Large induced subgraphs via triangulations and CMSO
LLarge induced subgraphs via triangulations and CMSO
Fedor V. Fomin , Ioan Todinca ∗ , and Yngve Villanger Department of Informatics, University of Bergen, Norway, [email protected],[email protected] LIFO, Univ. Orl´eans, France, [email protected]
November 10, 2018
Abstract
We obtain an algorithmic meta-theorem for the following optimization problem. Let ϕ be a Counting Monadic Second Order Logic (CMSO) formula and t ≥ G = ( V, E ), the task is to maximize | X | subject to the following: there is aset F ⊆ V such that X ⊆ F , the subgraph G [ F ] induced by F is of treewidth at most t ,and structure ( G [ F ] , X ) models ϕ , i.e. ( G [ F ] , X ) | = ϕ . Special cases of this optimizationproblem are the following generic examples. Each of these special cases contains variousproblems as a special subcase: • Maximum Induced Subgraph with ≤ (cid:96) copies of F m -cycles , where for fixednonnegative integers m and (cid:96) , the task is to find a maximum induced subgraph of agiven graph with at most (cid:96) vertex-disjoint cycles of length 0 (mod m ). For example,this encompasses the problems of finding a maximum induced forest or a maximumsubgraph without even cycles. • Minimum F -Deletion , where for a fixed finite set of graphs F containing a planargraph, the task is to find a maximum induced subgraph of a given graph containingno graph from F as a minor. Examples of Minimum F -Deletion are the problemsof finding a minimum vertex cover or a minimum number of vertices required todelete from the graph to obtain an outerplanar graph. • Independent H -packing , where for a fixed finite set of connected graphs H , thetask is to find an induced subgraph F of a given graph with the maximum numberof connected components, such that each connected component of F is isomorphicto some graph from H . For example, the problem of finding a maximum inducedmatching or packing into nonadjacent triangles, are the special cases of this problem.We give an algorithm solving the optimization problem on an n -vertex graph G in time O ( | Π G | · n t +4 · f ( t, ϕ )), where Π G is the set of all potential maximal cliques in G and f is a function of t and ϕ only. We also show how similar running time can be obtainedfor the weighted version of the problem. Pipelined with known bounds on the numberof potential maximal cliques, we derive a plethora of algorithmic consequences extendingand subsuming many known results on algorithms for special graph classes and exactexponential algorithms. ∗ Partially supported by the ANR project AGAPE. a r X i v : . [ c s . D S ] S e p Introduction
We provide a generic algorithmic result concerning induced subgraphs with properties ex-pressible in some logic. The main applications of our result can be found in two areas ofgraph algorithms: polynomial time algorithms on special graph classes and exponential timealgorithms.
Graph classes.
The algorithmic study of graphs with particular structure can be traced tothe introduction of perfect graphs by Berge in the beginning of 1960s. Most of the researchin this area focuses on graph algorithms exploiting the structure of the input graph. Manyproblems intractable on general graphs were shown to be solvable in polynomial time ondifferent classes of graphs like interval or chordal graphs. The book of Golumbic [44] providesalgorithmic studies of fundamental classes of perfect graphs while the book of Brandst¨adt et al.[15] gives an extensive overview of different classes of graphs. By the seminal work of Gr¨otschelet al. [47], the weighted versions of
Maximum Independent Set , Maximum Clique , Coloring , and
Minimum Clique Cover are solvable in polynomial time on perfect graphs.There are two natural research directions in this area extending the limits of tractability.One direction is to identify graph classes beyond perfect graphs, where a specific problemlike
Maximum Independent Set , can still be solved efficiently. The second direction is toidentify more general problems which still can be solved in polynomial time on subclasses ofperfect graphs.As an example, let us take
Maximum Induced Forest , which can be seen as a naturalextension of Maximum Independent Set , where instead of maximum edgeless graph one isseeking for a maximal acyclic graph. It easy to notice that the problem is NP-complete beingrestricted to bipartite, and thus to perfect, graphs. On the other hand, for other classes ofgraphs the problem is solvable in polynomial time. Yannakakis and Gavril [73] have shownhow to find in polynomial time a maximum induced forest and tree on chordal graphs. Infact, they show polynomial time solvability of more general problem of finding maximumand connected maximum k -colorable subgraphs in chordal graphs, where k is a constant.When k is a part of the input, they showed that on chordal graphs both problems are NP-compete. Other graph classes where Maximum Induced Forest was known to be solvablein polynomial time include circle n -gon graphs, circle trapezoid, circle graphs, and bipartitechordal graphs [41, 42, 53]. The containment relations between these classes of graphs is givenin Fig 1. According to the database on special graph classesthe complexity of (weighted) Maximum Induced Forest on weakly chordal is open.Another example of a well-studied problem on special graph classes is
Maximum InducedMatching . Here the task is to find a maximum induced subgraph such that every connectedcomponent of this graph is an edge. The complexity of this problem on different graphclasses was investigated in [17, 19, 20, 45]. Cameron and Hell in [18] introduced the followinggeneralization of
Maximum Induced Matching . Let H be a finite set of connected graphs.An H -packing of a given graph G is a pairwise vertex-disjoint set of subgraphs of G , eachisomorphic to a member of H . An independent H -packing of a given graph G is an H -packing,i.e. a set of pairwise vertex-disjoint set of subgraphs of G , each isomorphic to a member of H ,such that no two subgraphs of the packing are joined by an edge of G . The task is to find the In the literature, the complementary minimization problem of deleting the minimum number of verticessuch that the remaining graphs has no cycles, is known as
Minimum Feedback Vertex Set . Since fromexact algorithms perspective maximization and minimization versions are equivalent, we will be discussingmostly maximization problems. eakly chordal chordal bipartite circle circular-arc chordal permutationinterval d-trapezoid proper interval Cographk-polygon proper circular-arcsplit distance-hereditary trapezoid polygon-circle Figure 1: Graph classes with a polynomial number of potential maximal cliques.maximum number of graphs contained in an independent H -packing. For example, when H consists of K this is Maximum Independent Set , and when H = { K } , this is MaximumInduced Matching . It has been shown in [18] that for many graph classes including weaklychordal and polygon-circle graphs, H -packing is solvable in polynomial time. Exact exponential algorithms.
The second application of our results can be found in thearea of exact exponential algorithms. The area of exact exponential algorithms is about solv-ing intractable problems faster than the trivial exhaustive search, though still in exponentialtime [31]. While for any graph property π testable in polynomial time, the problem of findinga maximum induced subgraph with property π is trivially solvable in time 2 n n O (1) , for severalfundamental problems much faster algorithms are known. A longstanding open question inthe area is if Maximum Induced Subgraph with Property π can be solved faster thanthe trivial O ∗ (2 n ) for every hereditary property π testable in polynomial time.For the simplest property π , being edge-less, the corresponding maximum induced sub-graph problem is Maximum Independent Set . A significant amount of research was alsodevoted to algorithms for this problem starting from the classical work of Moon and Moser[59] (see also Miller and Muller [58]) from the 1960s [69, 50, 66, 30, 14, 54]. To the best of ourknowledge, the fastest known algorithm of running time O (1 . n ) is due to Robson [66].For Maximum Induced Forest an algorithm of running time O (1 . n ) was known [28].This result was improved and generalized by a subset of the authors, who have shown thatfor any fixed t , the maximum induced subgraph of treewidth at most t can be computed intime O (1 . n ) [35]. There is also a relevant work of Gupta et al. [48] who gave algorithmsfor Maximum Induced Matching and
Maximum 2-Regular Induced Subgraph , withrunning times time O (1 . n ) and O (1 . n ), respectively.Our main theorem is based on developments from two research areas: the theory ofminimal triangulations and logic. Minimal triangulations.
A triangulation of a graph G is a chordal (no induced cycle2f length at least four) supergraph of G . A triangulation H of G is minimal, if no propersubgraph of H is a triangulation of G . Triangulations are closely related to fundamentalproblems arising in sparse matrix computations which were studied intensively in the past[60, 67]. The survey of Heggernes [49] gives an overview of techniques and applications ofminimal triangulations. It appeared in 1990s that minimal separators play important rolein obtaining minimal triangulations with certain properties. Techniques based on minimalseparators were used to obtain polynomial algorithm computing the treewidth and minimumfill-in for different classes of graphs [10, 52, 51]. These results were extended by Bouchitt´e andTodinca in [12, 13], who also introduced the notion of a potential maximal clique, which is aset of vertices of a graph that is a clique in some minimal triangulation. Potential maximalcliques appeared to be a handy tool for computing the treewidth of a graph [32, 37]. Recentlypotential maximal clique based machinery was used to obtain a subexponential parameterizedalgorithm finding a minimum fill-in of a graph [36]. The work which is most relevant to ourresults is the work of a subset of the authors [35], where potential maximal cliques were used tofind maximum induced subgraphs of treewidth at most t . We build on the previous techniquesexploiting the structure of minimal triangulations, minimal separators and potential maximalcliques but to use the framework of minimal triangulations in full generality, we have tocombine it with the powerful tools from logic. Algorithmic applications of logic.
