Baker  devised to polynomial-time approximation schemes for a range of problems (including maximum independent set, minimum dominating set, largest -matching, minimum vertex cover, and many others) when restricted to planar graphs. Her technique (formulated in modern terms) is based on the fact that if is a partition of vertices of a connected planar graph according to their distance from an arbitrarily chosen vertex, then for any positive integer and , the treewidth of is bounded by a function of (more specifically, it is at most ), and hence many natural problems can be exactly solved for these subgraphs in linear time .
It is natural to ask whether these algorithms can be extended to larger classes of graphs. A layering of a graph is a function such that for every edge ; one should visualize the vertices of partitioned into layers for , with edges of only allowed inside the layers and between the consecutive layers. Let us say that a class has bounded treewidth layerings if for some function , each graph has a layering such that has treewidth at most for any finite interval of consecutive integers. Baker’s technique directly extends to all such graph classes (assuming that a suitable layering can be found in polynomial time). A natural obstruction for the existence of bounded treewidth layerings is as follows: let be the graph obtained from the grid by adding a universal vertex adjacent to all other vertices. Then each layering of has at most three non-empty layers, and . Hence, if a class has bounded treewidth layerings, then it contains only finitely many of the graphs . Conversely, Eppstein  proved that minor-closed classes that do not contain all such graphs have bounded treewidth layerings.
As another obstruction, let denote the graph obtained from an grid by adding all diagonals to its unit subcubes. Although the graphs have bounded maximum degree and very simple structure, Dvořák et al.  proved that for every integer , there exists such that for all , the vertex set of cannot be partitioned into two parts both inducing subgraphs of treewidth at most . This prevents existence of bounded treewidth layerings of
, since otherwise the partition of the graph into odd and even numbered layers would give a contradiction.
Of course, one can work around these obstructions:
Each of the graphs contains the universal vertex such that the class has bounded treewidth layerings; and for many optimization problems, one can devise a reduction from the problem in to a variant of the problem in (possibly encoding the neighborhood of by coloring vertices of ).
The grids have the property that in their distance layering, the unions of bounded numbers of layers induce subgraphs which themselves have bounded treewidth layerings, making it possible to iterate Baker’s technique.
Let be a class of graphs. Informally, we will say that is a Baker class if each graph from can be reduced to an empty graph by a constant number of iterations of these operations (removal of vertices, arbitrary choice of a bounded number of consecutive layers in a layering). To enable our intended application to proper minor-closed classes, we need to allow the number of iterations depend not only on , but also on the number of layers we select from each layering (which in itself depends on the considered optimization problem and the desired precision of the approximation). This becomes problematic if the number of iterations is large compared to , as the errors accumulate in each iteration. To deal with this issue, we allow the number of layers selected from the layering to grow in each iteration. As the resulting definition is rather technical, we postpone it to Section 2. Let us remark that the iteration in the definition of a Baker class ends in an empty graph (rather than a graph of bounded treewidth). Stopping when we reach graphs of bounded treewidth would not result in a more general property, since classes with bounded treewidth are themselves Baker.
Building upon the work of Dawar et al. , we show that monotone optimization problems expressible in the first-order logic admit Polynomial-Time Approximation Schemes on Baker classes. Throughout the paper, we work with first-order formulas on graphs, i.e., formulas using a single irreflexive symmetric binary predicate (interpreted by the adjacency in a graph), any number of unary predicates (interpreted as colors on vertices of the graph), quantification on variables for vertices of the graph, equality, and standard logic operators , , and (with other operators such as expressed in terms of these basic operators). A graph language consists of the binary predicate symbol and a finite set of unary predicate symbols. An -interpretation consists of a graph and a set of vertices of for each unary symbol ; the binary symbol is naturally interpreted as the set of all pairs of adjacent vertices of , while for each unary symbol and vertex , we have if and only if . Otherwise, the semantics of first-order formulas is defined in the usual way.
Consider a first-order formula using a unary predicate . We say that is -positive if all appearances of in are within the scope of an even number of negations, and -negative if all appearances of in are within the scope of an odd number of negations. Suppose ; note that if is -positive, then implies ; and if is -negative implies .
