Lower Bounds on the Complexity of Model-Checking 111A preliminary short version of this paper appeared in the Proceedings of STACS’12.
One of the most important algorithmic meta-theorems is a famous result by Courcelle, which states that any graph problem definable in monadic second-order logic with edge-set quantifications (i.e., model-checking) is decidable in linear time on any class of graphs of bounded tree-width. Recently, Kreutzer and Tazari KT10b () proved a corresponding complexity lower-bound—that model-checking is not even in XP wrt. the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH).
In this paper we present a closely related result. We show that even model-checking with a fixed set of vertex labels, but without edge-set quantifications, is not in XP wrt. the formula size as parameter for graph classes which are subgraph-closed and whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari; we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and we assume a different set of problems to be efficiently decidable, namely -definable properties on vertex labeled graphs instead of -definable properties on unlabeled graphs.
Our result has an interesting consequence in the realm of digraph width measures: Strengthening the recent result Ganianetal10 (), we show that no subdigraph-monotone measure can be “algorithmically useful”, unless it is within a poly-logarithmic factor of undirected tree-width.
graph MSO logic; tree-width; digraph width; intractability
A famous result by Courcelle, published in 1990, states that any graph property definable in monadic second-order logic with quantification over vertex- and edge-sets () can be decided in linear time on any class of graphs of bounded tree-width Cou90 (). More precisely, the model-checking problem for a graph of tree-width and a formula , i.e. the question whether , can be solved in time . In the parlance of parameterized complexity, this means that model-checking is fixed-parameter tractable (FPT) with respect to the tree-width as parameter.
This result has a strong significance. As logic can express many interesting graph properties, we immediately get linear-time algorithms for important -hard problems, such as Hamiltonian Cycle, Vertex Cover, and 3-Colorability, on graphs of bounded tree-width. Such a result is called an algorithmic meta-theorem, and many other algorithmic meta-theorems have since appeared for other classes of graphs—see e.g. Gro08 (); Kre11 () for a good survey.
As can be seen, Courcelle’s theorem is a fast and relatively easy way of establishing that a problem can be solved efficiently on graphs of bounded tree-width. However, one may ask how far this result could be generalized. That is, is there another reasonable graph class of unbounded tree-width such that model-checking remains tractable on this class? Considering how important this question is for theoretical understanding of what makes some problems on certain graph classes hard, it is surprising that until recently there has not been much research in this direction.
1.1 Related prior work
The first “lower bound” to Courcelle’s theorem, by Makowski and Mariño, appeared in mm03 (). In that paper the authors show that if a class of graphs has unbounded tree-width and is closed under topological minors, then model-checking for is not fixed-parameter tractable unless . More recently, a stronger lower bound result by Kreutzer—not requiring the class to be closed under minors—appeared in Kre09CSL (). In that paper, Kreutzer used the following version of “unbounding” the tree-width of a graph class:
Definition 1.1 (Kreutzer and Tazari Kre09CSL (); KT10b ()).
The tree-width of a class of graphs is strongly unbounded by a function if there is and a polynomial s.t. for all there is a graph with the following properties:
the tree-width of is between and and is greater than , and
given , the graph can be constructed in time .
The degree of the polynomial is called the gap-degree of (with respect to ). The tree-width of is strongly unbounded poly-logarithmically if it is strongly unbounded by , for all .
In other words, saying that tree-width of is strongly unbounded means that
there are no big gaps between the tree-width of witness graphs (those certifying that the tree-width of -vertex graphs in is greater than ), and
we can compute such witnesses effectively—in sub-exponential time wrt. .
The main result of Kre09CSL () is the following theorem:
Theorem 1.2 (Kreutzer Kre09CSL ()).
Let be a fixed set of (at least two) colours, and be a class of graphs such that
the tree-width of is strongly unbounded poly-logarithmically;
is closed under -colourings (i.e., if and is obtained from by colouring some vertices or edges by colours from , then ); and,
is constructable (i.e., given a witness graph in , a certain substructure can be computed in polynomial time).