Algorithmic meta-theorems are algorithmic resultswhich can be applied to large families of combinatorial problems, instead of just specificproblems. Such theorems provide a better understanding of the scope of general algorith-mic techniques and the limits of tractability. Usually meta-theorems are based on the deeprelations between logic and combinatorial structures, which is a fundamental issue of compu-tational complexity [46, 56]. A typical example of a meta-theorem is the celebrated Courcelle’stheorem [23] which states that all graph properties definable in Monadic Second Order Logiccan be decided in linear time on graphs of bounded treewidth. More recent examples of suchmeta-theorems state that all first-order definable properties on planar graphs can be decidedin linear time [38], that all first-order definable optimization problems on classes of graphswith excluded minors can be approximated in polynomial time to any given approximationratio [26], and that all parameterized problems with finite integer index and additional “com-pactness” or “bidimensional” combinatorial property, admit linear kernels on planar graphs[9, 34]. As it often happens with meta-theorems, a combination of logic and graph theorynot only give a uniform explanation to tractability of many algorithmic problems but alsoprovide a variety of new results. There are several extensions of Courcelle’s theorem known inthe literature. In particular, for a counting variant of MSO, Counting Monadic Second OrderLogic (CMSO), where we are allowed to have sentences testing if a set is equal to q modulo r, for some integers q and r , and analogue of Courcelle’s theorem was obtained by Borie et al.[11] and Lagergren and Arnborg [57]. Our proof is using the framework of Borie et al. [11]. Our results. A property P ( G, X ) on graphs, where G is a graph and X is a vertex subset ofits vertices, associates to each graph G and each vertex subset X of G a boolean value. Borieet al. [11] defined regular properties , which definition we postpone till the next section. Forall our applications, we need only the fact from Borie et al. [11] that every property P ( G, X )expressible by a CMSO-formula is regular. Then our result can be stated as follows. Let ϕ bea CMSO-formula, G = ( V, E ) be a graph, and t ≥ | X | subject to There is a set F ⊆ V such that X ⊆ F ;The treewidth of G [ F ] is at most t ;( G [ F ] , X ) | = ϕ. (1)For example, Maximum Independent Set can be encoded by (1) by taking t = 0, and ϕ expressing that X = F and the absence of edges in G [ F ]. For another example, consider Independent Cycle Packing , where the task is to find an induced subgraph with maximumnumber of connected components such that each component is a cycle. In this case, t = 2and ϕ expresses the property that each connected component is a cycle and that X is a setof vertices containing exactly one vertex from each cycle.Let Π G be the set of all potential maximal cliques in G . Our main result is that (1) issolvable in time O ( | Π G | · | V | t +4 · f ( t, ϕ )) for some function f . Moreover, within the samerunning time one can find the corresponding sets X and F . Also it is easy to extend ouralgorithm to solve within the same running time weighted and annotated versions of (1).Many well studied graph classes have the following property: there is a polynomial func-tion p , depending only on the graph class, such that for every graph G from the class, thenumber of potential maximal cliques in G is at most p ( n ), see Fig 1 for examples of suchclasses. Moreover, if the number of potential maximal cliques in a graph is bounded by somepolynomial of n , then all potential maximal cliques can be enumerated in polynomial time[13]. Our algorithm implies directly that every problem expressible in the form of (1) issolvable in polynomial time on such graph classes. We discuss in details the bounds on thenumber of potential maximal cliques for different graph classes in Section 5. Interestinglyenough, while recognition of several of graph classes, like polygon-circle or d -trapezoid, canbe NP-complete, our algorithm is still able either to solve the problem, or to report that theinput graph does not belong to the specified graph class. Such algorithms were called robust by Raghavan and Spinrad [62]. To the best of our knowledge, very few robust algorithmswere known in the literature prior to our work.Another direct consequence of our algorithm is that because every n -vertex graph has O (1 . n ) potential maximal cliques [35], many intractable problems concerning maximuminduced subgraphs with different properties expressible in the form of (1), can be solvedsignificantly faster than by the trivial O (2 n )-time brute-force algorithm. We are not aware ofany algorithmic meta-result of this flavor in the area of exact algorithms.We mention below the most interesting special cases of the optimization problem (1). Eachof these special cases contains various problems as a special subcase, we discuss subcases afterintroducing each of the problems. For some of these cases, expressibility in the form of (1)is trivial but for some it is non-obvious and requires deep results from Graph Theory. Wediscuss these issues in more details in Section 4.Let F m be the set of cycles of length 0 (mod m ). Let (cid:96) ≥ Maximum Induced Subgraph with ≤ (cid:96) copies of F m -cycles Input:
A graph G . Task:
Find a set F ⊆ V ( G ) of maximum size such that G [ F ] contains at most (cid:96) vertex-disjoint cycles from F m . 4 aximum Induced Subgraph with ≤ (cid:96) copies of F m -cycles encompasses severalinteresting problems. For example, when (cid:96) = 0, the problem is to find a maximum inducedsubgraph without cycles divisible by m . For (cid:96) = 0 and m = 1 this is Maximum InducedForest .For integers (cid:96) ≥ p ≥
3, the problem related to
Maximum Induced Subgraphwith ≤ (cid:96) copies of F m -cycles is the following. Maximum Induced Subgraph with ≤ (cid:96) copies of p -cycles Input:
A graph G . Task:
Find a set F ⊆ V ( G ) of maximum size such that G [ F ] contains at most (cid:96) vertex-disjoint cycles of length at least p .Next example concerns properties described by forbidden minors. Graph H is a minor ofgraph G if H can be obtained from a subgraph of G by a (possibly empty) sequence of edgecontractions. A model M of minor H in G is a minimal subgraph of G , where the edge set E ( M ) is partitioned into c-edges (contraction edges) and m-edges (minor edges) such thatthe graph resulting from contracting all c-edges is isomorphic to H . Thus, H is isomorphicto a minor of G if and only if there exists a model of H in G . For an integer (cid:96) a finite set ofgraphs F , we define he following generic problem. Maximum Induced Subgraph with ≤ (cid:96) copies of Minor Models from F Input:
A graph G . Task:
Find a set F ⊆ V ( G ) of maximum size such that G [ F ] contains at most (cid:96) vertexdisjoint minor models of graphs from F .Even the special case with (cid:96) = 0, this problem and its complementary version called the Minimum F -Deletion , encompass many different problems. In the literature, the case (cid:96) = 0was studied from parameterized and approximation perspective [33].When F = { K } , a complete graph on two vertices, this is Maximum Independent Set ,the problem complementary to the
Minimum Vertex Cover problem. When F = { C } , acycle on three vertices, this is Maximum Induced Forest . Case F = { K } of MaximumInduced F -free Subgraph corresponds to maximum induced serial-parallel graph, F = { K , K , } to maximum induced outerplanar, and case when F consists of a diamond graph,which is K minus one edge, is to find a maximum induced cactus subgraph. Maximuminduced pseudo-forest is the case of F containing the diamond and butterfly graphs, which isobtained by identifying one vertex of two triangles. Maximum Apollonian graph correspondsto the situation with F consisting of the complete graph K , the complete bipartite graph K , , the graph of the octahedron, and the graph of the pentagonal prism. A fundamentalproblem, which is a special case of Minimum F -Deletion , is Minimum Treewidth η -Deletion or η -Transversal which is to delete minimum vertices to obtain a graph oftreewidth at most η . Since by the result of Robertson and Seymour [63] every graph oftreewidth η excludes a ( η + 1) × ( η + 1) grid as a minor, we have that the set F of forbiddenminors of treewidth η graphs contains a planar graph. Similarly, for (cid:96) > MaximumInduced Subgraph with ≤ (cid:96) copies of Minor Models from F generalizes problemslike finding a maximum induced subgraph containing at most (cid:96) vertex-disjoint cycles, at most (cid:96) vertex-disjoint outerplanar graphs, at most (cid:96) vertex-disjoint subgraphs of treewidth t , etc.For some graph classes, like circular-arc and weakly chordal, we show that even more generalcases of Minimum F -Deletion , when F is not requested to contain a planar graph, are still5olvable in polynomial time.Let t ≥ ϕ be a CMSO-formula. Let G ( t, ϕ ) be a class of connectedgraphs of treewidth at most t and with property expressible by ϕ . Our next example is thefollowing problem. Independent G ( t, ϕ ) -Packing Input:
A graph G . Task:
Find a set F ⊆ V ( G ) with maximum number of connected components such thateach connected component of G [ F ] is in G ( t, ϕ ).In other words, the task is to find a maximum vertex-disjoint packing in G of subgraphsfrom G ( t, ϕ ) such that no two subgraphs of the packing are joined by an edge of G . Thisproblem trivially generalizes several well studied problems. For example, Maximum InducedMatching is to find a maximum induced matching which was studied intensively for differentgraph classes. Similarly, when class G ( t, ϕ ) consists of one graph K , then Maximum Induced G ( t, P ) -Packing is induced triangle packing. This problem, under the name IndependentTriangle Packing was studied by Cameron and Hell [18]. Recall that Cameron and Helldefined more general problem, namely,
Independent H -Packing , where for a finite set ofconnected graphs H , the task is to find a maximum number of disjoint copies of graphs from H such that there is no edges between the copies. Since every finite set of graphs is triviallyin G ( t, P ) for some t and P , Independent H -Packing is a special case of Independent G ( t, ϕ ) -Packing . Another studied variant of the problem is Induced Packing of OddCycles introduced by Golovach et al. in [43], where the task is to find the maximum numberof odd cycles such that there is no edge between any pair of cycles.The next problem is an example of annotated version of optimization problem (1). k -in-a-Graph From G ( t, ϕ ) Input:
A graph G , with k terminal vertices. Task:
Find an induced graph from G ( t, ϕ ) containing all k terminal vertices.It is also easy to handle variants of this problem where terminal vertices have specificproperties, like being the endpoints of the path if G ( t, ϕ ) is the class of paths. Many variants of k -in-a-Graph From G ( t, ϕ ) can be found in the literature, like k -in-a-Path , k -in-a-Tree , k -in-a-Cycle . k -in-a-Path is clearly solvable in polynomial time for k = 2. For k = 3the problem is NP-complete already on graph of maximum vertex degree at most three [27].Bienstock [6] have shown that the following cases of k -in-a-Graph From G ( t, ϕ ) are NP-hard: finding an induced odd cycle of length greater than three, passing through a prescribedvertex and finding an induced odd path between two prescribed vertices. Polynomial timealgorithms for the odd path problem are known for several graph classes, including chordal[1] and circular-arc graphs [2]. Chudnovsky and Seymour have shown that k -in-a-Tree for k = 3 is solvable in polynomial time [21]. The complexity of the case k = 4 is open.Let us remark that because of the power of CMSO, different modifications of the problemsmentioned above, with additional requirements on the induced subgraph like being connected,constrains on vertex degree and parities of connected components, can be easily handled.6 Preliminaries
We denote by G = ( V, E ) a finite, undirected and simple graph with | V | = n vertices and | E | = m edges. Sometimes the vertex set of a graph G is referred to as V ( G ) and its edge setas E ( G ). A clique K in G is a set of pairwise adjacent vertices of V ( G ). The neighborhood of a vertex v is N ( v ) = { u ∈ V : { u, v } ∈ E } . For a vertex set S ⊆ V we denote by N ( S )the set (cid:83) v ∈ S N ( v ) \ S .The notion of treewidth is due to Robertson and Seymour [63]. A tree decomposition of agraph G = ( V, E ), denoted by
T D ( G ), is a pair ( X, T ), where T is a tree and X = { X i | i ∈ V ( T ) } is a family of subsets of V , called bags , such that(i) (cid:83) i ∈ V ( T ) X i = V ,(ii) for each edge e = { u, v } ∈ E ( G ) there exists i ∈ V ( T ) such that both u and v are in X i , and(iii) for all v ∈ V , the set of nodes { i ∈ V ( T ) | v ∈ X i } induces a connected subtree of T .The maximum of | X i | − i ∈ V ( T ), is called the width of the tree decomposition. The treewidth of a graph G , denoted by tw( G ), is the minimum width taken over all tree decom-positions of G . Counting Monadic Second Order Logic.
