Parameterized complexity of length-bounded cuts and multi-cuts Research was supported by the project SVV-2014-260103.

Parameterized complexity of length-bounded cuts and multi-cuts thanks: Research was supported by the project SVV-2014-260103.

Pavel Dvořák and Dušan Knop Author supported by the project Kontakt LH12095, project GAUK 1784214 and project CE-ITI P202/12/G061 Department of Applied Mathematics, Charles University, Prague {koblich,knop}@kam.mff.cuni.cz
Abstract

We show that the Minimal Length-Bounded -But problem can be computed in linear time with respect to and the tree-width of the input graph as parameters. In this problem the task is to find a set of edges of a graph such that after removal of this set, the shortest path between two prescribed vertices is at least long. We derive an algorithm for a more general multi-commodity length bounded cut problem when parameterized by the number of terminals also.

For the former problem we show a -hardness result when the parameterization is done by the path-width only (instead of the tree-width) and that this problem does not admit polynomial kernel when parameterized by tree-width and We also derive an algorithm for the Minimal Length-Bounded Cut problem when parameterized by the tree-depth. Thus showing an interesting paradigm for this problem and parameters tree-depth and path-width.

Keywords:
l

ength bounded cuts, parameterized algorithms, -hardness

1 Introduction

The study of network flows and cuts begun in 1960s by the work of Ford and Fulkerson [10]. It has many generalizations and applications now. We are interested in a generalization of cuts related to the flows using only short paths.

Length bounded cuts

Let be two distinct vertices of a graph – we call them source and sink, respectively. We call a subset of edges of an -bounded cut (or -cut for short), if the length of the shortest path between and in the graph is at least . We measure the length of the path by the number of its edges. In particular, we do not require and to be in distinct connected components as in the standard cut, instead we do not allow and to be close to each other. We call the set a minimum -cut if it has the minimum size among all -bounded cuts of the graph .

We state the cut problem formally:

PROBLEM: Minimum Length Bounded Cut (MLBC)
Instance: graph , vertices and integer
Goal: find a minimum -bounded cut

Length bounded flows were first considered by Adámek and Koubek [1]. They showed that the max-flow min-cut duality cannot hold and also that integral capacities do not imply integral flow. Finding a minimum length bounded cut is -hard on general graphs for as was shown by Itai et al. [14]. They also found algorithms for finding -bounded cut with in polynomial time by reducing it to the usual network cut in an altered graph. The algorithm of Itai et al. [14] uses the fact that paths of length and are edge disjoint from longer paths, while this does not hold for length at least .

Baier et al. [2] studied linear programming relaxation and approximation of MLBC together with inapproximability results for length bounded cuts. They also showed instances of the MLBC having integrality gap for their linear programming approach, which are series-parallel graphs and thus have constant bounded tree-width. The first parametrized complexity study of this and similar topics was made by Golovach and Thilikos [11] who studied parametrization by paths-length (that is in our setting the parameter ) and the size of the solution for cuts. They also proved hardness results – finding disjoint paths in graphs of bounded tree-width is a -hard problem.

The MLBC problem has its applications in the network design and in the telecommunications. Huygens et al. [13] use a MLBC as a subroutine in the design of -edge-connected networks with cycles at most long. The MLBC problem is called hop constrained in the telecommunications and the number is so called number of hops. The main interest is in the constant number of hops, see for example the article of Dahl and Gouveia [5].

Note that the standard use of the Courcelle theorem [3] gives for each fixed a linear time algorithm for the decision version of the problem. But there is no apparent way of changing these algorithms into a single linear time algorithm. Moreover there is a nontrivial dependency between the formula (and thus the parameter ) and the running time of the algorithm given by Courcelle theorem.

Now we give a formal definition of a rather new graph parameter, for which we give one of our results:

Definition 1 (Tree depth [16])

The closure of a forest is the graph obtained from by making every vertex adjacent to all of its ancestors. The tree-depth of a graph is one more than the minimum height of a rooted forest such that

Our Contribution

Our main contribution is an algorithm for the MLBC problem, its consequences and an algorithm for a more general multi-terminal version problem.