A positive FO minimization problem given by an -positive first-order sentence over a graph language seeks, for an input -interpretation , to find a set of the minimum size such that . Let the minimum size of such set be denoted by . The basic example is the minimum dominating set problem, given by the sentence
other examples include the minimum vertex cover and minimum-size set intersecting all triangles.
A negative FO maximization problem given by an -negative first-order sentence over a graph language seeks, for an input -interpretation , to find a set of the maximum size such that . Let the maximum size of such set be denoted by . The basic example is the maximum independent set problem, given by the sentence
other examples include the maximum -independent set for any fixed integer (i.e., the largest subset of vertices at distance greater than from each other) and largest induced subgraph of maximum degree at most for any fixed integer .
Note that if is the -positive sentence obtained from an -negative sentence by negating every appearance of , then ; nevertheless, in case that , even a very precise approximation of does not give a good approximation of , and thus from the approximation perspective, the two problems are distinct. We give a PTAS for both variants (see Section 2.6 for the definition of an -efficiently Baker class).
Suppose is an -efficiently Baker class of graphs. There exists an algorithm that given a first-order -positive sentence over a graph language , an integer and an -interpretation with vertices such that , in time finds a set satisfying
and , or determines no such set exists.
Suppose is an -efficiently Baker class of graphs. There exists an algorithm that given a first-order -negative sentence over a graph language , an integer and an -interpretation with vertices such that , in time finds a set satisfying
and , or determines no such set exists.
As we alluded to before, we prove that any proper minor-closed class of graphs is -efficiently Baker, and thus Theorems 1.1 and 1.2 give PTASes with time complexity for proper minor-closed classes. Unlike us, Dawar et al.  only give algorithms with time complexity . More importantly, their argument uses the minor structure theorem , and thus it is specific only to proper minor-closed classes and the multiplicative constants hidden in the O-notation are huge. Our results are in the setting of more general Baker classes, and the proof that proper minor-closed classes are Baker is direct, without using the minor structure theorem; consequently, the multiplicative constants of the O-notation are more manageable (although still impractically large).
The paper is organized as follows: In Section 2, we give the definition of Baker classes via Baker games introduced in Section 2.2, and work out the properties of this concept, culminating in the proof that proper minor-closed classes are Baker in Section 2.5. We discuss the algorithmic version of the concept in Section 2.6, and another Baker class in Section 2.7. In Section 3, we give polynomial-time approximation schemes for Baker classes: we prove Theorem 1.1 in Section 3.1, Theorem 1.2 in Section 3.2, and discuss problems not expressible in the first-order logic in Section 3.3.
1.1 Related results
As we already mentioned in the introduction, Eppstein  and Demaine and Hajiaghayi  showed that Baker’s technique generalizes to all proper minor-closed classes that do not contain all apex graphs. Going beyond the apex graph boundary, Grohe , Demaine et al.  and Dawar et al.  generalized the approximation algorithms to all proper minor-closed classes using the tree decomposition from the minor structure theorem. Baker’s technique also applies to other geometrically defined graph classes, such as unit disk graphs  and graphs embedded with a bounded number of crossings on each edge .
To go beyond classes with bounded treewidth layerings, Dvořák  introduced a weaker notion of fractional treewidth-fragility: instead of requiring the existence of a layering where consecutive layers induce subgraphs of bounded treewidth, fractional treewidth-fragility only requires the existence of many nearly-disjoint subsets of vertices whose removals results in a graph of bounded treewidth. This notion is sufficient to obtain polynomial-time approximation schemes for some graph parameters such as the independence number or the size of the largest -matching, but fails for others (minimum dominating set, distance constrained versions of the independence number). On the other hand, all proper minor-closed classes are fractionally treewidth-fragile , and so are all subgraph-closed graph classes with bounded maximum degree and strongly sublinear separators . Dvořák  later introduced a stronger notion of thin systems of overlays, also applicable to these classes (via the minor structure theorem) and sufficient to obtain PTASes for more problems (although not all that can be handled using our approach). Note it is easy to see every Baker class has thin systems of overlays, and in particular this gives a proof that proper minor-closed classes have thin systems of overlays and are fractionally treewidth-fragile without using the minor structure theorem.
