An FPT 2-Approximation for
Tree-Cut Decomposition††thanks: The research of the last author was
co-financed by the European Union (European Social Fund ESF) and Greek national
funds through the Operational Program “Education and Lifelong Learning” of the
National Strategic Reference Framework (NSRF), Research Funding Program: ARISTEIA II.
The second author was supported by Basic Science Research
Program through the National Research Foundation of Korea (NRF)
funded by the Ministry of Science, ICT & Future Planning
(2011-0011653). ††thanks: An extended abstract of this article will appear in the Proceedings of the 13th Workshop on Approximation and Online Algorithms (WAOA), Patras, Greece, September 2015.
The tree-cut width of a graph is a graph parameter defined by Wollan [J. Comb. Theory, Ser. B, 110:47–66, 2015] with the help of tree-cut decompositions. In certain cases, tree-cut width appears to be more adequate than treewidth as an invariant that, when bounded, can accelerate the resolution of intractable problems. While designing algorithms for problems with bounded tree-cut width, it is important to have a parametrically tractable way to compute the exact value of this parameter or, at least, some constant approximation of it. In this paper we give a parameterized -approximation algorithm for the computation of tree-cut width; for an input -vertex graph and an integer , our algorithm either confirms that the tree-cut width of is more than or returns a tree-cut decomposition of certifying that its tree-cut width is at most , in time . Prior to this work, no constructive parameterized algorithms, even approximated ones, existed for computing the tree-cut width of a graph. As a consequence of the Graph Minors series by Robertson and Seymour, only the existence of a decision algorithm was known.
Keywords: Fixed-Parameter Tractable algorithm; tree-cut width; approximation algorithm.
One of the most popular ways to decompose a graph into smaller pieces is given by the notion of a tree decomposition. Intuitively, a graph has a tree decomposition of small width if it can be decomposed into small (possibly overlapping) pieces that are altogether arranged in a tree-like structure. The width of such a decomposition is defined as the minimum size of these pieces. The graph invariant of treewidth corresponds to the minimum width of all possible tree decompositions and, that way, serves as a measure of the topological resemblance of a graph to the structure of a tree. The importance of tree decompositions and treewidth in graph algorithms resides in the fact that a wide family of -hard graph problems admits FPT-algorithms, i.e., algorithms that run in steps, when parameterized by the treewidth of their input graph. According to the celebrated theorem of Courcelle, for every problem that can be expressed in Monadic Second Order Logic (MSOL)  it is possible to design an -step algorithm on graphs of treewidth at most . Moreover, towards improving the parametric dependence, i.e., the function , of this algorithm for specific problems, it is possible to design tailor-made dynamic programming algorithms on the corresponding tree decompositions. Treewidth has also been important from the combinatorial point of view. This is mostly due to the celebrated “planar graph exclusion theorem” [14, 15]. This theorem asserts that:
(*) Every graph that does not contain some fixed wall111We avoid the formal definition of a wall here. Instead, we provide the following image that, we believe, provides the necessary intuition. as a topological minor222A graph is a topological minor of a graph if a subdivision of is a subgraph of . has bounded treewidth.
The above result had a considerable algorithmic impact as every problem for which a negative (or positive) answer can be certified by the existence of some sufficiently big wall in its input, is reduced to its resolution on graphs of bounded treewidth. This induced a lot of research on the derivation of fast parameterized algorithms that can construct (optimally or approximately) these decompositions. For instance, according to , treewidth can be computed in steps where while, more recently, a 5-approximation for treewidth was given in  that runs in steps.
Unfortunately, the aforementioned success stories about treewidth have some natural limitations. In fact, it is not always possible to use treewidth for improving the tractability of NP-hard problems. In particular, there are interesting cases of problems where no such an FPT-algorithm is expected to exist [7, 10, 6]. Therefore, it is an interesting question whether there are alternative, but still general, graph invariants that can provide tractable parameterizations for such problems.