Theorem 1.1

Let be a graph of tree-width . Let and be two distinct vertices of . Then for any an minimum -cut between and can be found in time .

Corollary 1

Let be a graph, and and be two distinct vertices of . Then for any an minimum -cut between and can be found in time , where is a computable function.

Proof

As is the tree depth of it follows that the length of any path in can be upper-bounded by . It is a folklore fact, that is also a upper-bound on the tree-width of . So we can use the Theorem 1.1 for and any other polynomial algorithm for minimum cut problem otherwise.

Corollary 2

Let be a graph of tree-width , and . There exists a computable function , such that a minimum -cut between and can be found in time , where .

Theorem 1.2

Minimal Length Bounded Cut parametrized by path-width is -hard.

Tree-width versus tree-depth

Admitting an algorithm for a problem when parameterized by the tree-width implies an algorithm for the problem when parameterized by the tree-depth, as parameter-theoretic observation easily shows. On the other hand, the algorithm parameterized by the tree-width usually uses exponential (in the tree-width) space, while the tree-depth version uses only polynomial space (in the tree-depth).

From this point of view, it seems to be interesting to find problems that are ”on the edge between path-width and tree-depth”. That is problems that admit an algorithm when parameterized by the tree-depth, but being [1]-hard when parameterized by the path-width.

The only other result of this type, we a re aware of in the time of writing this article is by Gutin et al. [12]. The Minimum Length Bounded Cut problem is also a problem of this kind—as Theorems 1.2 and 1 demonstrate.

Theorem 1.1 gives us that the MLBC problem is fixed parameter tractable () when parametrized by the length of paths and the tree-width and that it belongs to when parametrized by the tree-width only (and thus solvable in polynomial time for graph classes with constant bounded tree-width).

Theorem 1.3

There is no polynomial kernel for the Minimum Length Bounded Cut problem parameterized by the tree-width of the graph and the length unless

We want to mention that our techniques apply also for more general version of the MLBC problem.

Length-bounded multi-cut

We consider a generalized problem, where instead of only two terminals, we are given a set of terminals. For every pair of terminals, we are given a constraint—a lower bound on the length of the shortest path between these terminals. More formally:

Let be a subset of vertices of the graph and let be a mapping. We call a subset of edges of an -bounded multi-cut if length of the shortest path between and in the graph is at least long for every . Again if has smallest possible size, we call it minimum -bounded -multi-cut. We call the vertices terminals. Finally, as there are only finitely many values of the mapping we write instead of , we also write instead of function . Let , we say that the problem is -limited.

PROBLEM: Minimum Length Bounded Multi-Cut (MLBMC)
Instance: graph , set and for all satisfying the triangle inequalities
Goal: find a minimum length bounded multi-cut
Theorem 1.4

Let be a graph of tree-width , with and let . Then for any and any -limited length-constraints on an minimum -bounded multi-cut can be computed in time .

2 Preliminaries

In this section we recall some standard definitions from the graph theory and state what a tree decomposition is. After this we introduce changes of the tree decomposition specific for our algorithm. We proceed by the notion of auxiliary graphs used in proofs of our algorithm correctness. Finally, in Section 2.1 we summarize the results allowing us to prove that it is unlikely for a parameterized problem to admit a polynomial kernelization procedure.

We use the notion of tree decomposition of the graph:

Definition 2

A tree decomposition of a graph is a pair where is a rooted tree and is a family of subsets of such that

  1. for each there exists an such that ,

  2. for each there exists an such that ,

  3. for each induces a subtree of

We call the elements the nodes, and the elements of the set the decomposition edges.

We define a width of a tree decomposition as and the tree-width of a graph as the minimum width of a tree decomposition of the graph . Moreover, if the decomposition is a path we speak about the path-width of , which we denote as .

