Let be the set of permutations of the set and be the set of strings in encoding permutations in . The search for minimum size grammars generating the language has sparked a lot of interest in the automata theory and in the complexity theory communities, both in the study of lower bounds [19, 32, 21], and in the study of upper bounds [26, 3, 2]. In particular, a celebrated result due to Ellul, Krawetz and Shallit  states that any context-free grammar generating the language must have size . In this work we complement this line of research by establishing upper bounds for the size of context-free grammars representing a given subgroup . These upper bounds are stated in terms of three structural parameters of connected graphs embedding : the number of vertices, the treewidth and the maximum degree.
We say that a permutation group can be embedded in a graph with vertex set , if and is equal to the restriction of the automorphism group of to its first vertices . A more precise definition of the notion of graph embedding is given in Section 3. For a given class of connected graphs , the -embedding complexity of , denoted by , is defined as the minimum such that can be embedded in an -vertex graph .
Given an alphabet , the symmetric grammar complexity (SGC) of a formal language measures the minimum size of a context-free grammar accepting a permuted version of
. As a matter of comparison, we note that online Turing machines working in spaceand with access to a stack have symmetric grammar complexity . In this setting, the machine reads the input string from left to right, one symbol at a time. While reading this string, symbols can be pushed into or popped from the stack. The transitions relation depends on the current state, on the symbol being read at the input, and on the symbol being read at the top of the stack. The caveat is that the number of symbols used in the stack (which can be up to ) is not counted in the space bound , which can be much smaller than (say ). The SGC of a language is also polynomially related to the minimum size of a read-once branching program with a stack accepting (see for instance ).
1.1 Our Results
We show that the automorphism group of any graph with vertices, maximum degree and treewidth has symmetric grammar complexity at most (Theorem 3). More generally, we show that the SGC of groups that can be embedded in -vertex graphs of maximum degree and treewidth is at most (Theorem 5).
In linear programming theory, it can be shown that there are interesting polytopes, which can only be defined with an exponential (in ) number of inequalities, but which can be cast as a linear projection of a higher dimensional polytope that can be defined with polynomially many variables and constraints. Such a polytope is called an extended formulation of
. Extended formulations of polynomial size play a crucial role in combinatorial optimization because they provide an unified framework to obtain polynomial time algorithms for a large variety of combinatorial problems. For this reason, extended formulations of polytopes associated with formal languages and with groups have been studied intensively during the past decades, both from the perspective of lower bounds[38, 22, 40, 35, 4, 14, 30], and from the perspective of upper bounds [10, 18, 9, 10, 36, 20, 40, 15, 16].
By combining our main theorem 5 with a connection established by Pesant, Quimper, Rousseau and Sellmann  between the extension complexity of a permutation group and the grammar complexity of a formal language, we show that any permutation group that can be embedded in a connected graph with vertices, treewidth , and maximum degree can be represented by polytopes of extension complexity (Theorem 16).
By combining our upper bound from Theorem 5 with the lower bound from , we obtain an interesting complexity theoretic trade-off relating the index of a permutation group with the size, treewidth and maximum degree of a graph embedding this group (Theorem 21). As a corollary of this trade-off, we show that subgroups of with index up to for some small constant have superpolynomial graph embedding complexity on classes of graphs with treewidth and maximum degree (Corollary 22). Additionally, this lower bound can be improved from super-polynomial to exponential on classes of graphs of treewidth (for ) and maximum degree (Corollary 23). In particular, Corollary 23 implies exponential lower bounds for minor-closed families of connected graphs (which have treewidth ).
1.2 Related Work
Proving lower bounds for the size of graphs embedding a given permutation group is a challenging and still not well understood endeavour. It is worth noting that it is still not known whether the alternating group can be embedded in a graph with vertices. We note that by solving an open problem stated by Babai in , Liebeck has shown that any graph whose automorphism group is isomorphic to the alternating group (as an abstract group) must have at least vertices . Nevertheless, a similar result has not yet been obtained in the setting of graph embedding of groups, and indeed, constructing an explicit sequence of groups that have superpolynomial graph embedding complexity is a long-standing open problem . Our results in Corollary 22 and Corollary 23 provide unconditional lower bounds for interesting classes of graphs for any group of relatively small index (index at most for some small enough constant ).
The crucial difference between the abstract isomorphism setting considered in  and our setting is in the way in which graphs are used to represent groups. In the setting of , given a group , the goal is to construct a graph whose automorphism group is isomorphic to . On the other hand, in the graph embedding setting, we want the group to be equal to the action of the automorphism group on its first vertices. In the abstract isomorphism setting it has been shown by Babai that any class of graphs excluding a fixed graph as a minor, there exists some finite group which is not isomorphic to the automorphism group of any graph in . Our Corollary 23 can be regarded as a result in this spirit in the context of graph embedding. While the lower bound stated in Corollary 23 also applies to graphs that are not minor closed, this lower boud is only meaningful for graphs of maximum degree at most .