Then , the model-checking problem on all -coloured graphs from , is not in (and hence not in —see Section 2.3 for a definition of these complexity classes), unless all problems in the polynomial-time hierarchy can be solved in sub-exponential time.
This would, of course, mean that the Exponential-Time Hypothesis (ETH) IPZ01 () fails. The results of Kre09CSL () have been improved by Kreutzer and Tazari in KT10a (), where the constructability requirement (3) was dropped.
Theorem 1.3 (Kreutzer and Tazari KT10b ()).
Let be a class of graphs such that
the tree-width of is strongly unbounded poly-logarithmically; and
is closed under taking subgraphs, i.e. and implies .
Then , the model-checking problem on , is not in unless all problems in the polynomial-time hierarchy can be solved in sub-exponential time.
Note that , to be closed under subgraphs, is a strictly weaker condition than previous , to be closed under -colourings (of edges, too).
1.2 New contribution
In this paper we prove a result closely related to Kreutzer–Tazari’s Kre09CSL (); KT10a (); KT10b () but for logic with a fixed set of vertex labels. The role of vertex labels in our paper is similar to that of colours in Kre09CSL (); KT10a (), but weaker in the sense that the labels are not assigned to edges.222The reason we use the term labels and not colours is to be able to clearly distinguish between vertex-labeled graphs and the coloured graphs used in Kre09CSL (); KT10a (), where colours are assigned to edges and vertices. In contrast to the work by Kreutzer and Tazari, we assume a different set of problems—those expressible by on graphs with vertex labels from a fixed finite set —to be efficiently solvable on a graph class in order to derive an analogous conclusion.
Before stepping further, we mention one more fact. There exist classes of -labeled graphs of unbounded tree-width on which , the model-checking problem on , is polynomial time solvable, e.g. classes of bounded clique-width or rank-width. But it is important to realize that these classes are not closed under taking subgraphs.
Our main result then reads—cf. Section 4:
Theorem 1.4 (reformulated as Theorem 4.1).
Assume a (suitable but fixed) finite label set , and a graph class satisfying the following two properties:
is closed under taking subgraphs and under -vertex-labelings,
the tree-width of is densely unbounded poly-logarithmically (see Def. 3.3).
Then , the model-checking problem on all -vertex-labeled graphs from , is not in unless the non-uniform Exponential-Time Hypothesis fails.
Kreutzer and Tazari require witnesses as in (ii) of Definition 1.1 of KT10b () to be computable effectively in their proofs. It is unclear how this can be done, and hence they simply add this as a natural requirement on . Furthermore, the construction of certain obstructions (grid-like minors) used in their proof requires an involved machinery KT10a (). We adopt a different position (note our “densely unbounded” in Definition 3.1 vs. “strongly unbounded”) and avoid both aspects by using a stronger complexity-theoretic assumption, namely the non-uniform ETH instead of the ordinary ETH. In this way, we can get the obstructions as advice “for free.”
Our result applies to model-checking on -vertex-labeled graphs, while the result of KT10b () applies to over unlabeled graphs. There are problems that can be expressed in and not in and vice versa (take Red-Blue Dominating Set vs. Hamiltonian Cycle, for instance). If, however, the set of labels is fixed for both, has much weaker expressive power than - due to missing edge-set quantifications (see Section 2). In particular, note that many of the existing algorithmic meta-theorems (e.g. Cou90 (); CMR00 ()) that deal with -definable properties handle unlabeled as well as (vertex-)labeled inputs with equal ease. However, extending e.g. the results of CMR00 () from to is not possible unless .
Finally, because of the free advice, our proof does not need technically involved machinery such as the simulation of a run of a Turing machine encoded in graphs KT10b (). This makes our proof shorter and exhibits its structure more clearly.
After all, Theorem 1.4 gives a good indication (II) that poly-logarithmically unbounded tree-width along with closure under subgraphs is a strong enough condition for even the bare model-checking to be intractable (modulo appropriate complexity-theoretic assumptions).
