Approximating the Minimum k-Section Width in Bounded-Degree Trees with Linear Diameter

08/21/2017 ∙ by Cristina G. Fernandes, et al. ∙ Universidade de São Paulo 0

Minimum k-Section denotes the NP-hard problem to partition the vertex set of a graph into k sets of sizes as equal as possible while minimizing the cut width, which is the number of edges between these sets. When k is an input parameter and n denotes the number of vertices, it is NP-hard to approximate the width of a minimum k-section within a factor of n^c for any c<1, even when restricted to trees with constant diameter. Here, we show that every tree T allows a k-section of width at most (k-1) (2 + 16n / diam(T) ) Δ(T). This implies a polynomial-time constant-factor approximation for the Minimum k-Section Problem when restricted to trees with linear diameter and constant maximum degree. Moreover, we extend our results from trees to arbitrary graphs with a given tree decomposition.

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1. Introduction

1.1. The Minimum -Section Problem

We start with a few definitions. A -section in a graph  is a partition  of its vertex set into  sets whose sizes are as close to equal as possible. The width of a -section  is the number of edges between the sets  and is denoted by . The minimum width among all -sections in a graph  is denoted by . The goal of the Minimum -Section Problem is to compute  and a corresponding -section for a given graph  and an integer . This problem has many applications, e.g. in parallel computing, when tasks have to be evenly distributed to processors while minimizing the communication cost.

The special case where , which is also called the Minimum Bisection Problem, is known to be NP-hard for general graphs, see [9]. Jansen et al. [11] use dynamic programming to compute a minimum bisection in a tree on  vertices in  time. Their method can be turned into an algorithm for computing a minimum -section in trees, whose running time is polynomial in  but not in . However, when  is part of the input,  cannot be approximated in polynomial time within any finite factor on general graphs  unless PNP, see [1]. The reduction presented there is from the strongly NP-hard 3-Partition Problem and it is easy to adjust it in order to show that the Minimum -Section Problem restricted to forests cannot be approximated within any finite factor unless PNP.

Feldmann and Foschini [5] studied minimum -sections in trees, and pointed out some counterintuitive behavior even on this rather restricted class of graphs. Moreover, they showed in [5] that the Minimum -Section Problem remains APX-hard when restricted to trees with maximum degree at most 7, and that, for any , it is NP-hard to approximate  within a factor of  for trees with constant diameter.

1.2. Results

Here, we study the Minimum -Section Problem in trees and focus on bounded-degree trees with linear diameter. Moreover, we extend our results to tree-like graphs. Our first result gives an upper bound on the width of a minimum -section in trees and a corresponding algorithm.

Theorem 1.

For every integer  and for every tree  on  vertices, a -section  in  with

can be computed in  time.

Here, as usual,  and  denote the maximum degree of  and the diameter of , respectively, where the latter is defined as the length of a longest path in . Obviously, for , any graph on  vertices has essentially only one -section, so we can assume without loss of generality that and, hence, the running time in Theorem 1 is always polynomial in the input length.

Let  and  be two constants. Then, for any tree  on  vertices with  and , the factor from the previous theorem is bounded by a constant that depends only on  and . As every -section of a tree has width at least , this yields a constant-factor approximation for  for such a class of trees.

Corollary 2.

For all  and , there is a constant  such that the following holds. Let  be a class of trees such that every tree  on  vertices satisfies  and . Then there is a -approximation for the Minimum -Section Problem restricted to the class . In particular, one can choose .

Next, we extend our focus in this paper in two ways. On the one hand, we can improve the upper bound so that it becomes polylogarithmic in . Moreover, we now move from trees to graphs with a given tree decomposition. Here, instead of bounding the width of a -section in terms of the diameter, we define a parameter  that roughly measures how close the tree decomposition  is to a path decomposition, which is defined as a tree decomposition where the decomposition tree is a path. For example, suppose that we were given a graph  with a path decomposition  of  of width . Then it is easy to see that  allows a bisection of width at most  by walking along the path  until we have seen  vertices of  in the clusters, and then bisecting  there. In other words, finding a -section of the graph  of small width becomes easier when there is a path in the tree decomposition of  whose clusters contain many of the vertices of .

For the precise definition of the parameter , consider a tree decomposition  of a graph  with . The relative weight of a heaviest path in  is denoted by

where  denotes the number of vertices of . Observe that . Moreover, define the size of  as , which roughly measures the encoding length of .