A promising candidate in this direction is the graph invariant of tree-cut width that was recently introduced by Wollan in . Tree-cut width can be seen as an “edge” analogue of treewidth. It is defined using a different type of decompositions, namely, tree-cut decompositions that are roughly tree-like partitions of a graph into mutually disjoint pieces such that both the size of some “essential” extension of these pieces and the number of edges crossing two neighboring pieces are bounded (see Section 2 for the formal definition). Our first result is that it is NP-hard to decide, given a graph and an integer , whether the input graph has tree-cut width at most . This follows from a reduction from the Min Bisection problem that is presented in Subsection 2.2. This encourages us to consider a parameterized algorithm for this problem.
Another tree-like parameter that can be seen as an edge-counterpart of treewidth is carving-width, defined in . It is known that a graph has bounded carving-width if and only if both its treewidth and its maximum degree are bounded. We stress that this is not the case for tree-cut width, which can also capture graphs with unbounded maximum degree and, thus, is more general than carving-width. There are two reasons why tree-cut width might be a good alternative for treewidth. We expose them below.
(1) Tree-cut width as a parameter. From now on we denote by (resp. ) the tree-cut width (resp. treewidth) of a graph . As it is shown in  . Moreover, in , it was proven that and in Subsection 2.3, we prove that the latter upper bound is asymptotically tight. The graph class inclusions generated by the aforementioned relations are depicted in Fig. 1. As tree-cut width is a “larger” parameter than treewidth, one may expect that some problems that are intractable when parameterized by treewidth (known to be W-hard or open) become tractable when parameterized by tree-cut width. Indeed, some recent progress on the development of a dynamic programming framework for tree-cut width (see ) confirms that assumption. According to , such problems include Capacitated Dominating Set problem, Capacitated Vertex Cover , and Balanced Vertex-Ordering problem. We expect that more problems will fall into this category.
(2) Combinatorics of tree-cut width. In  Wollan proved the following counterpart of (*):
(**) Every graph that does not contain some fixed wall as an immersion333A graph is an immersion of a graph if can be obtained from some subgraph of after replacing edge-disjoint paths with edges. has bounded tree-cut width.
Notice that (*) yields (**) if we replace “topological minor” by “immersion” and “treewidth” by “tree-cut width”. This implies that tree-cut width has combinatorial properties analogous to those of treewidth. It follows that every problem where a negative (or positive) answer can be certified by the existence of a wall as an immersion, can be reduced to the design of a suitable dynamic programming algorithm for this problem on graphs of bounded tree-cut width.
Computing tree-cut width.
It follows that designing dynamic programming algorithms on tree-cut decompositions might be a promising task when this is not possible (or promising) on tree-decompositions. Clearly, this makes it imperative to have an efficient algorithm that, given a graph and an integer , constructs tree-cut decompositions of width at most or reports that this is not possible. Interestingly, an -time algorithm for the decision version of the problem is known to exist but this is not done in a constructive way. Indeed, for every fixed , the class of graphs with tree-cut width at most is closed under immersions . By the fact that graphs are well-quasi-ordered under immersions , for every , there exists a finite set of graphs such that has tree-cut width at most if and only if it does not contain any of the graphs in as an immersion. From , checking whether an -vertex graph is contained as an immersion in some -vertex graph can be done in steps. It follows that, for every fixed , there exists a polynomial algorithm checking whether the tree-cut width of a graph is at most . Unfortunately, the construction of this algorithm requires the knowledge of the set for every , which is not provided by the results in . Even if we knew , it is not clear how to construct a tree-cut decomposition of width at most , if one exists.
In this paper we make a first step towards a constructive parameterized algorithm for tree-cut width by giving an FPT 2-approximation for it. Given a graph and an integer , our algorithm either reports that has tree-cut width more than or outputs a tree-cut decomposition of width at most in steps. The algorithm is presented in Section 3.