Moreover, if we assume that the label set is potentially unbounded, then we obtain a stronger result (getting us even “closer” to KT10b ()):
Theorem 1.5 (reformulated as Theorem 5.4).
model-checking with vertex labels ( depending on the formula size) is not tractable for a graph class satisfying (a) and (b) of Theorem 1.4 unless every problem in the polynomial-time hierarchy is in .
Finally, as a corollary, we obtain an interesting consequence in the area of directed graph (digraph) width measures, improving upon Ganianetal10 ().
Theorem 1.6 (reformulated as Theorem 6.2).
Assume a (suitable but fixed) finite label set , and a digraph width measure such that
is monotone under taking subdigraphs and -vertex-labelings, and
, the model-checking problem on all -vertex-labeled digraphs from , is in wrt. and as parameters.
Then, unless the non-uniform ETH fails, for all the tree-width of the class , the underlying undirected graphs of digraphs of -width at most , is not densely unbounded poly-logarithmically.
Informally, a digraph width measure that is subdigraph-monotone and algorithmically “powerful” is at most a poly-logarithmic factor of the tree-width of the underlying undirected graph—cf. Section 6.
In Section 2 we overview some standard terminology and notation. Section 3 then includes the proof outline and the core technical concepts: unbounding tree-width (Definition 3.3), the grid-like graphs of Reed and Wood RW08 () (Proposition 3.6), and a new way of interpreting arbitrary graphs in labeled grid-like graphs of sufficiently high order (Lemma 3.8). These then lead to the proof of our main result, equivalently formulated as Theorem 4.1, in Section 4. Two extensions of the main result appear in Section 5; the first one discussing a stronger collapse of PH (under allowing non-fixed labeling of graphs), and the second one considering classes of (just) poly-logarithmically unbounded tree-width, i.e. those which may not be strongly/densely unbounded. A consequence for directed width measures is then discussed in Section 6, followed by concluding remarks in Section 7.
The graphs we consider in this paper are simple, i.e. they do not contain loops and parallel edges. Given a graph , we let denote its vertex set and its edge set. A path of length in is a sequence of vertices such that all are pairwise distinct and for every . Let be a family of sets for . Then the intersection graph on is the graph where and iff .
Let be a set of labels. A -vertex-labeled graph, or -graph for short, is a graph together with a function , assigning each vertex a set of labels, and we write to denote this graph. For a graph class , we shortly write for the class of all -graphs over , i.e. contains all where and is an arbitrary -vertex-labeling of . Note that, unlike in e.g. Kre09CSL (), we do not allow labels for edges, which is in accordance with our focus on logic of graphs.
2.2 logic on graphs
Monadic second-order logic () is an extension of first-order logic by quantification over sets. On the one-sorted adjacency model of graphs it reads as follows:
The language of , monadic second-order logic of graphs, contains the expressions built from the following elements:
variables for vertices, and for sets of vertices,
the predicates and with the standard meaning,
equality for variables, the connectives and the quantifiers .
Note that we do not allow quantification over sets of edges (as edges are not elements). If we considered the two-sorted incidence graph model (in which the edges formed another sort of elements), we would obtain aforementioned , monadic second-order logic of graphs with edge-set quantification, which is strictly more powerful than , cf. EF99 (). Yet even has strong enough expressive power to describe many common problems.
The 3-Colouring problem can be expressed in as follows:
The logic can naturally be extended to -graphs. The monadic second-order logic on -vertex-labeled graphs, denoted by , is the natural extension of with unary predicates for each label , such that holds iff .
2.3 Parameterized complexity and model-checking
Throughout the paper we are interested in the problem of checking whether a given input graph satisfies a property specified by a fixed formula . This problem can be thought of as an instance of a problem parameterized by , as studied in the field of parameterized complexity (see e.g. fg06 () for a background on parameterized complexity).