We use Counting Monadic Second Order Logic(CMSO), an extension of MSO, as a basic tool to express properties of vertex/edge sets ingraphs.The syntax of Monadic Second Order Logic (MSO) of graphs includes the logical con-nectives ∨ , ∧ , ¬ , ⇔ , ⇒ , variables for vertices, edges, sets of vertices, and sets of edges, thequantifiers ∀ , ∃ that can be applied to these variables, and the following five binary relations:1. u ∈ U where u is a vertex variable and U is a vertex set variable;2. d ∈ D where d is an edge variable and D is an edge set variable;3. inc ( d, u ) , where d is an edge variable, u is a vertex variable, and the interpretation isthat the edge d is incident with the vertex u ;4. adj ( u, v ) , where u and v are vertex variables and the interpretation is that u and v areadjacent;5. equality of variables representing vertices, edges, sets of vertices, and sets of edges.In addition to the usual features of monadic second-order logic, if we have atomic sentencestesting whether the cardinality of a set is equal to q modulo r, where q and r are integerssuch that 0 ≤ q < r and r ≥ , then this extension of the MSO is called the counting monadicsecond-order logic . So essentially CMSO is MSO with the following atomic sentence for a set S : card q,r ( S ) = true if and only if | S | ≡ q (mod r ) . We refer to [3, 22, 24] and the book of Courcelle and Engelfriet [25] for a detailed introductionon CMSO. In [25], the CMSO is referred to as
CM S .7 .1 Treewidth, t -terminal recursive graphs and regular properties We use one of the (many) alternative definitions of treewidth, based on terminal graphs . A t -terminal graph G = ( V, T, E ) is a graph with an ordered set T ⊆ V of at most t distinguishedvertices, called terminals . Denote by τ ( G ) the number of terminals of graph G .A t -terminal graph ( V, T, E ) is a base graph if V = T . We define composition operations over the set of t -terminal graphs. A composition operation f is of arity 1 or 2. When f is ofarity 2, it acts on two t -terminal graphs G , G and produces a t -terminal graph G = f ( G , G )as follows. It first makes disjoint copies of the two graphs, then “glues” some terminals ofgraphs G and G . Operation f is represented by a matrix m ( f ). The matrix has 2 columnsand τ ( G ) ≤ t lines, its values are integers between 0 and t . At line i of the matrix, elements m ij ( f ) indicate which terminals of graphs G j are identified to terminal number i of G . If m ij ( f ) = 0 it means that no terminal of G j was identified to terminal number i of G . Aterminal of G j can be identified to at most one terminal of G (a column j cannot contain twoequal, non-zero values). Note that if m i ( f ) = 0 and m i ( f ) = 0 it means that terminal i of G is a new vertex.When f is of arity 1, its matrix m ( f ) has only one column. The t -terminal graph G = f ( G ) is obtained from graph G and matrix m ( f ) as above, by identifying terminal m i ( f )to terminal number i in G .Observe that the number of possible composition operations over t -terminal graphs isbounded by some function of t . We say that a t -terminal graph G is t -terminal recursive if itcan be obtained from t -terminal base graphs through a sequence of composition operations.This sequence is called the t -expression of graph G . Proposition 1 ([8]) . For any ( t + 1) -terminal recursive graph H = ( V, T, E ) , there is a treedecomposition of ( V, E ) of width at most t , with a bag containing T . Conversely, for any treedecomposition of width t of graph G = ( V, E ) and any bag W of the decomposition, ( V, W, E ) is a ( t + 1) -terminal recursive graph.Proof. Assume that (
V, T, E ) can be obtained recursively, through composition operations,from ( t + 1)-terminal base graphs. The expression constructing this graph can be representedas a tree, the leaves being the base graphs, each internal node corresponding to a composi-tion operation. The tree decomposition of G is simply obtained by following this tree andputting, in each node, a bag corresponding to the terminals of the graph represented by thecorresponding sub-expression. The bags are clearly of size at most t + 1. One can easily checkthat the set of bags satisfies the conditions of a tree decomposition.The other direction is proved in [8], Theorem 40.Consider a property P ( G, X ) on graphs depending on a vertex subset X . That is, property P associates to each graph G and each vertex subset X of G a boolean value. By thecelebrated results of [22, 3, 11], it is well-known that if the property can be expressed bya CMSO-formula, there exists a linear-time algorithm taking as input a ( t + 1)-terminalrecursive graph G = ( V, T, E ) and computing a maximum (or minimum) size vertex set X such that P ( G, X ). Many natural problems like
Maximum Independent Set or MinimumDominating Set can be expressed in this setting.Typical algorithms for such problems proceed by dynamic programming. When browsingthe ( t + 1)-expression of G , the algorithm stores in each node a table of classes (sometimescalled characteristics ) depending on the branch of the current sub-expression and the partial8olutions (i.e., possible subsets of X ) encountered so far. Let G be such a sub-expressionand let X be a subset of vertices that we aim to extend into the solution X . The intuitionis that if the class of ( G , X ) is the same as the class of some other pair ( G , X ), then wecan replace the branch of G by an expression of G , and the new graph G (cid:48) is such that X extends into a solution X ∪ Y of G if and only if X extends into a solution X ∪ Y of G (cid:48) .In order to efficiently solve our problem, we need an efficient computation of classes forbase graphs, as well as an efficient computation of the classes for compositions of graphs andpartial solutions.We give a formal definition of these “good” properties; the vocabulary is inspired by Borie et al. [11].Let now G = ( V, T, E ) be a ( t +1)-terminal recursive graph. For any composition operation f , let ◦ f denote the composition operation over pairs ( G, X ), where f extends in a naturalway over the values of vertex sets. If G = f ( G ) then ◦ f (( G , X )) = ( G, X ). If G = f ( G , G )then ◦ f (( G , X ) , ( G , X )) = ( G, X ), the operation being valid only if, for each terminal of G , either we have mapped terminals from both G and G , contained in both X and X , orwe have not mapped any terminal belonging to X or X . Then X is obtained from X and X by merging those vertices corresponding to terminals that have been mapped on a sameterminal of G . Definition 1 (Regular Property) . Consider a property P ( G, X ) over graphs and correspond-ing vertex subsets. Property P is called regular if, for every t , there exists a finite set C , ahomomorphism h associating to each ( t + 1) -terminal recursive graph G and every X ⊆ V ( G ) a class h ( G, X ) ∈ C , and an update function (cid:12) f : C × C → C for each composition operation f of arity (resp. (cid:12) f : C → C for each composition operation f of arity ), satisfying: • (property P is preserved) If h ( G , X ) = h ( G , X ) then P ( G , X ) = P ( G , X ) . • (integrity of operations) For any composition operation f , we have that h ( ◦ f (( G , X ) , ( G , X ))) = (cid:12) f ( h ( G , X ) , h ( G , X )) if f is of arity , and h ( ◦ f ( G , X )) = (cid:12) f ( h ( G , X )) if f is of arity . We point out that the homomorphism class h ( G, X ) depends on G and on the value of X .Typically the class of h ( G, X ) encodes, among other informations, the intersection of X withthe set of terminals. For example, if the composition operation ◦ f (( G , X ) , ( G , X )) is notvalid, then (cid:12) f ( c , c ), where c and c are the respective homomorphism classes of ( G , X )and of ( G , X ), is also undefined.Note that for any fixed t and any regular property P , the number of classes is constant.Nevertheless, this constant depends on t and on the property P . For algorithmic purposes,given t and P , we need an explicit algorithm computing the homomorphism class of a givenbase graph, and an algorithm computing the update functions (cid:12) f . I.e., we need an algorithmthat takes as input a composition operation f and one or two classes c , c ∈ C and computesthe class (cid:12) f ( c , c ) if f is of arity 2 (resp. (cid:12) f ( c ) if f is of arity 1). Eventually, we mustknow the set of accepting classes , that is the set of classes c such that h ( G, X ) = c impliesthat P ( G, X ). 9s an example, consider the property 3