Another algorithmic approach for proper minor-closed graph classes is through the bidimensionality theory, bounding the treewidth of the graph in terms of the size of the optimal solution and exploiting the arising bounded-size balanced separators to obtain approximate solutions. Demaine and Hajiaghayi  and Fomin et al.  use this approach to construct polynomial-time approximation schemes on all proper minor-closed classes for many minor-monotone problems (e.g., minimum vertex cover) and on apex-minor-free classes for contraction-monotone problems (e.g., minimum dominating set).
A very different approach is taken by Cabello and Gajser  for proper minor-closed classes and more generally by Har-Peled and Quanrud  for classes of graphs with polynomial expansion (which by the result of Dvořák and Norin  is equivalent to having strongly sublinear separators). They showed that the trivial local search algorithm (performing bounded-size changes on an initial solution as long as it can be improved by such a change) gives polynomial-time approximation schemes for maximum independent and minimum dominating set, as well as many other related problems. It is an open problem whether some variation on this approach can give PTAS for all monotone FO optimization problems.
Let us remark that since the approximation factor does not affect the exponent in the complexity in Theorems 1.1 and 1.2, we also obtain fixed-parameter tractability for all the considered problems when parameterized by the order of the optimum solution. However, the fixed-parameter tractability of these problems has been established in greater generality, see [12, 19]. Our approach was in part inspired by ; to obtain their result, they introduce Splitter games, and a winning strategy for a Baker game translates into a winning strategy for the Splitter game (but not vice versa).
2 Baker game
2.1 Layerings and ordered graphs
Let us start with some preliminaries.
Let be a graph and consider a function . If is connected, , and is equal to the distance from to in for each , then is a layering; we call this layering the BFS layering starting from . If has components , …, and for some positive integer , holds for every and , then also is a layering, which we call the -spread componentwise layering. For a layering , the width of is defined as .
For any two vertices and of a graph , let denote the distance between and in . For , a layering of is -geodesic if for every . We need the following observation on extendability of layerings.
Let be a graph and let . A layering of extends to a layering of if and only if is -geodesic.
Any layering of satisfies for every ; hence, if extends to a layering of , then is -geodesic.
Let us now argue that every -geodesic layering of extends to a layering of . We prove the claim by induction of . The case being trivial, we can assume there exists a vertex . Let us define . By definition, we have , and thus for every . Also, there exists such that . Consider any ; by the triangle inequality we have , and since is -geodesic, we conclude . Consequently, with this definition of , we have for every . In particular, if for , then , and thus is a -geodesic layering of . By the induction hypothesis, extends to a layering of . ∎
An ordered graph is a graph together with a linear ordering of its vertices. If is a subgraph of an ordered graph , the ordering of vertices of is naturally the ordering of vertices of restricted to .
2.2 Rules of the game
For an infinite sequence and an integer , let denote the sequence , let , and let . A sequence is dominated by if for every .
The Baker game between two players (Destroyer and Preserver) is defined as follows. The state of the game is a pair , where is an ordered graph and is an infinite sequence. If , then the game stops. Otherwise, Destroyer chooses one of the following actions:
Destroyer deletes the smallest vertex from the graph; Preserver takes no action and the game proceeds with the state .
Destroyer selects a layering of . Preserver selects an interval of at most consecutive integers and the game proceeds with the state . That is, Preserver selects consecutive layers and deletes the rest of the graph.
Destroyer seeks to minimize the number of rounds of this game; we say that Destroyer wins in rounds on the state if regardless of Preserver’s strategy, the game stops after at most rounds.
Let us remark that in the Delete action, we could more generally allow the Destroyer to delete any vertex, not just the smallest one; however, throughout this paper we only deal with the “monotone” strategies where the vertices are deleted in order, and it is more convenient to formulate the game already including this assumption rather than repeating it everywhere. Furthermore, since the Delete action does not depend on the sequence , it is tempting not to consume its first element, and proceed with the state rather than . However, in that setting we would run into problems in some of the arguments below, in particular in the proof of Lemma 2.3.