We observe that in Theorem 5 an exponential dependence on the maximum degree parameter is unavoidable. Indeed, as stated above, the symmetric grammar complexity of the language is . On the other hand, for each , the symmetric group can be embedded in the star graph with vertex set , and edge set , which is a connected graph of treewidth . Nevertheless, it is not clear to us whether the logarithmic factor can be shaved from the exponent of the upper bound . We also note that the connectedness requirement is also crucial for our upper bounds since can be embedded in the discrete graph with vertex set , and edge set .
We let denote the set of non-negative integers and denote the set of positive integers. For each , we let . For each finite set we let denote the set of all subsets of . For each set and each , we let be the set of subsets of of size and the set of subsets of size at most . For a function and a set , we denote by the restriction of to , i.e. the function with for each .
Prefix Closed Sets. For each , we let be the set of all strings over , including the empty string . Let and be strings in . We say that is a prefix of if there exists such that . Note that is a prefix of itself, and that the empty string is a prefix of each string in . A non-empty subset is prefix closed if for each , each prefix of is also in . We note that the empty string is an element of any prefix closed subset of . We say that is well numbered if for each and each , the presence of in implies that also belong to .
Tree-Like Sets. We say that a subset is tree-like if is both prefix-closed and well-numbered. Let be a tree-like subset of . If , then we say that is a child of , or interchangeably, that is the parent of . If for , then we say that is a descendant of . For a node we let denote the set of all descendants of . Note that is a descendant of itself and therefore, . A leaf of is a node without children. We let be the set of leaves of , and be the set of leaves which are descendants of .
Terms. Let be a finite set of symbols. An -ary term over is a function whose domain is a tree-like subset of . We denote by the set of all terms over . If are terms in , and , then we let be the term in which is defined by setting and for each and each .
3 Embedding Permutation Groups in Graphs
For each finite set , we let be the group of permutations of .
If and , then we say that
stabilizes setwise if . Alternatively, we say
that is invariant under . We let be the permutation in
which is defined by setting for each . In other words,
is the restriction of to .
If is a subgroup of , then we let be the set
of permutations in that stabilize setwise. We say that a group stabilizes
if . Alternatively, we say that is invariant under .
We let be the set of restrictions of permutations in
In what follows, for each we write to denote .
Let . An -vertex graph is a pair ,
Isomorphisms and Automorphisms. If and are two -vertex graphs, then an isomorphism between and is a permutation such that for each , if and only if . An automorphism of is an isomorphism between and . We let denote the set of all isomorphisms between and , and let be the set of automorphisms of . If is invariant under then we define .
Let be a subgroup of and be a connected -vertex graph where . We say that is embeddable in if .
In other words, is embeddable in if the image of action of the automorphism group of on its first vertices is equal to . We note that the requirement that the graph of Definition 1 is connected is crucial for our applications.
Let be a class of connected graphs and be a subgroup of . We say that is -embeddable if there exists some graph such that is embeddable in . The embedding complexity of , denoted by is the minimum such that is embeddable in a graph with at most vertices. If no such a graph exists, then we set .
4 Using Grammars to Represent Finite Permutation Groups
A context-free grammar is a -tuple where is a finite set of symbols, is a finite set of variables, is a finite set of production rules, and is the initial variable of . The notion of a string generated by can be defined with basis on the notions of -parse-tree and yield of a -parse-tree, which are inductively defined as follows.
For each the term which sets is a -parse-tree. Additionally, .
If is the empty symbol then the term which sets is a -parse-tree. Additionally, .
If are -parse-trees and is a production rule in , then the term is a -parse-tree. Additionally,
In other words, the yield of is the concatenation of the yields of the subterms .
We say that a -parse-tree is accepting if . We say that a string is generated by if there is an accepting -parse-tree with . The language generated by is the set of strings generated by . The size of is defined as
is the number of symbols/variables in , is the number of elements in and
is the number of elements in .
We denote by the set of context-free grammars over the alphabet .
A context-free grammar is said to be regular if each production rule is either
of the form for some and , or of the form
for some and some . We denote by
the set of regular context-free grammars over the alphabet .
Complexity Measures. If and then we let be the string obtained by permuting the positions of according to . If then we let . In other words, is the language obtained by permuting the positions of each string according to . The symmetric grammar complexity of a language is defined as the minimum size of a context-free grammar generating for some . More precisely,
Analogously, the symmetric regular grammar complexity of a language is defined as the minimum size of a regular grammar generating for some .