COL ( G, X ) which is true only if G [ X ] is 3-colourable. We show that it is regular. Let P ( t ) be the set of partitions of subsets of { , . . . , t + 1 } into three parts. The set C of homomorphism classes is P ( t ). Considera ( t + 1)-terminal recursive graph G = ( V, T, E ) and let X ⊆ V . For each 3-partition( X , X , X ) of the vertex subset X into three independent sets, let p ( X , X , X ) ∈ P ( t ) bethe 3-partition of T ∩ X corresponding to ( T ∩ X , T ∩ X , T ∩ X ); here, for T ∩ X i , we onlykeep the ranks of the terminals of T ∩ X i in the ordered set T . The class h ( G, X ) will be { p ( X , X , X ) | ( X , X , X ) is a partition of V into three independent sets } . In particular,the unique non-accepting class is ∅ . It is not hard to see that, for fixed t , the class of everybase graph can be computed in constant time, and that for any composition operation f the update function (cid:12) f exists and can also be computed in constant time. The number ofclasses is constant even though the number of subsets X is arbitrarily large. When solvingthe problem max | X | : 3 COL ( G, X ) on a ( t + 1)-terminal recursive graph G , we must store, ineach node u of the ( t + 1)-expression, for each class c , the size of the maximum vertex subset X u of the current graph G u such that h ( G u , X u ) = c . The overall solution is the maximumone among the accepting classes of the root node.We say that a CMSO-formula ϕ expresses a property P ( G, X ) if P ( G, X ) is true if andonly if (
G, X ) models ϕ (i.e., the formula is true exactly on graphs G and vertex subsets X such that P ( G, X ) is true).
Proposition 2 (Borie et al. [11]) . Any property P ( G, X ) expressible by a CMSO-formula isregular. Moreover, the result of Borie et al. [11] is constructive in the sense that, given a CMSO-formula, it provides the homomorphism classes C , the subset of accepting classes and thealgorithms computing the classes of base graphs as well as the update functions for theregular property P on ( t + 1)-terminal recursive graphs. The regularity is actually provenin [11] for all properties expressible by CMSO-formulae, which allows an arbitrary numberof free variables over vertices, edges, vertex sets and and edge sets. For our applications, itis sufficient to consider properties over graphs and one vertex set, corresponding to formulaewith a unique free variable, which is a set of vertices.To our knowledge, the question whether all regular properties are CMSO-expressible isstill open. Chordal graphs and clique trees
A graph H is chordal (or triangulated ) if every cycleof length at least four has a chord, i.e., an edge between two nonconsecutive vertices of thecycle. By a classical result due to Buneman and Gavril [16, 40], every chordal graph G has atree decomposition such that each bag of the decomposition is a maximal clique of G . Sucha tree decomposition is referred as a clique tree of the chordal graph G . Minimal triangulations, potential maximal cliques and minimal separators A triangulation of a graph G = ( V, E ) is a chordal graph H = ( V, E (cid:48) ) such that E ⊆ E (cid:48) . Graph H is a minimal triangulation of G if for every edge set E (cid:48)(cid:48) with E ⊆ E (cid:48)(cid:48) ⊂ E (cid:48) , the graph F = ( V, E (cid:48)(cid:48) ) is not chordal. It is well known that for any graph G , tw( G ) ≤ k if and only ifthere is a triangulation H of G of clique size at most k + 1.10et u and v be two non adjacent vertices of a graph G . A set of vertices S ⊆ V is a u, v -separator if u and v are in different connected components of the graph G [ V ( G ) \ S ]. Aconnected component C of G [ V ( G ) \ S ] is a full component associated to S if N ( C ) = S .Separator S is a minimal u, v -separator of G if no proper subset of S is a u, v -separator.Notice that a minimal separator can be strictly included in another one, if they are minimalseparators for different pairs of vertices. If G is chordal, then for any minimal separator S and any clique tree T G of G there is an edge e of T G such that S is the intersection of themaximal cliques corresponding to endpoints of e [16, 40]. We say that S corresponds to e in T G .We will need the following result of Berry et al. [5]. Proposition 3 ([5]) . There is an algorithm listing the set ∆ G of all minimal separators ofan input graph G in time O ( n | ∆ G | ) . A set of vertices Ω ⊆ V ( G ) of a graph G is called a potential maximal clique if there is aminimal triangulation H of G such that Ω is a maximal clique of H . Proposition 4 ([13]) . Let Π G denote the set of all potential maximal cliques of graph G . Wehave | Π G | ≤ n | ∆ G | + n | ∆ G | + 1 , and the set Π G can be listed in time O ( n m | ∆ G | ) . We also have:
Proposition 5 ([35]) . The set of potential maximal cliques can be listed in time O (1 . n ) . Let Ω be a potential maximal clique. By [12], a subset S ⊆ Ω is a minimal separator of G if and only if S is the neighborhood of a connected component of G [ V ( G ) \ Ω].For a minimal separator S and a full connected component C of G [ V ( G ) \ S ], we say that( S, C ) is a full block associated to S . We sometimes use the notation ( S, C ) to denote theset of vertices S ∪ C of the block. It is easy to see that if X ⊆ V corresponds to the set ofvertices of a block, then this block ( S, C ) is unique: indeed, S = N ( V \ X ) and C = X \ S .For convenience, the couple ( ∅ , V ) is also considered as a full block. For a minimal separator S , a full block ( S, C ), and a potential maximal clique Ω, we call the triple (
S, C, Ω) good if S ⊆ Ω ⊆ C ∪ S . By [32], the number of good triples is at most n | Π G | .The following proposition was obtained by Fomin and Villanger [35]. Proposition 6 ([35]) . Let G [ F ] be an induced subgraph of a graph G , let T F be a minimaltriangulation of G [ F ] . There exists a minimal triangulation T G of G such that T F is aninduced subgraph of
T G .Equivalently, for every clique K G of T G , the set K G ∩ F is a (possibly empty) clique of T F . Moreover, they consider the problem of finding a maximum induced subgraph of treewidthat most t : Proposition 7 ([35]) . Given a graph G and with its set Π G of potential maximal cliques, prob-lem Maximum Induced Subgraph of Treewidth ≤ t can be solved in time O ( | Π G | n t +4 ) . By Propositions 7, 4 and 5, we deduce that for fixed t the problem can be solved in O (1 . n ) time for arbitrary graphs, and in polynomial time for classes of graphs withpolynomial number of minimal separators. 11 Optimal induced subgraph for P and t Let t ≥ P ( G, X ) be a property. We define the following generic problem.
Optimal Induced Subgraph for P and t Input:
A graph G Task:
Find sets X ⊆ F ⊆ V such that X is of maximum size, the induced subgraph G [ F ]is of treewidth at most t and P ( G [ F ] , X ) is true.Let us give two examples of problems that are particular cases of Optimal InducedSubgraph for P and t , when P ( G, X ) is a regular property.1. Let F be a finite family of graphs containing at least one planar graph. The problem Maximum induced F -minor free graph takes as input a graph G and asks for aninduced subgraph G [ F ] such that G [ F ] contains no minor from F , and F is of maximumsize for this property. As we shall see in details in Section 4, the property P ( G [ F ] , X )expressing the fact that G [ F ] is F -minor free and X = F is the vertex set of G [ F ] canbe expressed by a CMSO. Since F contains a planar graph, G [ F ] must be of treewidthat most t for some constant t depending only on F [64]. Therefore, this problem (or theequivalent problem Minimum F -Deletion ) is a particular case of Optimal InducedSubgraph for P and t .2. The problem Independent H -Packing was introduced by Cameron and Hell [18].Here H denotes a finite set of connected graphs, and the task is to find, in an inputgraph G , a maximum number of disjoint copies of graphs from H such that there isno edges between the copies. Clearly these copies induce a subgraph G [ F ] of boundedtreewidth. We will give a CMSO-formula expressing the property P ( G [ F ] , X ), which istrue if and only if G [ F ] is a collection of copies of H , and X has exactly one vertex ineach connected component of G [ F ]. This problem, generalizing the Maximum InducedMatching , is again a particular case of
Optimal Induced Subgraph for P and t .We prove here the main theorem of this article. Theorem 1.