Theorem 3.

For every integer , for every graph  and every tree decomposition  of  of width at most , a -section  in  with

can be computed in  time, when the tree decomposition  is provided as input.

Again, as in the case of trees in Theorem 1 and Corollary 2, a constant-factor approximation for a certain class of tree-like graphs is obtained. More precisely, fix  and define . Consider a class  of connected graphs such that every graph  satisfies  and allows a tree decomposition  of width at most  and . Then, there is a -approximation for the Minimum -Section Problem restricted to the graph class .

1.3. Related Work

The results presented here rely on our earlier work concerning the Minimum Bisection Problem, in particular the following theorem.

Theorem 4 (Theorem 1 in [8]).

For every tree  a bisection  in  with

can be computed in  time.

Although Theorem 1 and Theorem 4 look quite similar, it does not seem possible to directly apply Theorem 4 to yield a recursive construction of a -section that satisfies the bound presented in Theorem 1. Indeed, it is known that, even when  is a power of , the natural approach to construct a -section in a graph by recursively constructing bisections can give solutions far from the optimum, even when a minimum bisection is used in each step, see [15]. Furthermore, in the setting that we are considering here, i.e., bounded-degree trees with linear diameter, nothing is known about the diameter in the two subgraphs that are produced by Theorem 4. So, after the first iteration, the diameter of one part of the bisection could be as low as , and indeed such parts can be produced by the algorithm contained in Theorem 4. For example, consider the tree  obtained from a perfect ternary tree  on  vertices and a path  on  vertices by inserting an edge joining the root  of  and a leaf of . The algorithm111To give some more details, this algorithm computes a longest path  in , which must contain all vertices from  as well as . Then, it computes a -labeling, as introduced in Section 3.2 ahead, which can label the vertices of  with  and the vertices in  with . Then, the algorithm checks whether there is a vertex  such that  where we identified the vertices with their labels. This is the case for  and, hence, the bisection  with  and  is output. For further details see Section 6.1 in [14]. contained in Theorem 4 can output the bisection  with  and , which is in fact the unique minimum bisection in . In the next round, a bisection in  is needed, which has width . Thus, a -section of width  is obtained, whereas creftype 1 promises that  allows a -section of constant width.

Observe that, when only the universal bound  is available for the diameter of a bounded-degree tree on  vertices, then creftype 4 yields a bound of  for the width of a bisection.

1.4. Further Remarks

The results presented in creftype 1 and creftype 3 do not only hold for -sections but also for cuts , i.e., partitions of the vertex set of the considered graph, where the sizes of the sets are specified as input. Furthermore, the bound in creftype 1 can be improved to

(1)

Observe that this is slightly stronger than the bound implied by creftype 3 and the fact that every tree  allows a tree decomposition  of width at most one with  and . Moreover, extensions to -sections in trees with weighted vertices have recently been investigated, see [10].

1.5. Organization of the Paper

Section 2 introduces some notation for cuts in general graphs as well as some tools for trees, which will then be used in Section 3 to show creftype 1. Moreover, in Section 3.4, it is argued that the bound on the width of the -section in creftype 1 can be improved as claimed in (1). Section 4 concerns tree-like graphs. First, in Section 4.1, the notation for tree decompositions is settled and selected tools for tree-like graphs are presented. Second, the proof for creftype 3 is given in Section 4.2-4.4. Since the proofs in Section 3 and Section 4 follow the same ideas, we do not repeat the full details for the case of tree-like graphs in Section 4 but focus on the aspects that become more involved when dealing with a tree decomposition and refer to Section 3 whenever possible.

2. Preliminaries

First, for , define . Moreover, for a real  denote by  the largest integer  and denote by  the smallest integer . Consider an arbitrary graph  on  vertices. For a set , we use  to denote the subgraph of  induced by  and for  we define . A cut in  is a partition  of , where  is arbitrary and empty sets are allowed. An edge  of  is cut by a cut  if there are distinct  with  and . The width of a cut  in  is defined as the number of edges of  that are cut by  and is denoted as . For , a -section  in  is a cut  in  with  for all . A -section  in  that satisfies  for all -sections  in  is called a minimum -section in  and its width is denoted by .