Nice tree decomposition

[15]  For algorithmic purposes it is common to define a nice tree decomposition of the graph. We naturally orient the decomposition edges towards the root and for an oriented decomposition edge from to we call the parent of and a child of . If there is an oriented path from to we say that is a descendant of .

We also adjust a tree decomposition such that for each decomposition edge it holds that (i.e. it joins nodes that differ in at most one vertex). The in-degree of each node is at most and if the in-degree of the node is then for its children holds that (i.e. they represent the same vertex set).

We classify the nodes of a nice decomposition into four classes—namely introduce nodes, forget nodes, join nodes and leaf nodes. We call the node an introduce node of the vertex , if it has a single child and . We call the node a forget node of the vertex , if it has a single child and . If the node has two children, we call it a join node (of nodes and ). Finally we call a node a leaf node, if it has no child.

Proposition 1

[15] Given a tree decomposition of a graph with vertices that has width and nodes, we can find a nice tree decomposition of that also has width and nodes in time .

So far we have described a standard nice tree decomposition. Now we change the introduce nodes. Let be an introduce node and its parent. We add another two copies of to the decomposition. We remove decomposition edge and add three decomposition edges and . Note that after this operation, is a leaf of the decomposition, remains an introduce node and is a join node. We call a sibling of . Note that by these further modifications we preserve linear number of nodes in the decomposition.

Figure 1: Change of Introduction nodes in the nice tree decomposition

Auxiliary subgraphs

Recall that for each edge there is at least one node containing that particular edge. Note that after our modification of the decomposition for each edge there is at least one leaf of the decomposition satisfying . To see this, suppose this is not true and that some edge, say , must be placed into a non-leaf node . We may suppose that is an introduce node (for join or forget node choose its descendant). However, in our construction any introduce node has a sibling such that is a leaf in the decomposition tree and their bags are equal.

Thus, for every edge we choose an arbitrary leaf node such that and say that the edge belongs to the leaf node . By this process we have chosen set for each leaf node . We further use the notion of auxiliary graph or for short. For a leaf node we set a graph . For a non-leaf node we set a graph , where and .

2.1 Preliminaries on refuting polynomial kernels

Here we present simplified review of a framework used to refute existence of polynomial kernel for a parameterized problem from Chapter 15 of a monograph by Cygan et al. [4].

In the following we denote by a final alphabet, by we denote the set of all words over and by we denote the set of all words over and length at most

Definition 3 (Polynomial equivalence relation)

An equivalence relation on the set is called polynomial equivalence relation if the following conditions are satisfied:

  1. There exists an algorithm such that, given strings resolves whether in time polynomial in .

  2. Relation restricted to the set has at most equivalence classes for some polynomial

Definition 4 (Cross-composition)

Let be an unparameterized language and be a parametrized language. We say that cross-composes into if there exists a polynomial equivalence relation and an algorithm called the cross-composition, satisfying the following conditions. The algorithm takes on input a sequence of strings that are equivalent with respect to , runs in polynomial time in and outputs one instance such that:

  1. for some polynomial and

  2. if and only if for all

With this framework, it is possible to refute even stronger reduction techniques—namely polynomial compression—according to the following definition:

Definition 5 (Polynomial compression)

A polynomial compression of a parameterized language into an unparameterized language is an algorithm that takes as an input an instance works in polynomial time in and returns a string such that:

  1. for some polynomial and

  2. if and only if

It is possible to refute existence of polynomial kernel using Definitions 3,4 and 5 with the help of use of the following theorems and a complexity assumption that is unlikely to hold—namely .

Theorem 2.1 ([7])

Let be two languages. Assume that there exists an AND-distillation of into Then

Theorem 2.2

Assume that an -hard language AND-cross-composes to a parameterized language Then does not admit a polynomial compression, unless

3 Minimal Length Bounded Multi-cuts

In this section we give a more detailed study of the length constraints for the length-bounded multi-cut and the triangle inequalities. From this we derive Lemma 1 for merging solutions for edge-disjoint graphs.

