. However, there is a recent result due to Babai  presenting a quasipolynomial time algorithm.
In our context, things are fundamentally different. Namely, when considering the connected components of and , the additional structure imposed by colors is so strong that every color and direction preserving isomorphism is uniquely determined by the image of one arbitrary vertex. This implies that testing for the existence of such an isomorphism can be done in polynomial time. The first algorithm for the -SCP in was proposed in 1977 by Fontet running in time . Five years later, the algorithm was independently rediscovered by Hoffmann . An important special case of the SCP occurs when the tuples and generate transitive permutation groups or, equivalently, when and are connected. This restricted problem, referred to as the transitive SCP, was considered by Sridhar in 1989 . However, his -time algorithm does not work correctly as we recently showed in . Moreover, in the same paper we also showed that the transitive SCP can be solved in subquadratic time in at a given ; more precisely, we developed an algorithm with the running time .
A natural question arises whether the -SCP in can also be solved in subquadratic time in at a given . The following main result answers the question affirmatively.
Given a positive integer , the -SCP in the symmetric group can be solved in time.
The main idea behind our approach is as follows. First, we define two extreme cases depending on the number of connected components on the one hand, and on the size of individual components on the other hand. Second, a combination of solutions to these two extreme cases then yields the desired result in general.
As for the extreme cases, we say that a connected component is large, if it consists of vertices, and small otherwise. The first extreme case is when a digraph consists of only large components (and consequently, there are of them). In the other extreme the digraph consists of only small components (and consequently, there are of them). In the first case, we simply consider each pair of connected components of the same size and test whether they are isomorphic by applying the above mentioned subquadratic algorithm from . As for the other case, a different specially tailored approach is used. To this end, we present a canonical-labeling-based algorithm that takes time; however, when both digraphs consist only of small connected components its running time decreases to subquadratic in at a given .
The structure of the paper is the following. Section 2 contains the necessary notation and basic definitions to make the paper self-contained. In Section 3 we present a canonical-labeling-based algorithm for the SCP. The main theorem is proven in Section 4. We conclude the paper by discussing some open problems in Section 5.
2 Permutation digraphs and colour-isomorphism
We establish some notation and terminology used in the paper. For the concepts not defined here see .
For and , we write for the image of under the permutation rather than by the more usual . Let be a -tuple of permutations in . The permutation digraph of is a pair , where is the set of vertices, and
is the set of ordered pairs, , called arcs. The size of is , while the degree of is . An arc has its initial vertex , terminal vertex , and color ; the vertex is also referred to as the out-neighbour of coloured . The vertices and are the end-vertices of .
A walk from a vertex to a vertex in a permutation digraph is an alternating sequence of vertices and arcs in such that for each , the vertices and are the end-vertices of the arc . If for any two vertices and in there is a walk from to , we say that is connected. Clearly, is connected if and only if the tuple generates a transitive subgroup of . A subdigraph of consists of a subset and a subset such that every arc in has both end-vertices in . A walk in a subdigraph of is a walk in consisting only of arcs from . If is not connected, its maximal connected subdigraphs are called the connected components of . Note that there are no arcs between connected components, and so the components are also permutation digraphs of degree .
A colour-isomorphism between two permutation digraphs and is a pair of bijections, where and such that , and for any arc . If there is a colour-isomorphism between and , we say that and are colour-isomorphic, and we write .
Let be the set of permutation digraphs of size and degree , and let be a set of strings over some fixed-sized alphabet. A labeling function for is a function . Such a function is canonical whenever for all a colour-isomorphism from onto exists if and only if . In this case, is the canonical label of .
3 A canonical labeling algorithm
We present an algorithm for finding a canonical label of a permutation digraph of size and degree based on publicly known techniques. It runs in time in general, but in the case when consists of only small connected components its running time decreases to subquadratic in at a given .
We first handle the case when is connected. For a fixed we relabel the vertices of in a breadth-first-search order starting at , see the algorithm Relabel. The out-neighbours of a current vertex are visited in the ascending order of colours of the respective out-going arcs (lines 7-11 in Relabel). Let be the resulting relabeling. The relabelled digraph induced by is , where (line 12 in Relabel).
The code of a permutation digraph is
which is a string of length over obtained by concatenating, in turn, the images of under the permutations . For a connected digraph , let denote the lexicographically smallest string from among codes , . We now prove that is the canonical label of .
Let be the set of all connected permutation digraphs of size and degree , and let be the set of all strings of length over . Then the function defined by is a canonical labeling function for .
We first show that if , then and are colour-isomorphic. Let be a vertex for which the permutation digraph returned by Relabel has code . Similarly, let be a vertex for which the permutation digraph returned by Relabel has code . By assumption, it follows that . Hence , and since and it follows that .
Conversely, let be a colour-isomorphism mapping onto , and let be a vertex for which the permutation digraph returned by Relabel has code . Next, let and consider the permutation digraph returned by Relabel. Note that and hence . It remains to prove that . Suppose to the contrary that for some returned by Relabel, the string is lexicographically smaller than the string . Consider now the permutation digraph returned by Relabel. Similarly as above, . Since is lexicographically smaller then , it follows that is lexicographically smaller than . A contradiction. ∎
Next, we bound the time complexity of computing .