Recall that the diameter of a tree  is the length of a longest path  in , i.e., . In the following, we need to compare the diameter of trees with different numbers of vertices and during our construction also non-connected forests may arise. Consider a forest  on  vertices and denote by  the connected components of . Then, the relative diameter of  is defined as

The term  denotes the number of vertices on a longest path in the component  and for a tree  the relative diameter equals the proportion of vertices on a longest path in . Using this notation, we can state the version of creftype 4 which we will employ in Section 3. It follows from Theorem 1 in [8] and the comments in Section 1.4 there. When considering a cut in a graph  with exactly two sets, we use  and  for these sets and refer to them as the black and the white set of the cut.

Theorem 5 (similar to Theorem 1 in [8]).

For every forest  on  vertices, for every , a cut  with  and

can be computed in  time.

The previous theorem allows to cut off an arbitrary number of vertices in a bounded-degree tree and guarantees a small cut width if the diameter is large. The next tool relaxes the size-constraint on the set . Let  be a graph. For an integer , a cut  in  is called an approximate -cut if . The next lemma states that every bounded-degree tree allows an approximate cut of small width, even when the diameter is small.

Lemma 6 (approximate cut in forests).

Let  be a tree on  vertices and fix a vertex . For every integer , an approximate -cut  with  and  can be computed in  time.

The previous lemma is similar to Lemma 7 in [8] but as the bound on  claimed here is smaller we present a sketch of its proof.

Sketch of Proof for Lemma 6..

Let , , , and  be as in the statement. If , define . Otherwise, root  in  and, for , denote by  the set of descendants of . It is easy to check that there is a vertex  such that  and  for all children  of . If there is a child  of  with , define . Otherwise, the set  can be constructed greedily from the sets  where  is a child of . Then, the cut  with  has the desired properties and can easily be computed in linear time. ∎

3. Minimum -Section in Trees

The aim of this section is to prove creftype 1 about -sections in trees. Section 3.1 introduces the main lemma that immediately implies the existence part of the desired theorem and Section 3.2 presents the proof of the main lemma. All algorithmic aspects of creftype 1 are presented in Section 3.3. Finally, Section 3.4 argues that the bound on the width of the -section in creftype 1 can be improved as stated in (1).

3.1. Proof of creftype 1

The aim of this section is to prove our main result for trees. As mentioned in Section 1.3, constructing a -section by bisecting the graph repeatedly does not yield the bound provided by creftype 1. So we follow a different approach: The main idea is to cut off one set for the -section at a time while ensuring that the relative diameter of the remaining forest does not decrease. This is made precise by the next lemma, which looks similar to creftype 5 but is more powerful, as it contains additional information on the relative diameter of the subgraph induced by the white set of the cut.

Lemma 7.

For every forest  on  vertices and for every , there is a cut  in  with , that satisfies  and

Observing that  now yields the existence part of creftype 1.

3.2. Proof of Lemma 7

Consider a forest  on  vertices and fix an integer . The main idea is to apply creftype 5 to a carefully chosen subgraph , which then yields the set  for the desired cut  in . On the one hand, needs to have large relative diameter such that the bound on the cut width provided by creftype 5 is low when applied to . On the other hand, the relative diameter of the graph induced by the white set of the computed cut will roughly be the relative diameter of , so needs to have a large relative diameter. Note that these two conditions compete against each other.

If , then  and a cut with the desired properties is easy to construct. So assume that . Without loss of generality, we may assume that  is connected as otherwise edges can be added to  to obtain a tree whose relative diameter is equal to  and maximum degree . Set , let  be a longest path in  and observe that . Denote by  and  the two leaves of . When removing all edges in  from , then  decomposes into trees, one tree  for every . For each , let . For , the unique vertex  with  is called the path-vertex of . When labeling the vertices of a graph with , we say that the vertices in  receive consecutive labels if there are  such that each vertex in  receives one label in  and every label in  belongs to exactly one vertex in . A -labeling of  is a labeling of the vertices of  with  such that the following holds:

  • For each , the vertices of  receive consecutive labels and  has the largest label among all vertices in .

  • For all with , if  is closer to  than to , then the label of  is smaller than the label of .