A parameterized problem is a subset of , where is a finite alphabet and . A parameterized problem is said to be fixed-parameter tractable if there is an algorithm that given decides whether is a yes-instance of in time where is some computable function of alone, is a polynomial and is the size measure of the input. The class of such problems is denoted by . The class is the class of parameterized problems that admit algorithms with a run-time of for some computable , i.e. polynomial-time for every fixed value of .
We are dealing with a parameterized model-checking problem where is a class of graphs; the task is to decide, given a graph and a formula , whether . The parameter is , the size of the formula . We actually consider the labeled variant for being a class of -graphs.
2.4 Interpretability of logic theories
One of our main tools is the classical interpretability of logic theories Rab64 () (which in this setting is analogical to transductions as used e.g. by Courcelle, cf. CE12 ()). To describe the simplified setting, assume that two classes of relational structures and are given. The basic idea of an interpretation of the theory into is to transform formulas over into formulas over in such a way that “truth is preserved”:
First, one chooses a formula intended to define in each structure a set of individuals (new domain) , where denotes the set of individuals (domain) of .
Then, one chooses for each -ary relational symbol from a formula , with the intention to define a corresponding relation and . With these formulas one defines for each the relational structure intended to correspond with structures in .
Finally, there is a natural way to translate each formula (over ) into a formula (over ), by induction on the structure of formulas. The atomic ones are substituted by corresponding chosen formulas (such as ) with the corresponding variables. Then one proceeds via induction simply as follows:
The whole concept is shortly illustrated in by the scheme in Figure 1.
Definition 2.3 (Interpretation between theories).
Let and be classes of relational structures. Theory is interpretable in theory if there exists an interpretation as above such that the following two conditions are satisfied:
For every structure , there is such that , and
for every , the structure is isomorphic to some structure of .
Furthermore, is efficiently interpretable in if the translation of each into is computable in polynomial time and the structure , where , can be computed from any in polynomial time.
2.5 Exponential-Time Hypothesis
The Exponential-Time Hypothesis (ETH), formulated in IPZ01 (), states that there exists no algorithm that can solve -variable 3-SAT in time . It was shown in IPZ01 () that the hypothesis can be formulated using one of the many equivalent problems (e.g. -Colourability or Vertex Cover)—i.e. sub-exponential complexity for one of these problems would imply the same for all the others.
ETH can be formulated in the non-uniform version: There is no family of algorithms (one for each input length) which can solve -variable 3-SAT in time . In theory of computation literature, “non-uniform algorithms” are often referred to as “fixed-sized input circuits” where for each length of the input a different circuit is used. Yet another way of thinking about non-uniform algorithms is as having an algorithm that is allowed to receive an oracle advice, which depends only on the length of the input. As mentioned in CSH08 (), the results of IPZ01 () hold also for the non-uniform ETH.
3 Key Technical Concepts
We are going to show via a suitable multi-step reduction, that the potential tractability of model-checking on our graph class (whose tree-width is densely unbounded poly-logarithmically), implies sub-exponential time algorithms for problems which are not believed to have one (cf. ETH). The success of the reduction, of course, rests on the assumptions of being subgraph-closed and of unbounded tree-width. So, at a high level, our proof technique is similar to that of Kreutzer and Tazari.
However, there are some crucial differences. While KT10b () uses the effectiveness assumption in Definition 1.1. ii and some further technically involved algorithms to construct a “skeleton” in the class suitable for their reduction, in our reduction we will obtain a corresponding labeled skeleton in the class “for free” from an oracle advice function which comes with the non-uniform computing model. That is why our complete proof is also significantly shorter than that in KT10b (). Additionally, our arguments shall employ a result on strong edge colourings of graphs in order to “simulate” certain edge sets within the language, thus avoiding the need for a more expressive logic such as .
3.1 Unbounding Tree-width
Following Definition 1.1, we aim to formally describe what it means to say that the tree-width of a graph class is not bounded by a function . Recall (see also Kre09CSL (); KT10b ()) that it is not enough just to assume for some sporadic values of with huge gaps between them, but a reasonable density of the surpassing tree-width values is also required. Hence we suggest the following definition as a weaker alternative to Definition 1.1:
Definition 3.1 (Densely unbounded tree-width).