The triangle inequalities

Note that the solution for MLBMC problem has to satisfy the triangle inequalities with respect to its instance. This means that for any three terminals and the distance function it holds that . Thus, we can restrict instances of MLBMC problem only to those satisfying these triangle inequalities.

Definition 6 (Length constraints)

Let be a graph, and let . We call a vector a length constraint if for every it holds that .

For our approach it is important to see the structure of the solution on a graph composed from two edge disjoint graphs.

Lemma 1

Let be edge disjoint graphs. Then for the graph and and an arbitrary length constraints it holds that the minimum length bounded multi-cut for is a disjoint union of the multi-cuts and for and .

Proof

First we prove that there cannot be smaller solution than . To see this observe that for every cut on it holds that is an cut on (and vice versa for ). Hence if would be a cut of smaller size then , we would get a contradiction with the minimality of choice of and , because we would have .

Now we prove that is a valid solution. To see this we prove that every path between two terminals is not short. We prove that there cannot be such by an induction on number of . If then because and are edge disjoint, we may (by symmetry) assume that , a contradiction with the choice of . If then there is a vertex such that the path is composed from two segments and , where is a path between and and is a path between and . And so we have , what was to be demonstrated.

4 Restricted bounded multi-cut

In this section we present our approach to the -bounded cut for the graphs of bounded tree-width. We present our algorithm together with some remarks on our results.

Recall that we use dynamic programming techniques on a tree decomposition of an input graph. First we want to root the decomposition in a node containing both source and sink of the -cut problem. This can be achieved by adding the source to all nodes on the unique path in the decomposition tree between any node containing the source and any node containing the sink. Note that this may add at most to the width of the decomposition.

Length vectors and tables

As it was mentioned in the previous section, we solve the -cut by reducing it to simple instances of generalized MLBMC problem. We begin with a mapping with meaning . For simplicity we represent the mapping by a vector, calling it a length vector and relax it for a node , where and . We reduce the problem to the -bounded multi-cut for terminals, where (the additional two is for changing the decomposition). Let us introduce a relation on length vectors on of the same size. We write , if for all .

Let the set of vertices , be a length vector, let and let . By we denote the length vector containing if and only if both and (in an appropriate order) – in this case we say is contracted on the set .

Recall that for each node we have defined the auxiliary graph (see Section 2 for the definition). With a node we associate the table . The table entry for constraints of (denoted by ) for the node contains the size of the -bounded multi-cut for the set in the graph Note that for two length vectors it holds that .

4.1 Node Lemmas

The leaf nodes are the only nodes bearing some edges. We use an exhaustive search procedure for building tables for these nodes. For this we need to compute the lengths of the shortest paths between all the vertices of the leaf node, for which we use the well known procedure due to Floyd and Warshall [8, 17]:

Proposition 2 ([8, 17])

Let be a graph with nonnegative length . It is possible to compute the table of lengths of the shortest paths between any pair with respect to in time .

Lemma 2 (Leaf Nodes)

For all -limited length vectors and a leaf node the table of sizes of minimum length-bounded multi-cuts can be computed in time , where .

Proof

Fix one -limited length vector . We run the Floyd-Warshall algorithm (stated as Proposition 2) for every possible subgraph of . As there are such subgraphs. This gives us a running time for a single length vector – we check all length constraints if each of them is satisfied and set

Finally there are -bounded length vectors, this gives our result.

We now use Lemma 1 to prove time complexity of finding a dynamic programming table for join nodes from the table of its children.

Lemma 3 (Join Nodes)

Let be a join node with children and , let be the limit on length vectors components and let . Then the table can be computed in time from the table and .

Proof

Recall that graphs and are edge disjoint and that we store sizes of -bounded multi-cuts. Note also that and so we can apply Lemma 1 and set for each satisfying the triangle inequalities.

As there are entries in the table we have the complexity we wanted to prove.