For any fixed t and any regular property P , the problem Optimal InducedSubgraph for P and t is solvable in | Π G | n t + O (1) time, when Π G is given in the input. Let us note that by Proposition 2, results of Theorem 1 hold for every property P ( G, X )expressible by a CMSO-formula. Combined with Propositions 4 and 5, we obtain the followingapplication of Theorem 1.
Corollary 1.
For any fixed t and regular property P , problem Optimal Induced Subgraphfor P and t can be solved in O (1 . n ) time for arbitrary graphs, and in polynomial timefor classes of graphs with polynomial number of minimal separators. Our algorithm proceeds by dynamic programming on blocks and on good triples. The generalstrategy of dynamic programming over blocks and good triples follows the ideas from [32] and[35] for computing the treewidth and subgraphs of bounded treewidth. However, the devil isin details, and we need more work to make this strategy applicable for our problem.12ecall that in our definition of ( t + 1)-terminal graphs, the set of terminals is ordered. Thevertices of our graph are numbered from 1 to n . An ordered set W of vertices corresponds tothis natural ordering over set W . Property P is regular, so notations C , h and (cid:12) f correspondto Definition 1.Let G [ F ] be an induced subgraph of G and let T F be a triangulation of G [ F ]. We saythat a minimal triangulation T G of G respects T F if, for any clique K of T G , its intersectionwith F is a clique in T F . By Proposition 6, if G [ F ] is of treewidth at most t , then thereexists a (minimal) triangulation T F of G [ F ] of width at most t , and a minimal triangulation T G of G respecting T F .The next definition and the following notations are crucial for our algorithm.
Definition 2 (Partial Compatible Solution) . Let ( S, C ) denote a full block and ( S, C, Ω) denote a good triple. Let W ⊆ S (resp. W ⊆ Ω) be a vertex subset of size at most t + 1 and c ∈ C be a homomorphism class for property P . We say that ( G [ F ] , X ) is a partial solutioncompatible with ( S, C, W, c ) (resp. with ( S, C, Ω , W, c ) ) if:1. F ⊆ S ∪ C and F ∩ S = W (resp. F ∩ Ω = W );2. the ( t + 1) -terminal recursive graph H = ( F, W, E ( G [ F ])) satisfies h ( H, X ) = c ;3. there is a triangulation T F of G [ F ] of width at most t and a minimal triangulation T G of G respecting T F , such that S is a minimal separator (resp. Ω is a maximal clique)of T G . The third condition implies that W is a clique in the triangulation T F of G [ F ].Let α ( S, C, W, c ) (resp. β ( S, C, Ω , W, c )) denote the size of a largest vertex subset X such that ( G [ F ] , X ) is a partial solution compatible with ( S, C, W, c ) (resp. compatible with(
S, C, Ω , W, c )). Observe that in the β function, W represents the intersection between thepartial solution and the potential maximal clique Ω, while in the definition of the α func-tion, W is the intersection of the partial solution with the minimal separator S . If partialcompatible solutions do not exist, we simply set α or β to −∞ . Our algorithm proceeds by dynamic programming on full blocks and good triples. By [32],the number of good triples is O ( n | Π G | ). The blocks are first sorted by size. For eachblock ( S, C ) by increasing size, we first compute the values β ( S, C, Ω , W, c ) from values α ( S i , C i , W i , c i ) corresponding to smaller blocks, then we compute the values α ( S, C, W, c )13rom values β ( S, C, Ω , W (cid:48) , c (cid:48) ), as described in Algorithm 1. Algorithm 1:
Optimal Induced Subgraph for P and t Input : graph G and Π G Output : sets X ⊆ F ⊆ V ( G ) such that G [ F ] has treewidth at most t , P ( G [ F ] , X ) istrue and X is of maximum size Order all full blocks by inclusion; forall the full blocks ( S, C ) in this order do forall the good triples ( S, C, Ω) , all W ⊆ Ω of size ≤ t + 1 and all c ∈ C do if Ω = S ∪ C then Compute β ( S, C, Ω , W, c ) using Equation 2; else Compute β ( S, C, Ω , W, c ) using Equations 4, 5, 6, and 7 ; forall the W ⊆ S of size ≤ t + 1 and all c ∈ C do Compute α ( S, C, W, c ) using Equation 3; Compute the optimal solution using Equation 8;Consider a ( t + 1)-terminal recursive graph D = ( V D , T, E D ) and let c be a homomor-phism class. Although this is not explicitly required by the definition of regular properties(Definition 1), we may assume w.l.o.g. that all sets Y such that h ( D, Y ) = c have the sameintersection with the set T of terminals. (Otherwise, if sets Y and Y (cid:48) have different intersec-tions with T but h ( D, Y ) = h ( D, Y (cid:48) ) = c , we can “split” class c in at most 2 t +1 classes, onefor each possible intersection between T and such a vertex subset Y .) Moreover the class c encodes the intersection of Y with the set of terminals of D , i.e., given the homomorphismclass c , we can retrieve the rank of the vertices of Y ∩ T .Therefore we assume that we have a function term ( c, T ), taking a class c and an ordered set T of terminals, and returning the terminals that belong to Y , for any Y such that h ( D, Y ) = c . The base case.
The base case consists in minimal full blocks (
S, C,
Ω), in which caseΩ = S ∪ C by [12]. In this situation, for any partial solution ( G [ F ] , X ) compatible with( S, C, Ω , W, c ) we must have F = W , hence G [ W ] corresponds to a base ( t + 1)-terminalgraph. Also, we must have X = term ( c, W ), so X is unique (or might not exist). β ( S, C, Ω , W, c ) = (cid:26) | X | if there is X ⊆ W such that h ( G [ W ] , X ) = c −∞ otherwise (2)The computation of each value β ( S, C, Ω , W, c ) corresponding to a base case takes O ( n )time, because we have to store the value in a table indexed by ( S, C, Ω , c ). The number ofgood triples is O ( n | Π G | ) so altogether these computations take O ( n t +3 | Π G | ) time. (Actually,one can prove by a more careful analysis that the number of good triples corresponding tobase cases is at most n .) Computing α from β . Our goal is to compute α ( S, C, W, c ) from values β ( S, C, Ω , W (cid:48) , c (cid:48) )such that ( S, Ω , C ) is a good triple and W = W (cid:48) ∩ S .Consider any partial solution ( G [ F ] , X ) compatible with ( S, C, W, c ). Let
T F be a trian-gulation of G [ F ] like in Definition 2 and let T G be a minimal triangulation of G respecting T F . Let Ω be the maximal clique of
T G such that S ⊆ Ω ⊆ S ∪ C (this clique is unique14y [12]) and take W (cid:48) = Ω ∩ F . Note that ( G [ F ] , X ) is also a partial compatible solutionfor ( S, C, Ω , W (cid:48) , c (cid:48) ) where c (cid:48) is the homomorphism class of h ( H (cid:48) , X ); here H (cid:48) is the ( t + 1)-terminal recursive graph ( F, W (cid:48) , E ( G [ F ])). Also observe that the ( t + 1)-terminal graph H = ( F, W, G [ F ]) is obtained from H (cid:48) by the unary composition operation f ( W (cid:48) , W ) thatconsists in removing W (cid:48) \ W from the set of terminals, and possibly renumbering the remainingterminals. Therefore (cid:12) f ( W (cid:48) ,W ) ( c (cid:48) ) = c .We claim that: Lemma 1. α ( S, C, W, c ) = max β ( S, C, Ω , W (cid:48) , c (cid:48) ) , (3) where the maximum is taken over potential maximal cliques Ω such that ( S, C, Ω) is a goodtriple, all subsets W (cid:48) ⊆ Ω of size at most t + 1 such that W (cid:48) ∩ S = W and all classes c (cid:48) ∈ C such that (cid:12) f ( W (cid:48) ,W ) ( c (cid:48) ) = c .Proof. By the above observation, α ( S, C, W, c ) is at most the right-hand side of the equality.Conversely, let (