Identify each vertex with its label and consider any number that differs by a multiple of  from a label in  to be the same as this label. When talking about labels and vertices, in particular when comparing them, we always refer to the integer in . For three vertices with , we say that  is between  and  if , , or if starting at  and going along the numeration given by the labeling reaches  before . If , then we say that  is between  and  if . For example, when , we say that  is between  and , and  is between  and . For technical reasons, we will refer to the pair  as an edge of , even though  does not contain such an edge. For a vertex , the vertex  is the vertex after  on  if the tree  contains the vertex . Then, is called the edge after  on . Similarly, in this case,  is called the edge before  on  and  is called the vertex before  on .

For two vertices , the -distance of  and  is defined as

It is easy to see that

(2)

Recall that  was defined to be an end of . Now, define  for all . Then, and

Thus, there are two vertices  with

The fact that  is an integer for all , and (2) implies that there is a vertex  with . Let  and  be the path-vertices of  and , respectively. Define . Set  and note that  or . Furthermore, as  is not counted in , we have . Define

In the following figures, the path  will be drawn in the top and the trees  for  are drawn underneath . The vertices in  that are counted in  will be colored gray.

a) Case 1, where  and .
b) Case 2a, where  and .
Figure 1. Construction of the black set in Case 1 and Case 2a.

Case 1: and .
Define  and . Then , see Figure 0a). Furthermore, and

Case 2: and .
Let  be the path-vertex of . Observe that  as otherwise  and , which contradicts . None of the edges in  is cut by  when , i.e., when  is the vertex before  on , and this case is treated separately for technical reasons.

Case 2a: .
Analogously to Case 1, the cut  with  and  satisfies all requirements, see Figure 0b).

Figure 2. Construction of  in Case 2b, where , , and . Note that  can also lie in .

Case 2b: .
First, observe that the cut  might cut too many edges in the tree . Moreover, the cut cuts few edges, but using creftype 5 to cut off  vertices from  might yield a too large bound. The reason for this is that the relative diameter of  can be much less than , for example when  contains  vertices and  is small. So, instead of using , we will now define a set  such that  cuts few edges, contains all vertices counted by , and , which will ensure that .

Let , which satisfies as . Lemma 6 guarantees an approximate -cut  in  with  and . Define and note that . Furthermore, as and , we have that . The graph  consists of at least two components as  and there are no edges between  and , see Figure 2. Therefore,

Now, creftype 5 guarantees a cut  in  with  and . Let  and . Every vertex from  that is not counted by  is in  by construction. Consequently,

which implies that .

To estimate the width of the cut 

, consider first the cut . The cut  cuts at most  edges within . Recall that . Now, if  is the vertex before  on , then and other than the edges in  only edges incident to  are cut by . Otherwise, at most  edges incident to  are cut by  as either  or the edge after  on  exists and is not cut. Then, from the edges incident to  that are cut by , only the edge before  on  is not yet counted. Consequently, . Using that  cuts at most  edges in  gives the desired bound on the number of cut edges.

Case 3: and .
This case is similar to Case 2b, but some arguments need to be adjusted as the labeling cannot simply be reversed to obtain a labeling with the same properties due to the requirement that each  receives the largest label among all vertices in . Denote by  the path-vertex of . As in Case 2b, let  and let  be an approximate -cut in  with  and . For technical reasons, the case when  is treated separately.

Case 3a: .
Then,  and . Define  and , which satisfy  and , see Figure 2a). Similarly to Case 2b, we can find the desired cut  with  by applying creftype 5 to .

a) Case 3a, where .
b) Case 3b, where .
Figure 3. Proof of Lemma 7, construction of  in Case 3, where  and . Note that, in both cases,  can also lie in .

Case 3b: .
Define

see Figure 2b). This definition of  is slightly different than in Case 2b, as here  but  is in  instead of , which will decrease the bound on the number of cut edges. Now, contains exactly  vertices of . One can argue similar to Case 2b that  as well as that  satisfies . Then, creftype 5 guarantees that there is a cut  in  with  and . Define  and . As mentioned before,  vertices of  are in . Therefore, at most  vertices of  are in  and, as in Case 2b, it follows that . Since  and , it follows that . Hence, the desired bound on  is obtained. This completes the proof of Lemma 7.

3.3. Algorithm for Trees

To achieve the running time in creftype 1, it suffices to argue that a cut with the properties claimed by Lemma 7 can be computed in linear time. More precisely, consider a forest  on  vertices and fix an integer . The aim is to compute a cut  in  in  time with  that satisfies  and . The algorithm described here follows the construction presented in Section 3.2, which works for trees with maximum degree at least . So assume for now that  has these properties.