The canonical label of a connected permutation digraph of size and degree can be computed in time.
One call of Relabel takes time in order to construct , while its code can also be computed in linear time. Since this has to be repeated for each , the total running time for constructing the codes is . Clearly, choosing the lexicographically smallest code does not increase this time bound. ∎
In the reminder of this section we consider the case when is not connected. Let us denote the connected components of by , and recall that each such component is a permutation digraph of degree . Further, let us concatenate the respective canonical labels in such an order that the resulting string is lexicographically smallest. The following result shows that is the canonical label of .
Let be the set of all permutation digraphs of size and degree , and let be the set of all strings of length over . Then the function defined by is a canonical labeling function for .
The proof follows from the description of above and Proposition 3.1. ∎
In the next section we will make use of the following result regarding permutation digraphs when all connected components have equal size.
If a permutation digraph of size and consists of precisely equal-sized connected components, then its canonical label can be computed in time.
Let be the connected components of . Since each component is of size we can find its canonical label , by Lemma 3.2, in time. Consequently, the total running time for constructing the canonical labels of all components is . To compute , all we need to do is to sort these labels. Using radix sort this can be done in time , which obviously does not increase the time bound . ∎
4 Proof of main result
Recall that a tuple is simultaneously conjugate to a tuple if and only if the permutation digraph is color-isomorphic to the permutation digraph . Before considering the general case when the digraphs and have variable size components, we deal with two extreme cases, namely, when the digraphs have either only equal-sized small components or only equal-sized large components.
Let and be permutation digraphs, each of size and degree , and let both and consist of only small equal-sized connected components. Then we can test whether and are colour-isomorphic in time at a given .
Let be the number of connected components. By Corollary 3.4 we can compute the canonical labels of both digraphs and hence perform the isomorphism test in time . Since all components are small, we have and consequently at a given . ∎
Let and be permutation digraphs, each of size and degree , and let both and consist of only large equal-sized connected components. Then we can test whether and are colour-isomorphic in time.
Let be the number of connected components. Since all components are large, we have . Obviously, at most pairs of components, each of size , need to be tested for isomorphism. By , this requires a total of time. ∎
We are now ready to prove the main result.
Proof of Theorem 1.1.
Finding the components of and requires time. If and do not have an equal number of components of the same size, they are not isomorphic, which can be tested by sorting the sizes of the components in time . So, let and have components of size , , where without loss of generality we may assume that components of sizes are large, and the remaining ones are small. Obviously, components of different sizes can be tested separately. By Lemma 4.2, we can test large components of size for isomorphism in time . On the other hand, by Corollary 3.4, we can test small components of size in time . Finally, for a large enough constant the total time is bounded from above by
The max-term is since for each large component as well as for each small component. The final result follows as . ∎
5 Concluding remarks
It remains an open problem whether for a given positive integer the -SCP in the symmetric group can be solved in a strongly subqudratic time in , that is, in time for some . Further, completely unanswered is the question of the problem’s lower bound, except for the trivial one, . The obvious question is whether it can be raised to reflecting erroneous Sridhar’s upper bound, or even to a higher bound by proving conditional lower bounds based on conjectures of hardness for well-studied problems, as it was already done for a number of other problems.
L. Babai, Graph isomorphism in quasipolynomial time [extended abstract]. In:
Proceedings of the Forty-eighth Annual ACM Symposium on Theory of Computing, STOC ’16, pages 684–697, New York, NY, USA, 2016.
-  A. Brodnik, A. Malnič, R. Požar, The simultaneous conjugacy problem in the symmetric group, arXiv:1907.07889v2.
-  R. Diestel, “Graph Theory”, Springer-Verlag, New York, 2005.
-  M. Fontet, Calcul de Centralisateur d’un Grupe de Permutatations, Bull. Soc. Math. France Mem. 49-50 (1977), 53–63.
-  C. M Hoffmann, Subcomplete Generalization of Graph Isomorphism. J. of Comp. and Sys. Sci. 25 (1982), 332–359.
-  A. Malnič, R. Nedela, M. Škoviera, Lifting graph automorphisms by voltage assignments. European J. Combin. 21 (2000), 927–947.
-  A. Malnič, R. Nedela, M. Škoviera, Regular homomorphisms and regular maps. European J. Combin. 23 (2002), 449–461.
-  P. Potočnik, R. Požar, Smallest tetravalent half-arc-transitive graphs with the vertex-stabiliser isomorphic to the dihedral group of order 8, J. Combin. Theory Ser. A 145 (2017), 172–183.
-  R. Požar, Computing stable epimorphisms onto finite groups. J. Symbolic Comput. 92 (2019), 22–30.
-  Á. Seress, “Permutation group algorithms”, Cambridge Tracts in Mathematics 152. Cambridge University Press, 2003.
-  M. A. Sridhar, A fast algorithm for testing isomorphism of permutation networks. IEEE Trans. Computers (TC) 38(6) (1989), 903–909.
-  A. Yavuz Oruç, M. Yaman Oruç, On testing isomorphism of permutation networks. IEEE Trans. Computers (TC) 34 (1985), 958-962.