As the forget node expects only forgetting a vertex and thus forgetting part of the table of the child. This is the optimizing part of our algorithm.

Lemma 4 (Forget Nodes)

Let be a forget node, its child, let be the limit on length vectors components and let . Then the table can be computed in time from the table .

Proof

Fix one length vector and compute the set of all -augmented length vectors. Formally if is a length vector satisfying the triangle inequalities for and . After this we set

There are at most of -augmented length vectors for each and this gives the claimed time.

Also the introduce node (as the counter part for the forget node) only adds coordinates to the table of its child. It does no computation as there are no edges it can decide about – these nodes now only add isolated vertex in the graph.

Lemma 5 (Introduce Nodes)

Let be an introduce node, its child, let be the limit on length vectors components and let . Then the table can be computed in time from the table .

Proof

Let be the vertex with the property . The key property is that the vertex is an isolated vertex in and thus we can set because is arbitrarily far from any vertex in , especially from the set .

4.2 Proofs of Theorems

We use Lemmas 234 and 5 to prove the theorem about computing -bounded -cut in graph of bounded tree-width. For this, note that we can use and put it into all the Lemmas as it is an upper bound on the size of any node in the decomposition.

We compute all the -bounded length-constraint that satisfy the triangle inequalities in advance. This takes additional time which can be upper-bounded by for and and so this does not make the overall time complexity worse.

Proof (Proof of Theorem 1.1)

As there are nodes in nice tree decomposition (by Proposition 1) and as we can upper-bound time needed to compute any type of node by , we have complexity proposed in state of the Theorem 1.1.

Let us now point out that the value of the parameter can be upper-bounded by the number of vertices of the input graph (in fact by as it is proved in [2]).

We now want to sum-up the key ideas leading to Theorem 1.1. First, it is the use of dynamic program for computing all options of cuts for bounded number of possible choices and for second it is the idea of creating a node that includes both the source and the sink while not harming the tree-width too much. On the other hand, we can use this idea to solve also the generalized version of the problem – length-bounded multi-cut – with the additional parameter the number of terminals. It is easy to see that in this setting that again it is possible to achieve node containing every terminal and thus this yields following Theorem 4.1.

Theorem 4.1

Let be a graph of tree-width , with and let . Then for any and any -limited length-constraints satisfying the triangle inequalities on an minimum -bounded multi-cut can be computed in time .

Proof

To see that Theorem 4.1 implies Theorem 1.4 note, that given any -limited length-constraints the minimum -bounded multi-cut must satisfy triangle inequalities. Thus, we find the minimum -bounded multi-cut among all . Note that we can do this in additional time , but this does not make the total running time worse.

5 Hardness of the -bounded cut

In this section we prove MLBC parametrized by path-width is -hard [9] by -reduction from -Multicolor Clique.

PROBLEM: -Multicolor Clique
Instance: -partite graph , where is independent set for every and they are pairwise disjoint
Parameter:
Goal: find a clique of the size

Denoting

In this section, sets are always partites of the -partite graph . We denote edges between and by . The problem is -hard even if every independent set has the same size and the number of edges between every and is the same. In whole Section 5 we denote the size of an arbitrary by and the size of an arbitrary by . For an -reduction from -Multicolor Clique to MLBC we need:

  1. Create an MLBC instance from the -Multicolor Clique instance where the size of is polynomial in the size of .

  2. Prove that contains a -clique if and only if contains an -bounded cut of the size where is a polynomial.

  3. Prove the path-width of is smaller than where is a computable function.

Our ideas were inspired by work of Michael Dom et al. [6]. They proved hardness of Capacitated Vertex Cover and Capacitated Dominating Set parametrized by the tree width of the input graph. We remarked that their reduction also proves hardness of these problems parametrized by path-width.