2 Problem definition and preliminary results
Unless specified otherwise, every graph in this paper is undirected and loopless and may have multiple edges. By and we denote the vertex set and the edge set, respectively, of a graph . Given a vertex , the neighborhood of is . Given two disjoint sets and of , we denote . For a subset of , we define .
2.1 Tree-cut width and treewidth
Tree-cut width. A tree-cut decomposition of is a pair where is a tree and such that
for all distinct and in ,
From now on we refer to the vertices of as nodes. The sets in are called the bags of the tree-cut decomposition. Observe that the conditions above allow to assign an empty bag for some node of . Such nodes are called trivial nodes. Observe that we can always assume that trivial nodes are internal nodes.
Let be the set of leaf nodes of . For every tree-edge of , we let and be the subtrees of which contain and , respectively.
We define the adhesion of a tree-edge of as follows:
For a graph and a set , the 3-center of is the graph obtained from by repetitively dissolving every vertex that has two neighbors and degree 2 and removing every vertex that has degree at most 2 and one neighbor (dissolving a vertex of degree two with exactly two neighbors and is the operation of removing and adding the edge – if this edge already exists then its multiplicity is increased by one).
Given a tree-cut decomposition of and node , let be the connected components of . The torso of at , denoted by , is a graph obtained from by identifying each non-empty vertex set into a single vertex (in this process, parallel edges are kept). We denote by the 3-center of . Then the width of equals
The tree-cut width of , or in short, is the minimum width of over all tree-cut decompositions of .
The following definitions will be used in the approximation algorithm. Let be a tree-cut decomposition of . It is non-trivial if it contains at least two non-empty bags, and trivial otherwise. We will assume that every leaf of a tree-cut decomposition has a non-empty bag. The internal-width of a non-trivial tree-cut decomposition is
If is trivial, then we set .
We decision problem corresponding to tree-cut width is the following:
Tree-cut Width Input: a plane graph and a non-negative integer . Question: ?
Treewidth. A tree decomposition of a graph is a pair such that is a tree and is a collection of subsets of where
for every edge there exists such that ; and
for every vertex the set of nodes induces a subtree of .
The vertices of are called nodes of and the sets are called bags. The width of a tree decomposition is the size of the largest bag minus one. The treewidth of a graph, denoted by , is the smallest width of a tree decomposition of .
2.2 Computing tree-cut width is Np-complete
We prove that Tree-cut Width is NP-hard by a polynomial-time reduction from Min Bisection, which is known to be NP-hard . The input of Min Bisection is a graph and a non-negative integer , and the question is whether there exists a bipartition of such that and .
Tree-cut Width is NP-complete.
It is easy to see that Tree-cut Width is in NP. We present a reduction from Min Bisection to Tree-cut Width (see Fig. 2). Let be an instance of Min Bisection on vertices. We may assume that since otherwise, the instance is trivially NO. We create an instance with as follows. The vertex set consists of a set of size , a set of size , and the set of size for every pair . Edges are added so that:
For every pair , all vertices of are adjacent with both and .
Each is adjacent with (arbitrarily chosen) vertices of .
We now proceed with the proof of the correctness of the above reduction. Suppose that is a Yes-instance to Min Bisection with a bipartition . Consider a tree-cut decomposition in which contains three nodes and some additional nodes. The tree forms a star with as the center and all other nodes as leaves. We have for , and each vertex of forms a singleton bag. It is not difficult to verify that is a tree-cut decomposition of whose width is . In particular, notice that and for .
Conversely, suppose that admits a tree-cut decomposition of width at most . Any two vertices must be in the same bag since they are connected by disjoint paths via . Hence, there exists a tree node, say , in such that .
Consider the set of the connected components of and let be the tree-edge between and . As , there are at most two tree-edges among such that . This means that there are at most two subtrees among such that . From the fact that , at least vertices of are not contained in and thus there exists at least one subtree such that . If there is such that , then , a contradiction. Hence, we conclude that there are exactly two subtrees, say and , in such that for and for , we have . This, together with the fact that , enforces that the sets and make a bipartition of into sets of equal size. Let us call this bipartition . Observe that , thus contains at least edges for . As , it follows . Therefore, is Yes-instance to Min Bisection which complete the proof.