2.3 Basic properties
We will often use the following observation.
Suppose Destroyer wins the Baker game on the state in rounds. If is a subgraph of and dominates a sequence , then Destroyer wins the Baker game on the state in rounds.
We prove the claim by induction on . If , then Destroyer wins on the state in rounds. Hence, suppose that , and thus also . Since Destroyer wins on the state in rounds, we have . We consider the first action of Destroyer on .
If he takes the Delete action, then on we also take the delete action. Let and be the smallest vertices of and , respectively. Note that if , then ; consequently, we have . By the induction hypothesis Destroyer wins on in rounds, and thus he wins on in rounds.
If Destroyer selects a layering of , we select a layering of . The Preserver answers by choosing an interval of at most consecutive integers. Destroyer wins on the state in rounds, and by the induction hypothesis, he also wins on the state in rounds. Consequently, Destroyer wins on in rounds. ∎
We say that is a Baker class if is a class of ordered graphs and for every sequence there exists an integer such that for each , Destroyer wins the Baker game on in rounds.
Let us now prove an important composition result. A partition of an ordered graph (with linear ordering of its vertices) is a sequence , …, of pairwise disjoint subsets of such that and all vertices and such that satisfy . For an integer , the partition is width- geodesic if for , has a -geodesic layering of width at most . Let denote the ordered graph obtained from by identifying all vertices in each part of to a single vertex and suppressing the arising loops and parallel edges, with the ordering of the vertices of matching the sequence of parts of . For a class and an integer , let denote the class of ordered graphs for which there exists a width- geodesic partition of such that .
If is Baker class, then is a Baker class for every integer .
By Lemma 2.2, we can without loss of generality assume that is subgraph-closed. For any sequence , let be an integer such that for every , Destroyer wins the Baker game on in rounds.
Consider any sequence . Let , and for , let . Let and let . We claim that for every ordered graph , Destroyer wins the Baker game on in rounds. More generally, we will prove the following claim for by induction. Suppose that is an ordered graph such that Destroyer wins the Baker game on in rounds. If has a width- geodesic partition such that , then Destroyer wins the Baker game on in rounds.
We identify the vertex set of with in the natural way. Note that
The claim is trivial for , and thus we can assume . We mimic the Destroyer’s strategy for as follows.
If Destroyer performs the Restrict action with layering , we let denote the layering of such that for every and . In the Baker game on , we perform the Restrict action with the layering , Preserver chooses an interval of at most consecutive integers and changes the state to , where . We follow up with Delete actions, either ending the game in the process or changing the state to for a subgraph of . In the Baker game on , we have Preserver also answer with , resulting in the state , where . Considering the partition , observe that and that is a width- geodesic partition of . Since Destroyer wins the Baker game on in rounds, by the induction hypothesis and Lemma 2.2 Destroyer wins the Baker game on in rounds, and thus Destroyer wins the Baker game on in rounds.
Next, suppose Destroyer performs the Delete action in the Baker game on , changing the state to , where is the smallest vertex of . We select a -geodesic layering of of width at most . By Lemma 2.1, extends to a layering of . We perform the Restrict action on with this layering . Preserver answers with an interval of at most consecutive integers, changing the state to . Note that . Next, we perform Delete actions, either ending the game in the process or changing the state to , where . Note that , and is a width- geodesic partition of . Since Destroyer wins the Baker game on in rounds, the induction hypothesis and Lemma 2.2 implies that Destroyer wins the Baker game on in round, and thus he also wins the Baker game on in rounds. ∎
We will also need another composition result based on clique-sums. Suppose and are classes of ordered graphs. Let denote the class of ordered graphs such that there exists a set satisfying the following conditions:
for each component of we have , the neighbors of vertices of in induce a clique , and all vertices of are smaller than all vertices of .
In this situation, we say that is the base of .
If and are Baker classes, then is a Baker class.
Consider a sequence , and let . Let be an integer such that for every , Destroyer wins the Baker game on in rounds. Let be an integer such that for every , Destroyer wins the Baker game on in rounds. By Lemma 2.2, we can without loss of generality assume that and are subgraph-closed.