5.1 Basic gadget

In the -Multicolor Clique problem we need to select exactly one vertex from each independent set and exactly one edge from each . Moreover, we have to make certain that if is the selected edge and are the selected vertices then . The idea of the reduction is to have a basic gadget for every vertex and edge. We connect gadgets for every in into a path . The path is cut in the gadget if and only if the vertex is selected into the clique. The same idea will be used for selecting the edges.

Definition 7

Let . Butte is a graph which contains paths of length and paths of length between the vertices and . The short paths (of length ) are called shortcuts, the long paths are called ridgeways and the parameter is called height.

A butte for is shown in Figure 2 part a. In our reduction all buttes will have the same parameter (it will be computed later). For simplicity we depict buttes as a dash dotted line triangles with its height inside (see Figure 2 part b), or only as triangles without the height if it is not important.

Figure 2: a) Example of a butte for and . b) Simply diagram for a butte of height .

Let be a butte. We denote by the parameters of butte and , respectively. We state easy but important observation about the butte path-width:

Observation 5.1

Path-width of an arbitrary butte is at most 3.

Proof

If we remove vertices and from we get paths from ridgeways and isolated vertices from shortcuts. This graph has certainly path-width 1. If we add and to every node of the path decomposition we get a proper path decomposition of with width 3.

Let butte be a subgraph of a graph . Let be vertices of and all paths between and goes through such that they enter into in and leave it in (see Figure 3). The important properties of the butte are:

  1. By removing one edge from all shortcuts of , we extend the the distance between and by . If we cut all shortcuts of butte we say the butte is ridged.

  2. Let size of the cut is bounded by and we can remove edges only from . If we increase to be bigger than then any path between and cannot be cut by removing edges from (only extended by ridging the butte ).

Figure 3: Example of a path going through a butte.

5.2 Butte path

In this section we define how we connect buttes into a path, which we call highland. The main idea is to have highland for every pair . In the highland for , there are buttes for every vertex and every edge . We connect vertex buttes and edge buttes into a path. Then we set the butte heights and limit the size of the cut in such way that:

  1. Exactly one vertex butte and exactly one edge butte have to be ridged.

  2. If a butte for a vertex is ridged, then only buttes for edges incident with can be ridged.

Formal description of the highland is in the following definition.

Definition 8

The highland is a graph containing 2 vertices and and buttes where:

  1. and for every .

  2. for .

  3. for .

  4. for every .

Let be a highland. We call buttes from low and buttes high (low buttes will be used for the vertices and high buttes for the edges). The vertex , where low and high buttes meet, is called the center of highland . Note that there can be more buttes with the same height among high buttes and they are not ordered by height as the low buttes. An example of a highland is shown in Figure 4.

Figure 4: Example of a highland .
Proposition 3

Let be a highland. Let . Let be the -cut of size , which cut all paths of length and shorter between and then:

  1. The cut contains only edges obtained by ridging the exactly two buttes , such that is low and is high.

  2. Let be the ridged low butte and be the ridged high butte. Then, .

Proof (Proof of Proposition 3)

Every butte has at least shortcuts and ridgeways. Therefore, can not cut all paths in between and and it is useless to add edges from ridgeways to the cut . Note that the shortest -path in has the length .

  1. If we ridge every low butte we extend the shortest -path by . However, it is not enough and at least one high butte has to be ridged. Two high buttes cannot be ridged otherwise the cut would be bigger then the bound. No high butte can extend the shortest -path enough, therefore at least one low butte has to be ridged. However, two low buttes and one high butte cannot be ridged because the cut would be bigger then the bound.

  2. Let be the set of removed edges from ridged buttes and . The height of is . Therefore, the length of the shortest -path after ridging and and the size of is . If then shortest -path is strictly shorter then . Thus, is not -cut. If then and thus is bigger than .

5.3 Reduction

In this section we present our reduction. Let be the input for -Multicolor Clique. As we stated in the last section, the main idea is to have a low butte for every vertex and a high butte for every edge . Vertex and edge is selected into the -clique if and only if the butte and the butte are ridged. From we construct MLBC input (the construction is quite technical, for better understanding see Figure 5):

  1. For every we create highland of buttes .

  2. Let . The vertex is represented by the low butte of the highland for every . Thus, we have copies of buttes (in different highlands) for every vertex. Hence, we need to be certain that only buttes representing the same vertex are ridged. Note that buttes representing the same vertex have the same height and the same distance from the vertex .