S, C, Ω , W (cid:48) , c (cid:48) ) be the quintuple realizing the maximum value of the right-handside expression. Let ( G [ F ] , X ) be a partial solution compatible with ( S, C, Ω , W (cid:48) , c (cid:48) ). Observethat ( G [ F ] , X ) is also a partial solution compatible with ( S, C, W, c ), hence α ( S, C, W, c ) ≥| X | . This proves the correctness of the formula computing α ( S, C, W, c ).For computing all values α ( S, C, W, c ) from values β ( S, C, Ω , W (cid:48) , c (cid:48) ), we proceed in aslightly different and more efficient way than the one described in the Algorithm 1. When β ( S, C, Ω , W (cid:48) , c (cid:48) ) is computed (lines 5 or 7 of the algorithm), if (cid:12) f ( W (cid:48) ,W ) ( c (cid:48) ) = c we sim-ply update the value of α ( S, C, W, c ) by taking the maximum between the previous valueand β ( S, C, Ω , W (cid:48) , c (cid:48) ). This only costs an extra O ( n ) for each quintuple ( S, C, Ω , W (cid:48) , c (cid:48) ).The number of such quintuples is O ( n t +2 | Π G | ), thus the total cost of these computations is O ( n t +3 | Π G | ). Computing β from α . We now compute β ( S, C, Ω , W, c ) from values α ( S i , C i , W i , c i ) where C i , 1 ≤ i ≤ p are the connected components of G [ C \ Ω], S i = N G ( C i ), W i = C i ∩ S i and c i are classes (still to be guessed). Recall that, by [12], ( S i , C i ) are full blocks.Intuitively, let ( G [ F ] , X ) be an optimal partial solution for β ( S, C, Ω , W, c ). We denote by H = ( F, W, E H ) the ( t + 1)-terminal recursive graph corresponding to G [ F ] with terminal set W , and let H i = ( F i , W i , E i ) be its trace on the smaller block ( S i , C i ). Hence F i = F ∩ ( S i ∪ C i ), W i = W ∩ S i and E i = E ( G [ F i ]). Also denote X i = X ∩ ( S i ∪ C i ). Observe that H is obtainedfrom the smaller H i s as follows: • on each H i , we introduce the terminals of W \ W i , obtaining a graph H + i = ( F i ∪ W, W, E + i ) with W as set of terminals and with E + i = E ( G [ F i ∪ W ]) as edge set. • we perform a sequence of joins, gluing one by one H +1 , H +2 , . . . , H + p on the same set ofterminals W .Formally, let us first define δ i ( S, C, Ω , W, c + i ) to be the size of the largest partial solution( G [ F + i ] , X + i ) compatible with ( S, C, Ω , W, c + i ) such that F + i ⊆ Ω ∪ C i . (This partial solutionwas denoted above by H + i , F + i corresponds to F i ∪ W , and X + i is X i ∪ ( X ∩ W ).) Consider thecomposition operation in ( W i , W ) which takes two ( t + 1)-terminal graphs, with terminal sets W i and W respectively, and composes them into a new ( t + 1)-terminal graph having W as set15f terminals. In the gluing operation, terminal number j of W i is glued on terminal number k of W if and only if they correspond to the same vertex of G . Hence, this compositionoperation in ( W i , W ) only depends on W i and W . Let X W ⊆ W , let G [ W ] denote the base( t + 1)- having W as set of terminals, and c W be the homomorphism class h ( G [ W ] , X W ). Lemma 2. δ i ( S, C, Ω , W, c + i ) = max c i ,c W s.t. (cid:12) in ( Wi,W ) ( c i ,c W )= c + i α ( S i , C i , W i , c i ) + | term ( c W , W ) \ term ( c i , W i ) | (4) over all classes c i and c W such that (cid:12) in ( W i ,W ) ( c i , c W ) = c + i and c W = h ( G [ W ] , X W ) for some X W ⊆ W .Proof. Let ( G [ F + i ] , X + i ) be a maximal partial solution compatible with ( S, C, Ω , W, c + i ) suchthat F + i ⊆ Ω ∪ C i . Denote F i = F + i ∩ ( S i ∪ C i ), X i = X + i ∩ ( S i ∪ C i ), X W = X ∩ W . Observe that( G [ F i ] , X i ) is a partial solution compatible with ( S i , C i , W i , c i ) for some class c i , that c W = h ( G [ W ] , X W ), and these classes must satisfy (cid:12) in ( W i ,W ) ( c i , c W ) = c + i . Hence δ i ( S, Ω , C, W, c + i )is at most equal to the right-hand side of the equation (note that term ( c W , W ) \ term ( c i , W i ) = X + i \ X i ).Conversely, let c i , c W be the classes maximizing the right-hand side of the equation. Takea maximum partial solution ( G [ F i ] , X i ) contained in S i ∪ C i , compatible with ( S i , C i , W i , c i ),where (cid:12) in ( W i ,W ) ( c i , c W ) = c + i . Then the graph ( F i ∪ W, W, E ( G [ F i ∪ W ])) together with thevertex subset X i ∪ term ( c W , W ) is a partial solution compatible with ( S, C, Ω , W, c + i ), andthe equality follows.We introduce another notation γ i ( S, C, Ω , W, c ), corresponding to the largest partial solu-tion compatible with ( S, C, Ω , W, c ), contained into Ω ∪ C ∪ · · · ∪ C i . It corresponds to thegluing of some partial solutions ( H +1 , X +1 ) , . . . ( H + i , X + i ). Lemma 3.
Function γ i is computed as follows. γ ( S, C, Ω , W, c ) = δ ( S, Ω , C, W, c ) (5) For any i , ≤ i ≤ p , γ i ( S, C, Ω , W, c ) = max c (cid:48) ,c (cid:48)(cid:48) γ i − ( S, C, Ω , W, c (cid:48) ) + δ i ( S, Ω , C, W, c (cid:48)(cid:48) ) − | term ( c (cid:48) , W ) | , (6) over all characteristics c (cid:48) , c (cid:48)(cid:48) ∈ C such that (cid:12) g ( W ) ( c (cid:48) , c (cid:48)(cid:48) ) = c , where g ( W ) is the compositionoperation corresponding to a join operation on W . I.e., the matrix m ( g ( W )) of g ( W ) has | W | rows, and m j, ( g ( W )) = m j, ( g ( W )) = j for each row j .Proof. The proof is trivial for γ .Now for any F ⊆ Ω ∪ C ∪ · · · ∪ C i , note that ( G [ F ] , X ) is a partial solution compatiblewith ( S, C, Ω , W, c ) if and only if ( G [ F \ C i ] , X \ C i ) (resp. ( G [ F \ ( C ∪ · · · ∪ C i − )] , X \ ( C ∪· · · ∪ C i − ))) are partial solutions compatible with ( S, Ω , C, W, c (cid:48)(cid:48) ) (resp. ( S, C, Ω , W, c (cid:48) )) and (cid:12) g ( W ) ( c (cid:48) , c (cid:48)(cid:48) ) = c . The term | term ( c (cid:48) , W ) | corresponds to X ∩ W and avoids over-counting ofthese vertices.The following result is a direct consequence of the definition of β and γ functions.16 emma 4. β ( S, C, Ω , W, c ) = γ p ( S, C, Ω , W, c ) . (7)We claim that computing, for a fixed quadruple ( S, C, Ω , W ), the values β ( S, C, Ω , W, c )from values α , takes O ( n ) time. Again by [32], the smaller blocks ( S i , C i ) can be listed in O ( m ) time. For each i , the computation of function δ i ( S, Ω , C, W, c + i ) takes O ( | S i | + | C i | )time, because we need to access the values α ( S i , W i , C i , c i ). The sum of these values is atmost n + m [32]. Computing γ i ( S, C, Ω , W, c ) from values γ i − and δ i can be done in O ( n )time for each i .Therefore the running time of the algorithm is the number of quintuples ( S, C, Ω , W, c )times n , that is O ( | Π G | n t +4 ). The global solution.
It can be obtained by considering the (special) full block ( ∅ , V ). Lemma 5.
The solution size is max c α ( ∅ , V, ∅ , c ) , (8) over all accepting classes c , i.e., classes such that ( h ( G, X ) = c ) implies that P ( G, X ) .Proof. By definition of regular properties and of α ( ∅ , V, ∅ , c ), our problem has a solution ofsize at least max c α ( ∅ , V, ∅ , c ) over accepting classes c .Let ( G [ F ] , X ) be a maximum size solution for our problem. By Proposition 6, this so-lution is compatible with α ( ∅ , V, ∅ , c ) for the class c of the ( t + 1)-terminal graph graph( F, ∅ , E ( G [ F ])), which achieves the proof of the lemma.This latter computation takes constant time.The total running time of the algorithm is O ( | Π G | n t +4 ). Note that, instead of keepingthe size of the largest solution ( G [ F ] , X ), we could explicitly store the vertex subsets ( F, X )of G . Theorem 1 can be extended to weighted and annotated versions of problem
Optimal InducedSubgraph for P and t , for any t ≥ P . Optimal Weighted Annotated Induced Subgraph for P and t Input:
A graph G = ( V, E ) a weight function w : V → R , a set U ⊆ V of annotatedvertices and a number t . Task:
Find sets X ⊆ F ⊆ V such that F contains U , the induced subgraph G [ F ] is oftreewidth at most t , property P ( G [ F ] , X ) is true and X is of maximum weight underthese conditions. Theorem 2.