2.3 Tree-cut width vs treewidth
In this section we investigate the relation between treewidth and tree-cut width. The following was proved in .
For a graph of tree-cut width at most , its treewidth is at most .
In the rest of this subsection we prove
that the bound of Proposition 1 is asymptotically optimal. For this we need some definitions.
Let be a graph. Two subgraphs and of touch each other if either or there is an edge with and . A bramble is a collection of connected subgraphs of pairwise touching each other. The order of a bramble is the minimum size of a hitting set of , that is a set such that for every , . In Seymour and Thomas , it is known that the treewidth of a graph equals the maximum order over all brambles of minus one. Therefore, a bramble of order is a certificate that the treewidth is at least .
We next define the family of graphs as follows. The vertex set of is a disjoint union of cliques, , each containing vertices. For each , the vertices of are labeled as , . Besides the edges lying inside the cliques ’s, we add an edge between and for every . Notice that the vertex does not have a neighbor outside . The graph is depicted in Fig. 3.
The tree-cut width of is at most .
Consider the tree-cut decomposition , in which is a star with as the center and as leaves. For the bags, we set , and for . It is straightforward to verify that the tree-cut width of is .
For any positive integer , the treewidth of is at least .
For notational convenience, we assume that is even. The argument can be easily extended to the case when is odd. For and a set , let denote the set . We define as
It is easy to verify that each subgraph of is connected. For any and such that , the number of cliques , , with which has non-empty intersection is at least . This means any two elements of touch each other, and thus is indeed a bramble. Henceforth, we show that the order of is at least .
Suppose that there is a hitting set of with . We define
Proof of the Claim: Suppose that the contrary holds. We use a counting argument to derive a contradiction. The set is partitioned into two sets: and . We have
contradicting to the assumption that .
There exists such that .
Proof of the Claim: Suppose the contrary, i.e. we have for every . Notice that the set is partitioned into and . Then,
where the last inequality follows from Claim 1. This contradicts the assumption that .
Consider some satisfying the condition of Claim 2. We observe that the set
contains at least vertices by the definition of and Claim 2. Pick any subset of of size exactly . To reach a contradiction, it suffices to show that . Indeed, from the fact that , it follows that
By the definition of it follows that which, implies that
For every there exists a graph such that .
3 The 2-approximation algorithm
We present a 2-approximation of Tree-cut Width running in time . As stated in Lemma 3 below, we first observe that computing the tree-cut width of reduces to computing the tree-cut width of 3-edge-connected graphs. This property can be easily derived from [23, Lemmas 10–11].
Given a connected graph , let be a partition of such that is a minimal cut of size at most two and let be a positive integer. For , let be the graph obtained from by identifying the vertex set into a single vertex . Then has tree-cut width at most if and only if both and have tree-cut width at most .
Recall that if admits an immersion of by [23, Lemma 11]. Hence, in order to prove the forward implication, it suffices to prove that is an immersion of , for . If , for each we can delete all vertices of except for the single vertex in and obtained . If , note that for each , is connected and thus can be obtained by deleting vertices, edges and lifting a sequence of edges along the path between two vertices in .
Conversely, let be a tree-cut decomposition of of width at most for , and consider the tree-cut decomposition such that and is obtained by the disjoint union of and after adding an edge between and , where is the tree node of containing , i.e. the vertex obtained by contracting . We remove and from the bags of .
We claim that is a tree-cut decomposition of width at most . Note first that the adhesion of is at most since and the adhesion of is at most for . From , it follows that for , the 3-center of of the tree-decomposition is the same as the 3-center of of the tree-decomposition . Therefore the width of is at most .
The proof of the next lemma is easy and is omitted.
Let be a graph and let be a vertex of with degree 1 (resp. 2). Let also be the graph obtained from after removing (resp. dissolving) . Then .