For a graph class , we say that the tree-width of is densely unbounded by a function if there is a constant such that, for every , there exists a graph whose tree-width is and . The constant is called the gap-degree of this property.
Comparing to Definition 1.1 one can easily check that if the tree-width of a class is strongly unbounded by a function , then the tree-width is densely unbounded by with the same gap-degree, and the witnessing graphs of Definition 3.1 can be computed for all efficiently—in sub-exponential time wrt. . Hence our definition is weaker in this respect.
For simplicity, we are interested in graph classes whose tree-width is densely unbounded by every poly-logarithmic function of the graph size. That is expressed by the following simpler definition:
Definition 3.3 (Densely unbounded tree-width II).
For a graph class , we say that the tree-width of is densely unbounded poly-logarithmically if it is densely unbounded by for every .
That is, for every the following holds: for all there exists a graph whose tree-width is and with size . (The gap-degree becomes irrelevant in this setting.)
3.2 Grid-like graphs
The notion of a grid-like minor has been introduced by Reed and Wood in RW08 (), and extensively used by Kreutzer and Tazari KT10a (); KT10b (). In what follows, we avoid use of the word “minor” in our definition of the same concept, since “-minors” where is grid-like are always found as subgraphs of the target graph, which might cause some confusion.
Definition 3.4 (Grid-like Rw08 ()).
A graph together with a collection of paths, formally the pair , is called grid-like if the following is true:
is the union of all the paths in ,
each path in has at least two vertices, and
the intersection graph of the path collection is bipartite.
The order of such grid-like graph is the maximum integer such that the intersection graph contains a -minor. When convenient, we refer to a grid-like graph simply as to .
Note that the condition (ii) is not explicitly stated in RW08 (), but its validity implicitly follows from the point to get a -minor in , cf. Theorem 3.6. Since the traditional square (and hexagonal, too) grids are grid-like with the horizontal and vertical paths forming the collection , the new concept of having a grid-like subgraph generalizes the traditional concept of having a grid-minor. See also Figure 2.
One can easily observe the following:
Let be a grid-like graph. Then the collection can be split into such that each , , consists of pairwise disjoint paths. Consequently, the maximum degree in is .
The next result is crucial for our paper (while we do not require constructability as in KT10b ()):
Theorem 3.6 (Reed and Wood Rw08 ()).
Every graph with tree-width at least contains a subgraph which is grid-like of order , for some constant .
3.3 interpretation on grid-like graphs
Now we prove the core new technical tool of our paper; showing how the subgraphs of of any grid-like graph can be efficiently -interpreted in itself with a suitable vertex labelling. First, we state a useful result about strong edge colourings of graphs—a strong edge-colouring is an assignment of colours to the edges of a graph such that no path of length three contains the same colour twice.
Theorem 3.7 (Cranston Cra06 ()).
Every graph of maximum degree has a strong edge-colouring using at most colours. This colouring can be found with a polynomial-time algorithm.
For a class of grid-like graphs , let denote the class of all subgraphs of their intersection graphs. Our core tool is the following lemma.
Let be any class of grid-like graphs. There exists a fixed finite set of labels, with , and a graph class , such that the following holds. The theory of has an efficient interpretation in the theory of —the class of all -vertex-labeled graphs over . Stated differently, any where is interpreted in some -graph of .
Note that the use of a class in the statement of the lemma is only a technicality related to (ii) of Definition 2.3. We are actually interested only in interpreting the graphs from , and then simply contains all the graphs that (also accidentally) result from the presented interpretation.
Hence we choose an arbitrary and . The task is to find a vertex labeling such that has an efficient interpretation in the labeled graph . By Theorem 3.7 (cf. also Proposition 3.5), let be a strong edge-colouring of the chosen graph . Let be the bipartition of the paths forming corresponding to the partite sets of . We call the paths of “white” and those of “black”. The remaining paths not in the vertex set of are irrelevant. The edges of white/black paths are also called white/black, respectively, with the understanding that some edges of may be both white and black. For , we let and . According to Proposition 3.5, and .
