Optimal Query Sets for Generic Automorphism Detection Chad Brewbaker Department of Electrical and Computer Engineering Iowa State University Ames, IA 50010 [email protected] November 19, 2007

Abstract Given a combinatorial object on n elements we classify the minimum set of permutations that need to be tested in order to detect a nontrivial automorphism, or prove that none exist. We show they have size n X

bnc

p X

p∈P RIM ES i=1

pi i!(p

n! . − ip)!(p − 1)

For problem sizes as small as n = 8 an optimal query set can reduce the search space by 90% compared to brute force permutation testing. This size of this set demonstrates that the Hidden Subgroup Problem (HSP) has no tractable solution for arbitrary groups, and domain knowledge of the underlying combinatorial object is required for faster algorithms.

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1

Background and Context

For the past few decades bounds on isomorphism an automorphism algorithms have been a central area of theoretical computer science research. Problems such as graph isomorphism (GI) and graph automorphim (GA)are not known to be NP-complete, however no polynomial time algorithms are known either, making them prime candidates to separate the classes P and NP. Further evidence such as having relatively easy counting complexity [8] is further proof that these problems might someday prove to be a keystone between tractable and untractable problems. With the advent of Shor’s quantum factoring algorithm [18] the there is also recent interest in attempts to abstract the result to find arbitrary hidden subgroups efficiently. In this paper we prove the best possible runtime for automorphism detection on a combinatorial object where no assumptions are made about it’s underlying structure. This both gives an upper bound on the runtime of automorphism computations and shows that the problem of finding hidden subgroups using only an oracle is intractable. For the group theorist our minimal detection sets are “trivally” single elements of the Abellian prime order subgroups of the symmetric group. However to date no enumeration of this set has appeared in the literature, and we have not been able to find a full description of the permutation structure which is critical from an algorithmic prospective. We pose the following problem. Problem 1 (Oracle Automorphism Detection). Given a combinatorial object on n elements, and an oracle to accept or reject whether a given permutation of the combinatorial object is an automorphism, what is the minimal set of queries needed to detect a non-trivial automorphism or show that none exist? Note that by “automorphism” we mean a permutation of the combinatorial object that leaves it structuraly unchanged, and by “non-trivial” we mean solutions not involving the identity permutation. For graphs and hypergraphs the best known algorithms are due to Babai and Luks. In [6] √ it was shown that graph isomorphism and automorphism are in O(exp(c nlogn)). Furthermore, in [14] it was shown that hypergraph isomorphism and automorphism are in O(cn ). This topic was been touched upon from a graph isomorphism setting in [1, 10, 15]. They used permutations composed of concatenated 2 and 3 2

cycles to generate asymptotic bounds. The harder problem of using an oracle to construct a complete generating set of the automorphism group was discussed from a computational learning theory perspective in Vindochandran’s thesis[19]. This work should not be confused with research on “black box group algorithms” pioneered by Szemeredi and Babai [3][7][20]. In their formulation one is presented with a generating set of some group, while in our case no such information is given.

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Terminology

Before proceding we outline our terminology. The framework for our proof will be based on a graph optimization problem. Definition 1 (MIN-DOMINATING-SET). A MIN-DOMINATING-SET is the smallest set D ⊆ V of verticies in a graph such that for each vertex v ∈ V , either v ∈ D or there is a (directed) edge from a vertex in D to v. Also, we will use a special relation between two permutations where having one implies the presence of the other. Definition 2 (Detection). We say that a permutation πi detects permutation πj if πi ∈ (πj )n for some natural number n, i.e. πi is in the cyclic group generated by πj . Thus, if πj is present in a group then πi will also be present. Using the detection relation we can form a useful graph. Definition 3 (Detection Graph of Sn ). Let Gn be a directed graph (without loops) whose vertices are labeled with a bijective mapping of the n! permutations of the symmetric group on n elements. Let πi be the permutation label of vertex i, and πj be the permutation label of vertex j. Place and edge from πi to πj if πi detects πj . Note that any MIN-DOMINATING-SET for the detection graph of Sn provides the minimum set of permutations needed to perform oracle automorphism detection. In the general case MIN-DOMINATING-SET is NP-hard [9]. For those studying MIN-DOMINATING-SET problems Niedemeier’s text[17] is useful as it contains polynomial time pre-processing algorithms to simplify the 3

Figure 1: The detection graph for S3 , the set of permutations on 3 elements ??

search. Also, of note is that this graph seems to exhibit exponential runtime behavior with NAUTY[16] version 2.2.