From now on, based on Lemmata 3 and 4, we assume that the input graph is -edge-connected. In this special case, the following observation is not difficult to verify. It allows us to work with a slightly simplified definition of the -centers in a tree-cut decomposition.
Let be a 3-edge-connected graph and let be a tree-cut decomposition of . Consider an arbitrary node of and let be the set containing every connected component of such that Then that is .
We observe that the proof of Lemma 3 provides a way to construct a desired tree-cut decomposition for from decompositions of smaller graphs. Given an input graph for Tree-cut Width, we find a minimal cut with and create a graph as in Lemma 3, with the vertex marked as distinguished. We recursively find such a minimal cut in the smaller graphs created until either one becomes 3-edge-connected or has at most vertices.
Therefore, a key feature of an algorithm for Tree-cut Width lies in how to handle 3-edge-connected graphs. Our algorithm iteratively refines a tree-cut decomposition of the input graph and either guarantees that the following invariant is satisfied or returns that .
Invariant: is a tree-cut decomposition of where .
Clearly the trivial tree-cut decomposition satisfies the Invariant. A leaf of such that is called a large leaf. At each step, the algorithm picks a large leaf and refines the current tree-cut decomposition by breaking this leaf bag into smaller pieces. The process repeats until we finally obtain a tree-cut decomposition of width at most , or encounter a certificate that .
3.1 Refining a large leaf of a tree-cut decomposition
A large leaf will be further decomposed into a star. To that aim, we will solve the following problem:
Constrained Star-Cut Decomposition
Input: An undirected graph , an integer , a set , and a weight function .
Output: A non-trivial tree-cut decomposition of such that
is a star with central node and with leaves for some ,
and for every leaf node , ,
or report that such a tree-cut decomposition does not exist.
Observe that a Yes-instance satisfies, for every , . We also notice that as the output of the algorithm is a non-trivial tree-cut decomposition, contains at least two nodes with non-empty bags and every leaf node is non-empty.
Given a subset , we define the instance of the Constrained Star-Cut Decomposition problem where for every , .
Let be a 3-edge-connected graph, , and let be a set of vertices such that and . If , then is a Yes-instance of Constrained Star-Cut Decomposition.
Let be a normalized tree-cut decomposition of of width at most . We extend the weight function on into on by setting for every and otherwise. Also, given a subtree of , we let . The idea is to identify a node of that will serve as the central node of the star decomposition. The leaves of the star decomposition will results from the contraction of the subtrees of containing bags that intersect the set . To find the node , we orient the edges of using the following two rules. Given an edge :
Rule 1: orient from to if .
Rule 2: orient from to if .
Let be the resulting orientation of . Observe that Rule 1 and 2 may leave some edges of non-oriented.
For every edge of , is oriented either in a single direction or not oriented in .
Proof of the Claim: Observe that if Rule 1 orients from to , neither Rule 1 nor Rule 2 may orient in the opposite direction. The former is an immediate consequence of the fact . Rule 2 does not orient from to either: if Rule 2 does so, we have and since the value is non-zero only when , we conclude that , a contradiction to the assumption that Rule 1 oriented from to . Moreover, the edge cannot be oriented in both directions by Rule 2 since is non-empty and thus at least one of the sets and intersects with .
By Claim 3, contains at least one node, say , which is not incident to an out-going edge in . Let be the connected components of containing a node such that . Observe that as and , cannot be included in a single bag of and thereby . Consider the following tree-cut decomposition of :
is a star with central node and leaf nodes ,
the bag of node is ,
for every leaf node , we set .
Observe that is a tree-cut decomposition of and since , it is non-trivial. By construction, as it is obtained from by contracting subtrees and removing vertices from bags, we have that . It remains to prove that and that for every leaf node . The former inequality directly follows from Observation 1 and the fact that is an optimal tree-cut decomposition of . The latter inequality follows from the fact that does not have an out-going edge in , in particular, Rule 1 does not orient any edge incident with outwardly from .