We claim that for every ordered graph , Destroyer wins the Baker game on in rounds. More generally, we will prove the following claim for by induction. Consider a graph with base . If Destroyer wins the Baker game on in rounds, then he also wins the Baker game on in rounds.
Destroyer first performs the Restrict action with the -spread componentwise layering. Preserver’s response then changes the state to , where either or is a component of ; we can assume the latter. Since and are subgraph-closed, is a base of .
If , then since is a base of and is connected, we conclude that . Destroyer performs (at most) Delete actions, resulting either in a victory or a state with . In the latter case, Destroyer wins the Baker game on in rounds. Hence, Destroyer wins the Baker game on in rounds, as required. Hence, we can assume . In particular, we have , since when , Destroyer wins the Baker game on in rounds, and thus ,
Since , Lemma 2.2 implies that Destroyer wins the Baker game on in rounds. Note that , and consider the first action of Destroyer on this state.
If the action is Restrict with layering , then let be defined as follows. Let for . For every component of , choose with a neighbor in (which exists since is connected) arbitrarily and let for every . Note that is a layering of : if has a neighbor , then since the vertices with a neighbor in form a clique, we have and . On the state , we preform the Restrict action with this layering and Preserver answers with an interval of at most consecutive integers, resulting in the state . Destroyer wins the Baker game on in rounds, and is a base of . By the induction hypothesis, Destroyer wins the Baker game on in rounds, and thus he also wins on in rounds.
Suppose now the action is Delete, hence changing the state to , where is the smallest vertex of ; Destroyer wins the Baker game from this state in rounds. Note that is also the smallest vertex of , since is a base of and is connected. Hence, the Delete action applied to deletes the same vertex, resulting in the state . Observe that is a base of . By the induction hypothesis, Destroyer wins the baker game on in rounds, and thus he also wins on in rounds. ∎
2.4 Bounded treewidth
An ordered graph is chordal if for every vertex , the neighbors of smaller than induce a clique in . The left-degree of is the number of such neighbors. Note that vertices of a graph can be linearly ordered so that the resulting ordered graph is chordal if and only if is chordal, i.e., contains no induced cycles of length at least . Furthermore, recall that graph has treewidth at most if and only if has a chordal supergraph with clique number at most , and thus the corresponding chordal ordered graph has maximum left-degree at most .
Let us start with a simple observation.
Let be a chordal ordered graph (with linear ordering of its vertices). If is an induced path in from to and is smaller than all other vertices of , then .
Suppose for a contradiction there exists such that , and choose smallest such index . By the minimality of , we have . Since is chordal, and are adjacent, contradicting the assumption that is an induced path. ∎
For a non-negative integer and a vertex of a graph , let denote the set of vertices of at distance at most from .
Let be a chordal ordered graph, let be the smallest vertex of , let be a non-negative integer, let be the vertex set of a component of , and let be the set of vertices in which have a neighbor in . Then induces a clique in , and all vertices in are smaller than all vertices in .
Suppose first that distinct vertices are not adjacent. Note that , as otherwise would contain a vertex at distance at most from . Let be the shortest path between and in , and let be a shortest path between and in . The concatenation of and is an induced cycle of length at least , contradicting the assumption that is chordal. Hence, is a clique.
Let be the largest vertex in , and let be the smallest vertex in . Suppose now for a contradiction that is smaller than . Let be a shortest path between and in . Note that is the smallest vertex of , and by Lemma 2.5, is the largest vertex of . Let be the neighbor of in . Since , we conclude that and there exists a shortest path from to (of length ) containing . By Lemma 2.5, is the largest vertex of . However, this is a contradiction, since and belong to both and . ∎
We are now ready to prove classes of graphs of bounded treewidth (or more precisely, classes of chordal ordered graphs with bounded maximum left-degree) are Baker. For an integer , let denote the class of chordal ordered graphs of maximum left-degree at most . Let , and for any integer , let .
For every integer , is a Baker class.
We prove the claim by induction on . The class consists of ordered graphs with no edges, and thus it is a Baker class. Hence, we can assume .