The key observation, derived directly from the definition of a strong edge-colouring, is that any edge is a white edge iff , and analogously for black edges. This allows us to speak separately about the white and black edges in using only the language of . Another easy observation is that the vertex sets of the paths in have a system of distinct representatives by Hall’s theorem. For if and contains white paths and black paths, then , proving Hall’s criterion. We assign a marker to each such that is the set of the representatives of white paths and is that of black paths (i.e., are not representatives). Finally, we assign another vertex marker to each vertex such that iff where and .
Hence the label set consists of “light” colours coming from values on white paths, another “dark” colours from black paths, and the three singletons described above (altogether binary labels). Note that the actual size of the needed label space over is even much smaller; at most . The label of a vertex then contains the disjoint union , the label if , and finally if .
Now, the interpretation of in is simply as follows: The domain, i.e. the vertex set of , is identified within by a predicate expressing that in . In formal logic language (cf. Section 2), it is . The relational symbol of is then replaced, for s.t. , with
and where () routinely expresses in the fact that belong to the same component induced by white (black) edges in . Precisely,
Clearly, in this interpretation thanks to our choice of . This completes the proof. ∎
Lemma 3.8 will be coupled with the next technical tool of similar flavor used in our previous Ganianetal10 (). We remark that its original formulation was even stronger, making the target graph class planar, but we are content with the following weaker formulation here. We call a graph -regular if all the vertices of have degree either one or three.
Lemma 3.9 ((Ganianetal10, , in Theorem 5.5)).
The theory of all simple graphs has an efficient interpretation in the theory of all simple -regular graphs. Furthermore, this efficient interpretation can be chosen such that, for every formula , the resulting property is invariant under subdivisions of edges; i.e. for every -regular graph and any subdivision of it holds iff .
4 The Main Theorem
Theorem 4.1 (cf. Theorem 1.4).
Let be a finite set of labels, . Unless the nonuniform Exponential-Time Hypothesis fails, there exists no graph class satisfying all the three properties
is closed under taking subgraphs,
the tree-width of is densely unbounded poly-logarithmically,
the model-checking problem is in , i.e., one can test whether in time for some computable function .
We will show that if there exists a graph class satisfying all three properties stated above, then we contradict the non-uniform ETH. Fix (to be determined later from Lemma 3.9) and any sufficiently large such that . By (b) and Definition 3.3, we have that for all there is such that and .
By Proposition 3.6, the graph contains a subgraph which is grid-like as of order , for all sufficiently large . Also by (a). We fix (one of) the -minor in , and denote by the partition of the vertex set of into connected subgraphs that define this minor. Furthermore, by Theorem 3.7, there exists a strong edge colouring of . Define an advice function that acquires the values (whenever is large enough for to be defined as above). Since and , our advice function is sub-exponentially bounded; .
Now we get to the core of the proof (cf. Figure 3). Assume that we get an arbitrary graph and any formula as input. We will show that the model-checking instance can be solved in sub-exponential time wrt. with help of our advice function . By Lemma 3.9, there is an interpretation such that there exists a -regular graph and . Moreover, since is efficient, we can compute efficiently and for a suitable fixed and sufficiently large . Then, we query the oracle advice value . Since our advice is a grid-like graph of order —i.e., its intersection graph has a -minor— has a minor isomorphic to , too. But is -regular and, in particular, has maximum degree three. Hence there exists a subgraph that is isomorphic to a subdivision of (in other words, is a topological minor of ). This subgraph can be straightforwardly computed from the advice over in polynomial time.
By Lemma 3.8 there is another efficient interpretation assigning to a labeling such that . This can actually be computed very easily with help of the advice from along the lines of the proof of Lemma 3.8, not even using the algorithmic part of Theorem 3.7. Finally, we compute in polynomial time the formula . According to Lemma 3.9, is invariant under subdivisions of edges, and so . Then, by the interpretation principle, . The final task is to run the algorithm of (c) on the instance . The run-time is for some depending only on , i.e. only on . Recall that and . Hence we get a solution to the model-checking instance in time for any fixed , with a sub-exponentially bounded oracle advice function .