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Structure of the Minimal Detection Sets

We would now like to show which sets of permutations form a MIN-DOMINATINGSET in our detection graph. Let Sn be the n! permutations for a set of n elements. Given an oracle that tells us if a permutation is in our hidden automorphism subgroup Hn ∈ Sn we would like to construct an efficent testing procedure to detect the non-trivial permutations in Hn . The following permutations will be of interest. Definition 4 (Concatinated Prime Cycle Permutation (CPCP)). A permutation composed of 1 or more prime length cycles, all disjoint, and all of the same size. Lemma 1. Only CPCPs detect CPCPs. Proof. Since the cyclic group of a CPCP only contains CPCPs nothing else can detect them. Lemma 2. Every permutation is detected by a CPCP. Proof. Let n be the size of the cyclic group of a given permutation π. If the permutation is a CPCP, then by the previous lemma we are done. If the cycle is not a CPCP then we can extract a CPCP that detects it by setting taking the element π i , where i is constructed by multiplying all but one prime of the factors of n together. The number i will be modulo zero for every one of its divisors except the prime we have left out, so all other cycles will go to the identity and we will have extracted a CPCP.

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From the previous two lemmas we can derive the following following theorem. Theorem 1. A minimal detection set for Sn consists of one element from each CPCP’s cyclic group.

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Enumeration of the Minimal Detection Sets

We now count the number of elements in the minimum detection set for Sn . The sum over all repeated prime cycles that can fit into the number n is bnc

n X

p X

.

p∈P RIM ES i=1

Now over-count and take all permutations of n composed of i disjoint p cycles. n! − ip)!

pi i!(p

Finally, we only want one element for each cyclic group generated by a CPCP. If the cycle has p elements then it will have period p, however we are not adding the identity permutation, so multiply by 1 . p−1 This gives us a total of n X

bnc

p X

p∈P RIM ES i=1

n! 1 . pi i!(p − ip)! p − 1

By collapsing the denominators we get the following theorem. Theorem 2 (Minimal Detection Set Enumeration). The size of a minimal query set for generic automorphism detection in Sn is n X

bnc

p X

p∈P RIM ES i=1

pi i!(p

5

n! . − ip)!(p − 1)

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Conclusion

The minimum query set can yield significant savings over checking all n! − 1 permutations. This is illustrated in Table 1.

n 3 4 5 6 7 8 9 10 11 12 13

Table 1: Savings for small n Percent Reduction Total Savings 20.0% 1 43.4782608695652% 10 65.546218487395% 78 78.9986091794159% 568 87.060924786664% 4387 93.3654108484834% 37644 97.084703165518% 352300 98.3721611475312% 3569728 98.2174522561291% 39205263 98.7097523238122% 472821292 98.7693990838716% 6150391024

For problem sizes as small as n = 8 there is a 90% reduction in the number of queries required. Even though minimal the minimal query sets are O(n!), their use can achieve significant improvement for tractable problem sizes over brute force testing. The size of the minimal detection sets for n = 1, ..., 19 have sizes {0, 1, 4, 13, 41, 151, 652, 2675, 10579, 59071, 711536, 6180307, 76629775, 873676259, 7496233396, 49493077951, 1571673343007, 24729597043375, 584039297226784}. The size of the minimal detection sets are asymptotically O(n)!. For prime n one element the n cycles must be checked is making the set larger than size (n − 2)!. This demonstrates that hidden subgroup detection is not tractable without additional structural knowledge of the combinatorial object under consideration.

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Acknowledgements

We would like to thank Johnathan D.H. Smith and Pavan Aduri for our useful discussions. Funding for this research was provided by US Dept. of Education GAANN fellowship. 6

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