Consider any sequence . By the induction hypothesis and repeated applications of Lemma 2.4, is a Baker class. Let be an integer such that for every , Destroyer wins the Baker game on in rounds. We claim that for every , Destroyer wins the Baker game on in rounds, by the following strategy.
First, Destroyer performs the Restrict action with the -spread componentwise layering; Preserver’s answer results in a state , where is connected. Next, Destroyer performs the Restrict action with the BFS layering starting from the smallest vertex of , and Preserver answers with an interval of at most consecutive integers, changing the state to , where .
Let be the smallest element of , and for , let . By induction on , we prove that . Let and consider the graph . If , then has only one vertex and . If , then note that each vertex has in a neighbor in , and by Lemma 2.6, is smaller than . Since has maximum left-degree at most , has maximum left-degree at most , and thus . In particular, the claim holds when , and thus we can assume . By the induction hypothesis, we have , and by Lemma 2.6, it follows that .
We conclude that , and thus Destroyer wins the Baker game on in rounds. Consequently, Destroyer wins the Baker game on in rounds. ∎
2.5 Forbidden minors
It is now easy to show that proper minor-closed classes are Baker. The inspection of the proof of Lemma 4.1 in  gives the following.
For every integer , every -minor-free graph has a linear ordering of vertices such that the corresponding ordered graph belongs to .
Let us remark that the proof of Lemma 2.8 is elementary and constructive, requiring at most breadth-first searches on to construct the required ordering and partition. Lemmas 2.3, 2.7 and 2.8 now give our first main result.
For every integer , there exists a Baker class such that every -minor-free graph together with some linear ordering of its vertices belongs to .
2.6 Algorithmic considerations
In the algorithmic application we develop in the next section, it is of course important not only that the Baker game can be won in a constant number of rounds for a particular class of graphs, but also that the next step in the strategy can be determined efficiently. Inspection of the proofs shows this is the case in all the discussed situations. Lemma 2.8 has algorithmic proof, returning for an -vertex -minor-free graph the corresponding ordering, width- geodesic partition, and layering of each of the parts in time . Lemma 2.3 then provides an explicit description of the strategy, in terms of the strategy for which in turn is explicitly provided by Lemmas 2.4 and 2.7; the most time-demanding step in determining the current action is the breadth-first search in Lemma 2.4, and thus the action can be determined in time . Applications of Lemma 2.2 throughout the proofs of other lemmas are dealt with by keeping track of the appropriate supergraph of the currently considered graph, determining the appropriate actions in this supergraph, and translating them to actions in the subgraph as described in the proof of Lemma 2.2.
For functions , we say that a class of graphs is -efficiently Baker if there exist algorithms and and for every sequence there exists an integer so that
for each , the algorithm determines in time an ordering of vertices of (and possibly other auxiliary information) so that for the resulting ordered graph,
the algorithm wins the Baker game on in rounds, using time to determine the action at each state of the game.
With this definition, the analysis presented at the beginning of this subsection implies the following.
For every integer , the class of -minor-free graphs is -efficiently Baker.
2.7 Graphs embedded with bounded distortion
Let us now give another example of Baker classes. For a real number , we say that a graph embeds in with distortion if there exists a mapping such that for every distinct , and for every . For example, the -dimensional grids with diagonals naturally embed in with distortion .
For , let denote the projection to the -th coordinate. Given a graph with embedding in with distortion , we for define a layering as . For a sequence , we can now apply the following strategy: First, we perform Restrict actions with layerings , …, . Any of the remaining vertices and then satisfy for . Since any distinct vertices satisfy , it follows that there are at most vertices left, and we can remove them by repeated Delete actions. Hence, Destroyer wins the Baker game in rounds (for any ordering of the vertices of ). We conclude the following holds.
Let be an integer, let be a real number, and let be a class of graphs such that for every , an embedding in with distortion can be found in time . Then is -efficiently Baker.
3 Polynomial-time approximation schemes
3.1 PTAS for positive FO minimization problems
For an integer , an -local formula is a formula with one free variable such that all quantifications in are of form or , where should be interpreted as the first-order formula describing that the distance between and is at most , that is, . That is, the validity of depends only on the neighborhood up to distance from the vertex interpreting in the graph.
A basic existential sentence is a sentence of form