First, the algorithm computes a longest path  in , which takes  time222The following well-known procedure due to Dijkstra computes a longest path in a tree . Root  in an arbitrary vertex  and compute a leaf  at maximum distance from . Root  in  and compute a leaf  at maximum distance from . Then, the unique -path in  is a longest path in .. Recall that the ends  and  of  were called  and  in Section 3.2. To compute a -labeling, the algorithm rearranges the adjacency list of  such that  is the first entry in the adjacency list of  for all . Then, the algorithm traverses the tree  with a depth-first search starting in  and labels each vertex when it turns black (i.e., when the processing of the vertex finishes, see the notation in [4]). It is easy to see that the computation of the -labeling takes time proportional to  and that it can be stored such that converting between vertices and labels and vice versa takes constant time. In the implementation, we do not change the vertex names or identify vertices with their labels as in Section 3.2. Furthermore, while computing the -labeling, the algorithm computes  for each vertex , i.e., the number of vertices of  that are already labeled right before  is labeled. Now, for all  where  and a vertex  with  and  or  can be computed in  time. Using that the algorithms contained in creftype 5 and Lemma 6 take linear time, it follows that a cut with the desired properties can be constructed in  time. For more details see Chapter 6.2 in [14].

To conclude, consider the case when  is not a tree with . Clearly, if , a cut in  with the desired properties can be computed in  time. If  is not connected and , the algorithm adds edges to  until a tree  with  and  is obtained. More precisely, these additional edges need to join the ends of two longest paths in different components of . As each connected component  of  is a tree, a longest path in  can be computed in  time as mentioned above. Since  holds for every cut  in , the procedure described above can be applied to  to obtain the desired cut in .

3.4. Remarks on Improving creftype 1

To improve the bound on the cut width in creftype 1 as stated in (1), recall that the proof of Lemma 7 uses creftype 5 to estimate the width of the cut in . The bound on the cut width in creftype 5 can be improved to

see Section 1.4 in [8] or Theorem 5.12 in [14]. Using this, the bound on the width of the cut in Lemma 7 improves to

and the improvement on the bound on the width of the -section in creftype 1 as stated in (1) follows.

Last but not least for -sections in trees, we mention that it does not seem to be obvious whether our algorithm or its analysis can be modified to obtain a linear time algorithm. More precisely, it is not clear how to reduce the dependency on in the running time.

4. Minimum -Section in Tree-Like Graphs

This section concerns the proof of creftype 3 about -sections in tree-like graphs. We begin with presenting the definition and some facts about tree decompositions as well as some tools for tree-like graphs in Section 4.1. As in the proof of creftype 1, Section 4.2 presents a lemma that immediately implies the existence part of creftype 3 and is proved in Section 4.3. The main idea is similar to the proof in Section 3.2. Hence, Section 4.3 focuses on the aspects that are more involved than in the case of trees and refers to Section 3.2 for steps that are analogous to the case of trees. All algorithmic aspects of creftype 3 are presented in Section 4.2. A detailed proof of creftype 3 can be found in Chapter 6.3 in [14].

4.1. Preliminaries for Tree Decompositions

Let us start by recalling the definition of a tree decomposition.

Definition 8.

Let  be a graph, be a tree, and  with  for each . The pair  is a tree decomposition of  if the following three properties hold.

  1. For every , there is an  such that .

  2. For every , there is an  such that .

  3. For all  and all  on the (unique) -path in , we have .

The width of  is defined as . The tree-width of , denoted by , is the smallest integer  such that  allows a tree decomposition of width .

Consider a graph  and a tree decomposition  with  of . To distinguish the vertices of  from the vertices of  more easily, we refer to the vertices of  as nodes. Furthermore, for , we refer to the set  as the cluster of  that corresponds to , or simply the cluster of  when the tree decomposition is clear from the context. It is easy to show that (T3) is equivalent to the following condition.

  1. For every , the graph  is connected, where .

Consider a graph  and a tree decomposition  with . In order to apply a procedure to a subgraph , it is often necessary to construct a tree decomposition of . For this purpose, the tree decomposition  induced by  in  is defined by  and  for all , where . Observe that  is indeed a tree decomposition of  as well as that the width and the size of  are at most the width and the size of , respectively. Usually, some clusters of an induced tree decomposition are empty, which can be avoided with the following concept. A tree decomposition  of a graph  with