We note that the symmetric regular grammar complexity of a language is polynomially related to the minimum size of an acyclic non-deterministic finite automaton accepting some permuted version of , or equivalently to the minimum size of a non-deterministic read-once oblivious branching program accepting . On the other hand, the symmetric context-free complexity of a language is polynomially related to the minimum size of a pushdown automaton accepting some permuted version of .
Let be a permutation in . We let
be the string associated with . For each group we let be the language associated with . The symmetric grammar complexity of is defined as . Analogously, the regular grammar complexity of is defined as .
If and are permutations in , then we let be the permutation that sends each to the number . If is a subset of , we let . Note that if is a subgroup of , is a subgroup of , and , then is a left coset of in . The following proposition, which will be used in the proofs of Lemma 18 and Theorem 5 follows from the fact that context-free languages are closed under homomorphisms.
Let , and be a permutation in . Let be a context-free grammar such that . Then for each permutation there is a context-free grammar of size generating .
Let be a context free grammar generating . Let be the extension of to the set which sets if and if . For each string let . Finally, let be the context-free grammar obtained from by replacing each production rule with the production rule . Clearly, we have that . Additionally, it is straightforward to verify that generates a string if and only if generates the string . Therefore, . ∎
The following theorem, which will be crucial to the proof of our main result (Theorem 5), upper bounds the symmetric grammar complexity of the automorphism group of a graph in terms of the number of its vertices, its maximum degree, and its treewidth. If the latter two quantities are bounded, then this upper bound is polynomial in the number of its vertices.
Let be a connected graph with vertices, treewidth and maximum degree . Then
Additionally, one can construct in time a permutation and a context-free grammar generating the language .
If the graph of Theorem 3 has pathwidth , then one may assume that is a regular grammar. In other words, in this case, .
Theorem 3 can be simultaneously generalized in two ways. First, by allowing grammars to represent not only the automorphism group of a graph, but also groups that can be embedded in the graph. Second, not only the groups themselves but also left cosets of such groups can be represented in the same way. The result of these generalizations is stated in the next theorem.
Let , and suppose that is embeddable on a graph with vertices (), maximum degree , and treewidth . Then, for each ,
Additionally, given and , one can construct in time a permutation (depending only on ) and a grammar generating the language .
If the graph of Theorem 5 has pathwidth , then one may assume that is a regular grammar. In other words, in this case, .
4.1 Proof of Theorem 5
In this section, we will prove Theorem 5, which establishes an upper bound for the symmetric grammar complexity of a permutation group in function of the size, treewidth and maximum degree of a graph embedding . On the way to prove Theorem 5, we will first prove Theorem 3. The proofs of Remarks 4 and 6 follow by small adaptations of the proofs of Theorems 3 and 5 respectively.
Subtree-Like Sets and Subterms. Let , be a tree-like set, and be the longest common prefix of and . Let and . The distance between and is defined as . We call a set subtree-like if there exists a , such that is a prefix of every , and if the set is prefix-closed. In particular, for each , the set is subtree-like. One can obtain from a tree-like set by making well-numbered in the obvious way. We call the tree-like set induced by . For a set , we call the smallest subtree-like set containing the closest ancestral closure of . For any subtree-like set , we call a subterm of . If is the induced tree-like set of , then we call the corresponding term with the -term induced by . For a position , we denote by the subterm of rooted at , i.e. we let .
Neighborhood of a Vertex, and Induced Subgraphs. Let be a -vertex graph. For a vertex , we let be the neighborhood of . If then we let be the neighborhood of . Finally, we let be the closed neighborhood of . The subgraph of induced by is defined as where .
Tree decomposition as Terms. If we regard the set as an alphabet, then each width- tree decomposition of a graph may be regarded as a term over . More precisely, let be an -vertex graph and . A width- tree decomposition (or simply tree decomposition, if is clear from the context) of is a term satisfying the following axioms.
For each vertex and each of its neighbors , there is a position such that .
For each vertex , the set induces a subterm of .
The treewidth of , is defined as the smallest non-negative integer such that admits a width- tree decomposition.
Annotated Tree Decompositions. Let be an -vertex graph, and be subsets of such that , and be a bijection. We say that is a partial automorphism of if is an isomorphism from the subgraph of induced by to the subgraph of induced by . Next, we define the notion of annotated tree decomposition of a graph . These are tree-decompositions whose bags are annotated with partial automorphisms.
Definition 7 (Annotated Bags).
Let be an -vertex graph and . A -annotated bag is a pair , where , and is a function satisfying the following two properties.