For any fixed t and any regular property P , the problem Optimal WeightedAnnotated Induced Subgraph for P and t is solvable in | Π G | n O (1) time, when Π G isgiven in the input.In particular the problem can be solved in O (1 . n ) time for arbitrary graphs, and inpolynomial time for classes of graphs with polynomial number of minimal separators. α and β functions. In order to forcethe annotated vertices to be in F , each value α ( S, C, W, c ) (resp. β ( S, C, Ω , W, c ) such that U ∩ S (cid:54)⊆ W (resp. U ∩ Ω (cid:54)⊆ W ) is immediately set to −∞ , meaning that such a partial solutionis rejected.In order to maximize the weight of the solution, the values α ( S, C, W, c ) (respectively β ( S, C, Ω , W, c )) will correspond to the maximum weight over partial solutions compatiblewith ( S, C, W, c ) (resp. (
S, C, Ω , W, c )). In the algorithm, we simply replace the cardinality ofsets (e.g., | X | in Equation 2, | term ( c (cid:48) , W ) | is Equation 6 and | term ( c W , W ) \ term ( c i , W i ) | inEquation 4) by the weights of these sets.We also point out that the weights can be negative. In particular, we can use Theorem 2to compute an induced subgraph G [ F ] of treewidth at most t and a subset X ⊆ F such that P ( G [ F ] , X ) is true, and X is of minimum size (or weight) under these conditions.One can imagine more extensions of Theorems 1 and 2. A natural one consists in findingsets X and F such that the size of X is exactly an input value v . For this purpose, wecan adapt our definitions of α and β to store, for each possible value v (cid:48) ≤ v , a boolean α ( S, C, W, c, v (cid:48) ) (resp. β ( S, C, Ω , W, c, v (cid:48) )), set to true if and only if there exists partial solution( G [ F (cid:48) ] , X (cid:48) ) compatible with ( S, C, W, c ) (resp. (
S, C, Ω , W, c ) such the size of X (cid:48) is exactly v (cid:48) . The computation of α and β is quite straightforward, by adapting Equations 2 to 8. Thecomplexity of the algorithm is multiplied by a polynomial factor.Even more involved, we can consider properties P ( G, X , . . . , X p , E , . . . , E q ), where each X i is a vertex subset and each E j is an edge subset of graph G . The notion of regularityextends in a very natural way to several variables. Recall that Borie et al. [11] proved that all properties expressible by CMSO-formulae are regular, so we are allowed to use any (fixed)number of free variables corresponding to vertex sets and edge sets.Let t ≥ P ( G, X , . . . , X p , E , . . . , E q ) be a regular property on graphsand vertex subsets X i and edge subsets E j . We define the following generic problem. Constrained Induced Subgraph for P and t Input:
A graph G , integer values v , . . . , v p ≤ n and w , . . . , w p ≤ n ( n − Task:
Find F ⊆ V , sets X i ⊆ F and E j ⊆ E ( G [ F ]) such that the induced subgraph G [ F ]is of treewidth at most t , P ( G, X , . . . , X p , E , . . . , E q ) is true, each set X i is of size v i and each set E j is of size w j .Since property P is regular, we need to adapt the definition of partial solutions to morevariables (again very naturally) and then we define as above boolean functions α ( S, C, W, c, v (cid:48) , . . . , v (cid:48) p , w (cid:48) . . . , w (cid:48) q ) , respectively β ( S, C, Ω , W, c, v (cid:48) , . . . , v (cid:48) p , w (cid:48) . . . , w (cid:48) q )to be true if there exists a partial solution ( G [ F (cid:48) ] , X (cid:48) , . . . , X (cid:48) p , E (cid:48) , . . . , E (cid:48) q ) compatible with( S, C, W, c ) (resp. (
S, C, Ω , W, c )) such that each X (cid:48) i is of size v (cid:48) i and each E (cid:48) j is of size w (cid:48) j .For computing the α and β values, we must again adapt Equations 2 to 8. Basically, for eachclass c , the function term ( c, W ) used in the equations for a homomorphism class c and anorder set of terminals W must now return each intersection of type X (cid:48) i ∩ W for vertex sets and E (cid:48) j ∩ G [ W ] for edge sets. These intersections will be used to avoid overcounting when glueingpartial solutions. The complexity of the algorithm becomes larger by a factor of n O ( p + q ) .18herefore we can solve problems like finding, among maximum induced subgraph oftreewidth at most t , the one with minimum dominating set. In this section we discuss several applications of Theorem 1. Our results are summarized inthe following theorem. Recall that the problems have been defined in the
Introduction . Theorem 3.
Let G be an n -vertex graph given together with the set of its potential maximalcliques Π G . Then • Maximum Induced Subgraph with ≤ (cid:96) copies of F m -cycles , • Maximum Induced Subgraph with ≤ (cid:96) copies of p -cycles , • Maximum Induced Subgraph with ≤ (cid:96) copies of Minor Models from F , where F contains a planar graph, • Independent G ( t, ϕ ) -Packing , and • k -in-a-Graph From G ( t, ϕ ) are solvable in time | Π G | · n O (1) . Here the hidden constants in O depend on m, p, (cid:96) , F , t , and ϕ . Combined with Proposition 5, Theorem 3 implies the following.
Corollary 2.
Let G be an n -vertex graph. All problems from Theorem 3 are solvable in time O (1 . n ) . The proof of Theorem 3 follows from Theorem 1 and Lemmata 6, 7, 8, 9, and 10.Let us remark that Theorem 3 also holds for different modifications of these problems, likerequirements of the maximum induced subgraph being connected, of maximum vertex degreeat most some constant ∆, etc. Such modifications easily capture problems like computing alongest induced path, cycle, or an induced tree with given maximum vertex degree.
Hitting and packing cycles of length m ) . We will need the following result ofThomassen.
Proposition 8 ([70]) . For every integers (cid:96), m > there exists an integer k ( (cid:96), m ) > suchthat the treewidth of a graph with at most (cid:96) vertex-disjoint cycles from F m is at most k ( (cid:96), m ) . With the help of Proposition 8, we obtain the following lemma.
Lemma 6.
Maximum Induced Subgraph with ≤ (cid:96) copies of F m -cycles is a specialcase of Optimal Induced Subgraph for P and t with t = f ( (cid:96), m ) , where f depends onlyon m and (cid:96) .Proof. For a graph G let F be the maximum vertex set such that G [ F ] has at most (cid:96) vertex-disjoint cycles from F m . We put f ( (cid:96), m ) = k ( (cid:96), m ), where k ( (cid:96), m ) is the integer from Propo-sition 8. By Proposition 8, the treewidth of G F is at most f ( (cid:96), m ).19hen Maximum Induced Subgraph with ≤ (cid:96) copies of F m -cycles is to maximize | X | for the following property P ( G [ F ] , X ) = { F = X and G [ F ] contains at most (cid:96) vertex-disjoint cycles from F m . } To show that P ( G [ F ] , X ) is regular, we observe that it is expressible by a CMSO-formula.Indeed, this formula expresses that for every partition of V ( G F ) into (cid:96) + 1 subsets, there is asubset containing no cycle from F m . Hitting long cycles.
We need the following result, which is due to Birmel´e et al.
Proposition 9 ([7]) . Graphs without (cid:96) disjoint cycles of length at least p are of treewidth O ( (cid:96) p ) . By making use of Proposition 9, it is easy to prove the following lemma.
Lemma 7.
Maximum Induced Subgraph with ≤ (cid:96) copies of p -cycles is a special caseof Optimal Induced Subgraph for P and t with t = O ( (cid:96) p ) .Proof. For a graph G let F be the maximum vertex set such that G [ F ] has at most (cid:96) vertex-disjoint cycles of length at least p . By Proposition 9, the treewidth of G [ F ] is at most O ( (cid:96) p ).Then we are maximizing | X | for the following property P ( G [ F ] , X ) = { F = X and G [ F ] contains ≤ (cid:96) vertex-disjoint cycles of length ≥ p. } To show that this property is regular, we observe that property of not having a cycle of lengthat least p is expressible in CMSO. Indeed, a property of a set C of vertices to induce a cycleis CMSO, and because p is fixed, the formula expressing the sentence that for every subset C inducing a cycle, the number of elements is at most p , is of constant length. Because (cid:96) is alsofixed, it is possible to express by a constant size CMSO-formula the sentence that for everypartition in (cid:96) + 1 subsets there is a subset inducing a subgraph without a cycle of length atleast p . Excluding planar minors.
The following proposition follows almost directly from theexcluded grid theorem of Robertson and Seymour [64], see also [65].
Proposition 10 ([64]) . For every integer (cid:96) > and family F containing a planar graph, thereexists an integer k ( (cid:96), F ) > such that the treewidth of a graph with at most (cid:96) vertex-disjointminor models from F is at most k ( (cid:96), F ) . Lemma 8. If F contains a planar graph, then Maximum Induced Subgraph with ≤ (cid:96) copies of Minor Models from F is a special case of Optimal Induced Subgraph for P and t with t = k ( (cid:96), F ) .Proof. For a graph G let F be the maximum vertex set such that G [ F ] has at most (cid:96) vertex-disjoint models of minors from F . By Proposition 10, the treewidth of G [ F ] is at most k ( (cid:96), F ). The property that a graph does not contain a fixed graph as a minor is known to beexpressible in CMSO. This implies that the property P ( G [ F ] , X ) = { F = X and G [ F ] has ≤ (cid:96) vertex-disjoint minor models from F } is regular. 20 ndependent packing.Lemma 9.