Given a -edge-connected graph, applying Lemma 5 on a large leaf of a tree-cut decomposition that satisfies the Invariant, we obtain:
Let be a 3-edge-connected graph such that , and let be a large leaf of a tree-cut decomposition satisfying the Invariant. Then is a Yes-instance of Constrained Star-Cut Decomposition.
The next lemma shows that if a large leaf bag of a tree-cut decomposition satisfying the Invariant defines a Yes-instance of the Constraint Tree-Cut Decomposition problem, then can be further refined.
Let be a 3-edge-connected graph and be tree-cut decomposition of satisfying the Invariant. If is a solution of Constrained Star-Cut Decomposition on the instance where is a large leaf of , then the pair where
, where is the unique neighbor of in and is the central node of ,
is a tree-cut decomposition of satisfying the Invariant. Moreover the number of non-empty bags is strictly larger in than in .
By construction, is a tree-cut decomposition of . The fact that is non-trivial implies that the number of non-empty bags is strictly larger in than in .
It remains to prove that . Since is a solution to , we have . As is edge -connected, by Observation 1, the torso size at in at most , which is at most . Let us verify that the adhesion of is at most . For this, it suffices to bound the value for the newly created edges , for all . We have
The inequality follows from that is a solution to . More precisely, is implied by the fact that . And is a consequence of .
Finally, as is a non-trivial tree-cut decomposition, the number of non-trivial nodes is strictly larger in than in .
3.2 An FPT algorithm for Constrained Star-Cut Decomposition
Lemma 1 provides a quadratic bound on the treewidth of a graph in term of its tree-cut width. This allows us to develop a dynamic programming algorithm for solving Constrained Star-Cut Decomposition on graphs of bounded treewidth. To obtain a tree-decomposition, we use the 5-approximation FPT-algorithm of the following proposition.
Proposition 2 (see )
There exists an algorithm which, given a graph and an integer , either correctly decides that or outputs a tree-decomposition of width at most in time .
A rooted tree decomposition is a tree decomposition with a distinguished node selected as the root. A nice tree decomposition (see ) is a rooted tree decomposition where is binary, the bag at the root is , and for each node with two children it holds , and for each node with one child it holds or for some . Notice that a nice tree decomposition is always a rooted tree decomposition. We need the following proposition.
Proposition 3 (see )
For any constant , given a tree decomposition of a graph of width and nodes, there exists an algorithm that, in time, constructs a nice tree decomposition of of width and with at most nodes.
Let be an input of Constrained Star-Cut Decomposition and let . There exists an algorithm that given outputs, if one exists, a solution of in steps.
Sketch of proof: From Proposition 3, we can assume that we are given a nice tree-decomposition of of width at most , which can be obtained in time because of Proposition 2. We describe dynamic programming tables. Let be the vertex set , where is the subtree of rooted at . For every , we need to compute a collection of pairwise disjoint subsets of (some of them may be empty sets) such that . The subset will play the role of the bag of the central node of the star-cut decomposition.
To guarantee that the specification of the problem can be checked, the dynamic programming table at node will store a collection of quadruples with the following specifications:
, indicating that belongs to ;
is a number indicating the size of the central bag in ;
is a function, indicating the weight ;
is a function, indicating ;
Suppose that we have constructed tables for all nodes of such that: for every node , a quadruple appears in the table at node if and only if there exists a collection meeting the specifications. It is not difficult to see that the instance is Yes if and only if the table at the root contains a quadruple such that . Furthermore, such tables can be constructed using standard dynamic programming in a bottom-up manner.
Observe that the size of the dynamic table at each node is dominated by the number of collections of pairwise disjoint subsets of , with , which is . Maintaining these tables follows by a standard dynamic programming algorithm.
3.3 Piecing everything together
We now present a 2-approximation algorithm for Tree-cut Width leading to the following result.
There exists an algorithm that, given a graph and a , either outputs a tree-cut decomposition of with width at most or correctly reports that no tree-cut decomposition of <