. In other words, the image of under is equal to the closed neighborhood of the image of under .
is a partial automorphism of .
We let be the set of all -annotated bags of . If is a -annotated bag in , then we denote the first coordinate of by and the second coordinate of by . In other words, . We let be the map that takes an annotated bag and sends it to the bag . In other words, the map erases the second coordinate of the annotated bag . We extend to terms in positionwise. More precisely, for each term , we let be the term in where and for each . We say that a term is an annotation of a term if . Note that a term may have many annotations.
In Definition 7, once a subset is fixed, there are at most choices for the image of under the partial isomorphism . Once such an image is fixed, for each vertex there are at most ways of mapping the neighbors of to the neighbors of . Hence there are at most choices for obtaining a partial automorphism for a fixed image of . Therefore, by noting that , we have the following observation.
Let be a graph of maximum degree and let . Then, .
Definition 9 (Annotated Tree Decomposition).
Let be a term in . We say that is an annotated width- tree decomposition if the following conditions are satisfied.
is a tree decomposition.
for each with children , and for each , the restriction of to is equal to the restriction of to .
Intuitively, the first condition states that if we take an annotated tree decomposition and forget annotation then the result is a tree-decomposition of . The second condition guarantees that the annotation is consistent along the whole tree decomposition, in the sense that for each vertex , if the partial automorphism of one bag sends to vertex , then the partial automorphism of each bag sends to . Each annotated tree decomposition gives rise to a map which sets for each . We call the map the annotation morphism of . The following lemma is the main technical tool of this section.
Let be an -vertex graph of treewidth and . Then, is an automorphism of if and only if there exists an annotated tree decomposition of such that .
(Only if direction.) First, we show that if is an automorphism of then there is an annotated tree decomposition such that . Let be a width- tree decomposition of and suppose is an automorphism. Then, we construct an annotated tree decomposition with by letting for each , and . In other words, the annotation of each bag is simply the restriction of to the closed neighborhood of . Clearly, by construction we have that . Therefore, it is enough to show that is an annotated tree decomposition. Since is an automorphism, one immediately verifies that for each , is an isomorphism from to , i.e. condition (2) of Definition 7 holds. Condition (2) of Definition 9 is satisfied as well, since by construction, for any pair of positions and , .
(If direction) Suppose that there is an annotated width- tree decomposition of such that . We show that is an automorphism of . First, we argue that is well-defined. Since is a tree decomposition, we have by 1 that for each , there is at least one position such that . By property 3 of tree decompositions and condition (2) of Definition 9, we can conclude that is assigned a unique value, and therefore that is a well defined function. It remains to argue that is surjective, injective, and indeed an automorphism.
Let . If and , then there exists a such that and .
From Claim 11, the following claim follows straightforwardly by induction on the length of paths.
Let . If and there exists a path from to , then there exists a vertex such that .
The proof is by induction on the length of paths. In the base case, the path has length . In this case, and the claim follows trivially by setting . Now, let , and assume that the claim is true for every path of length at most . Let be a path of length from to . Then, by the induction hypothesis, there exists such that . Now since is an edge in , by Claim 11, we have that there exists a such that . ∎
Since has at least one vertex, we have that there exist vertices such that . Now, since is connected, there is a path from to any other vertex in . Therefore, from Claim 12 and from the fact that is well-defined, we can conclude that is surjective.
Since is a surjective map whose domain and codomain have the same size, we can infer that is injective as well.
Now, suppose . By property 2 of tree decompositions, there is a position such that . Let and . By condition (2) of Definition 7, is an isomorphism from to , so we know that . Since and , we have that . On the other hand, if , then by Claim 11, .
This concludes the proof that is an automorphism. ∎
A tree decomposition is called permutation yielding, if there is a bijection such that for each leaf , .
In other words, a tree decomposition is permutation yielding if each vertex occurs in precisely one leaf bag. The next lemma shows that any tree decomposition can be transformed in polynomial time into a permutation yielding tree decomposition of same width. We note that a statement analogous to Lemma 14 can also be obtained by observing that tree-decompositions can be converted in polynomial time into branch decompositions of roughly the same width . We include a proof of Lemma 14 for completeness.
Let be an -vertex graph, , and a width- tree decomposition of . Then, one can construct from in polynomial time a permutation yielding width- tree decomposition.
For each position , let be the number of children of in . We can assume that for each , there is at least one position such that , otherwise we could pick an arbitrary position with and add a leaf and set . Hence, we can assume that there is a set containing precisely one leaf with for each . Let be the closest ancestral closure of in . We let be the -term induced by the closest ancestral closure of . Note that .
By construction, there is a bijection with for each . Let