In particular, if expresses the fact that a graph is 3-colourable (Example 2.2), then this shows that , contradicting non-uniform ETH. ∎
5 Extending the Main Theorem
We can strengthen Theorem 4.1 by showing that even every problem in the Polynomial-Time Hierarchy (PH) Sto76 () is in , i.e., admits subexponential-sized circuits. This stronger new conclusion comes at the price of a stricter assumption on the graph class ; we assume that the model-checking problem is in for every finite set of labels such that , i.e., wrt. the formula size as a parameter determining also the label set . Note that in Theorem 4.1, was a fixed finite set of labels.
We also study what happens if we drop the condition that is densely unbounded, and only require that does not have poly-logarithmically bounded tree-width (i.e., there might be arbitrarily large gaps between the graphs witnessing large tree-width in ). Then we can show that all problems in the Polynomial-Time Hierarchy would admit robust simulations FS11 () using subexponential-sized circuits.
5.1 PH collapse result
Our strategy to prove this result is as follows. We first define a problem which we call and show it to be complete for , the -th level of PH. The problem turns out to be expressible in for each , though, the required set of labels depends on . Now any language in PH reduces to for some and hence it is sufficient to show that for all . We show this by mimicking the proof of Theorem 4.1. We start by defining the problem .
For a graph and a set , a function is called a precolouring of on iff the induced subgraph is properly three-coloured. For two precolourings , , with , we let be defined as such that for all , iff .
Definition 5.1 (Alternating colouring, and ).
Let be a graph, an odd positive integer, be a partition of , and be a precolouring of on . A -alternating colouring for is a function such that
is a precolouring for ; and
if , for all such that is a precolouring for , there exists a -alternating colouring for , where and .
For any odd , the problem is defined as follows: Given a graph , a partition , and a precolouring , decide whether there is a -alternating colouring for .
Recall that a polynomial-time many-one honest reduction from to is a polynomial-time computable function such that iff and for some integer DF03 ().
Note that for and , , the problem is the classical 3-Colouring problem and hence complete for . More generally:
For each odd positive integer , the problem is complete for under honest polynomial-time many-one reductions.
Containment follows from the existence of an alternating Turing machine that guesses the colouring of vertices in the respective sets . For hardness, consider the problem (also known as ) which is the set of true quantified Boolean formulas with quantifier alternations beginning with an -quantifier, such that the formulas are in CNF for odd and in DNF for even . By Sto76 (); Wra76 (), for each , is complete for under honest polynomial-time many-one reductions. We give a polynomial-time many-one reduction from to by extending the standard reduction from SAT to 3-Colouring. Given an input to , where is a Boolean formula in CNF and is a partition of the variables in such that a variable in is existentially quantified if is odd and universally otherwise, we create a graph as follows:
First, we create a triangle with distinct vertices (“true”), (“false”), and (“forbid”), and
for each variable , we create an edge between two distinct vertices and , and connect both vertices to . The result is depicted in Figure 4.
For each CNF clause , we use of the OR-gadgets depicted in Figure 4. The output vertex of each OR-gadget is connected to . The output of the final OR-gadget for each clause is additionally connected to .
We let and be defined as , , and .
For each , we let , and additionally let contain all OR-gadgets.
It is not hard to see that this reduction takes polynomial time. We induct over and show that for every even the following holds: Let be an assignment to the variables in . Then , iff there is a -alternating colouring for , where and with and for all variables .
The base case of induction is . Suppose . Then there is an assignment to the variables of , such that . We need to show that there is a -alternating colouring for , i.e., a precolouring , such that is a proper three-colouring of the graph. Let for each , and . Then, using the same arguments as for the standard reduction, is a three-colouring of the graph. For the converse direction, suppose there is a precolouring