Independent G ( t, ϕ ) -Packing is a special case of Optimal Induced Sub-graph for P and t .Proof. For a graph G let F be a vertex set such that G F = G [ F ] has the maximum numberof connected components, and each of the components is in G ( t, ϕ ). Because the treewidthof every component does not exceed t , the treewidth of G [ F ] does not exceed t . We use cc ( G [ F ]) to denote the set of connected components of G [ F ]. Because the property thatevery connected component has regular is also regular, we have that the following propertyis regular P ( G [ F ] , X ) = { [ X ⊆ V ( G F )] ∧ [ ∀ C ∈ cc ( G F )( C ∈ G ( t, ϕ ) ∧ | X ∩ C | = 1)] . } k -in-a-graph. Because in k -in-a-Graph From G ( t, ϕ ), k is part of the input we need theannotated variant of the main theorem (Theorem 2). The following lemma follows from thedefinition of the problems. Lemma 10. k -in-a-Graph From G ( t, ϕ ) is a special case of Optimal Weighted Anno-tated Induced Subgraph for P and t . In this section we discuss the consequences of Theorem 3 for special graph classes. In par-ticular, by Proposition 4, every class of graphs with polynomially many minimal separatorsalso has polynomially many potential maximal cliques. For example, every n -vertex weaklychordal graph, i.e. graph with no induced cycle or its complement of length greater than four,has O ( n ) minimal separators [12]. This class of graphs is a generalization of many graphclasses intensively studied in the literature like chordal, split, and interval graphs. Anotherclass of graphs of this type is the class of circular-arc graphs, intersection graphs of a set ofarcs on the circle. Every circular-arc with n vertices has at most 2 n − n minimal separators[52]. The class of d -trapezoid graphs is defined as follows. Let L , . . . , L d be d parallel linesin the plane. A d -trapezoid is the polygon obtained by choosing an interval I i on every line L i and connecting the left, respectively, right endpoint of I i with the left, respectively, rightendpoint of I i +1 . A graph is a d -trapezoid graph if it has an intersection model consisting of d -trapezoids between d parallel lines. Every d -trapezoid graph has at most (2 n − d − minimalseparators [55], see also [15]. An intersection graph of polygons enclosed by a bounding circleis is know as a polygon-circle graph . As it was observed by Suchan in [68], every polygon-circlewith n vertices has O ( n ) minimal separators. See Fig 1 of the Introduction for the relationsbetween most known classes of graphs with polynomially many minimal separators. We referto the encyclopedia of graph classes [15] for definitions of different graphs from Fig 1.Let us remark that the only information for our algorithms we need is the bound on thenumber of minimal separators in the specific graph class. While many of the algorithms fromthe literature for intersection classes of graphs strongly use the intersection model this is notnecessary for our algorithms—they produce correct output regardless of whether the inputactually belongs to the specific class of graphs. If the number of minimal separators and thus21otential maximal cliques is bounded, our algorithm correctly solves the problem. Otherwise,the algorithm correctly reports that the given input is not from the restricted domain. Suchtype of algorithms were called robust by Raghavan and Spinrad [62]. For example, whilerecognition of d -trapezoid and polygon-circle graphs is NP-complete [72, 61], our algorithmeither correctly solves the problem or outputs that the input graph is not d -trapezoid orpolygon-circle. Corollary 3.
All problems from Theorem 3 are solvable in polynomial time on classes ofgraphs from Fig 1.
On several classes of graphs even more general problems can be solved. The observationhere is that for many classes of graphs from Fig 1, the treewidth of a graph is upper boundedby some function of other parameters like the maximum clique-size or maximum degree.For example, Yannakakis and Gavril [73] have shown that for every fixed χ , a maximuminduced subgraph of a chordal graph colorable in χ colors can be found in polynomial time.To see why this result follows as a corollary of our theorem, let us observe that for chordalgraphs, as for all perfect graphs, the chromatic number is equal to the maximum clique size,see e.g. [44]. On the other hand, the treewidth of a chordal graph is known to be equal to themaximum clique size minus one. Thus every induced χ -colorable subgraph of a chordal graphis of treewidth at most χ −
1. Since colorability in a constant number of colors is expressiblein CMSO, the result follows.For other variant of colorings, we need the the following proposition due to Gaspers et al.
Proposition 11 ([39]) . Let G be a graph of maximum vertex degree at most D . Then thetreewidth of G is at most • D , if G is a circle graph, • D , if G is a weakly chordal graph or a circular-arc graph. Combined with Proposition 11, Theorem 3 allows us to show that on several graph classes,in addition to problems encompassed by Corollary 3, even larger class of problems can besolved efficiently. For example, edge coloring of a graph is an assignment of colors to theedges of the graph so that no two adjacent edges have the same color. The chromatic index of a graph is the minimum number of colors required for edge coloring. By Vizing’s theorem,for every graph with maximum vertex degree D , its chromatic index is either D or D + 1.Since edge coloring in a constant number of colors is expressible in CMSO, we conclude thatthe problem of finding a maximum induced edge-colorable in k colors subgraph (for a fixedconstant k ) is solvable in polynomial time on circle, weakly chordal and circular-arc graphs.Similarly, the problems like for a fixed constant k finding a maximum induced (connected)subgraph of maximum vertex degree at most k are also solvable in polynomial time on theseclasses of graphs.The next lemma provides a different set of applications of the main theorem for specialgraph classes. Lemma 11.
Let G be a graph excluding some fixed graph H as a minor. Then the treewidthof G is at most • f ( H ) for some function f of H only, if G is a weakly-chordal graph, and | V ( H ) | if G is a circular-arc graph.Proof. Let G be a weakly chordal graph excluding H as a minor. By a theorem from [29],there is a constant c H such that every H -minor-free graph of treewidth at least c H k canbe transformed by making only edge contractions either to a planar triangulation Γ k of a( k × k )-grid, or to Π k , which is a graph obtained from G k by adding a universal vertex. Sinceboth Γ k and Π k for k ≥ G does not exceed some constant depending only on H . Indeed, otherwise acontraction of G , and hence G too, would contain an induced cycle of length more than 4.For circular-arc graphs, we can prove the statement of the lemma by using the observationfrom [52] that every potential maximal clique of a circular-arc graph is the union of at mostof three cliques. Thus every circular-arc graph of treewidth at least 3 | V ( H ) | should contain apotential maximal clique of size at least 3 | V ( H ) | , and hence a clique of size at least | V ( H ) | .Thus every circular-arc graph of treewidth at least 3 | V ( H ) | contains H as a minor.By combining Lemma 11 with Theorem 3, we obtain that Minimum F -Deletion issolvable in polynomial time on circular-arc and weakly chordal graphs for every finite family F of graphs. The requirement that F contains a planar graph can be omitted in this case. While regular properties and CMSO capture many interesting problems, it seems that theapproach based on minimal triangulations is not restricted by these settings. Take for examplethe following problem.
Minimum Induced Disjoint Connected (cid:96) -Subgraphs
Input:
A graph G , and a collection { T , T , . . . , T p } of terminal vertices, T i ⊆ V ( G ), ofsize at most (cid:96) . Task:
Find a set F ⊆ V ( G ) of minimum size such that G [ F ] has connected components C , C , . . . , C p and for every 1 ≤ i ≤ p , T i ⊆ C i .This problem is a generalization of the Induced Disjoint Paths , where for a given setof p pairs of terminals x i , y i , 1 ≤ i ≤ p , the task is to find a set of paths connecting terminalssuch that the vertices from different paths are not adjacent. Belmonte et al. [4] have shownthat Induced Disjoint Paths is solvable in polynomial time on chordal graphs. Because p is part of the input and not fixed, this problem cannot be expressed by a CMSO-formula ofconstant size. On the other hand, by applying a modification of the dynamic programmingalgorithm over potential maximal cliques and minimal separators, it is possible to show thatthis problem is solvable in time proportional to the number of potential maximal cliques, upto polynomial factor n t + O (1) .Another example can be the following problem. Let t be an integer. Homomorphism from t -Treewidth Subgraph Input:
Graph G and H Task:
Find a set F ⊆ V ( G ) of maximum size such that the treewidth of G [ F ] is at most t and there is a homomorphism from G [ F ] to H .By the classical result of Yannakakis and Gavril [73], for every fixed χ , a maximuminduced subgraph of a chordal graph colorable in χ colors can be found in polynomial time.23ecause coloring into χ colors is homomorphism in a complete graph on χ vertices, andbecause the treewidth of a χ -colorable chordal graph is at most χ − Homomorphismfrom t -Treewidth Subgraph extends this problem. However, the property of having ahomomorphism to H is not CMSO-expressible because H is part of the input. Moreover,it is easy to see that already very special case of graph homomorphism problem, where weare asked for a homomorphism from a clique of size k (and thus of treewidth k −
1) to H is equivalent to deciding if H has a clique of size at least k , which is W[1]-hard. Thushomomorphism from G to H parameterized by the treewidth of G is W[1]-hard. But on theother hand, dynamic programming over potential maximal cliques and minimal separatorsshows that Homomorphism from t -Treewidth Subgraph is solvable in time proportionalto the number of potential maximal cliques, up to polynomial factor n O ( t ) .Both examples indicate that even more general framework capturing problems solvablein time proportional to the number of potential maximal cliques can exist. Defining such ageneral framework is an interesting open question.Another open question concerns counting problems. Our approach does not work forcounting problems due to potential double counting in the process of computing functions α and β . We do not exclude a possibility that with additional (clever) ideas the main algorithmof the paper can also count maximum sets with regular properties but we do not know howto do it, and leave it as an interesting open question.Another problem which seems to be very much related but still cannot be handled directlyby our approach is Connected Feedback Vertex Set , where we are asked to find aminimum feedback vertex set inducing a connected subgraph. Interestingly, out approachworks without problems for
Maximum Induced Tree , where the task is to find a minimumfeedback vertex set such the remaining graph is connected, i.e. a tree.
Acknowledgements
We thank Bruno Courcelle, Daniel Lokshtanov, Mamadou Kant´e,Dieter Kratsch, Saket Saurabh, Bich Dao and Dimitrios M. Thilikos for fruitful discussionsand useful suggestions on the topic of the paper.24 eferences [1]
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