  3. Let . Edge is represented by the high butte of the highland and by the high butte of the highland . Note that two buttes represented the same edge has same distance from the vertex . Let be the heights of buttes representing the vertices and , respectively. We set the buttes heights:

  4. We add edge for every and .

  5. We add paths of length connected and for every and .

  6. We set to .

We call paths between highlands in Items 4 and 5 the valley paths.

Figure 5: Some part of the graph . All vertices labeled and are actually two vertices and in the graph . We divided them for better illustration. Highlands and have also high buttes, but we omitted them.
Observation 5.2

Graph has a polynomial size in the size of the graph .

Theorem 5.3

If graph has a clique of size then has an -cut of size .

Proof

Let has a -clique where for every and . For every we ridge all buttes representing the vertex in . And for every we ridge both buttes representing the edge .

We claim that set of removed edges from ridged buttes forms the -cut. Let be an arbitrary highland. There is no -path shorter than in . Let where is arbitrary butte representing the vertex . By construction of , the high butte representing the edge in has height . Thus, ridged buttes in extend the shortest -path by and it has length . Buttes representing the vertex have same height. Thus, a path through the low buttes of highlands using some valley path is always longer than path going through low buttes of only one highland. Therefore, it is useless to use valley paths among low buttes for the shortest -path.

Other situation is among high buttes because buttes representing the same edge have different heights. The butte representing the vertex extend the shortest path at least by . The butte representing the edge extend the shortest at least by . However, if then and have to be in different highlands. Therefore, the -path going through and has to use a valley path between high buttes, which has length . Hence, any -path has the length at least .

We remove edges from each highland and there are highlands in . Therefore, has -cut of the size .

Theorem 5.4

If has an -cut of size then has a clique of size .

Proof

Let be an -cut of . Every shortest -path going through every highland has to be extended by . By Proposition 3 (Item 1), exactly one low butte and exactly one high butte of each highland has to be ridged. We remove from every highland in . Therefore, there can be only edges from ridged buttes in .

For fixed , highlands are the highlands which low buttes represent vertices from . We claim that ridged low buttes of represent the same vertex. Suppose for contradiction, there exists two low ridged buttes of and of which represent different vertex from . Without loss of generality and are next to each other (i.e. ) and the distance from to is smaller than the distance from to . Let be a butte of such that it has the same distance from as the butte (see Figure 6). The path does not go through any ridged low butte. Therefore, this path is shorter than , which is contradiction. We can use the same argument to show that there are not two high ridged buttes of highland and which represent different edges from .

Figure 6: How to miss every ridged low butte if there are ridged two low buttes representing two different vertices from one color class. Ridged butte is depicted as triangle without hypotenuse.

We put into the -clique the vertex if and only if an arbitrary butte representing the vertex is ridged. We proved in the previous paragraph that exactly one vertex from can be put into the clique . Let be an edge represented by ridged high buttes. We claim that . Let be a butte representing with height . Then by Proposition 3 (Item 2), butte of height has to be ridged. By construction of , only buttes representing edges incident with have such height. Therefore, chosen edges are incident with chosen vertices and they form the -clique of the graph .

Observation 5.5

Graph has path-width in .

Proof

Let be a graph created from by replacing every butte by a single edge and contract the valley paths between high buttes into single edges, see Figure 7 transformation . Let be a vertex set containing , and every highland center. Let be a graph created from by removing all vertex from , see Figure 7 transformation .

Figure 7: The transformation replace all buttes in by single edges and contract long valley paths into single edges. The transformation removes vertices and and all highland centers (highlighted by dotted ellipsis) from .

Graph is unconnected and it contains grids of size and grids of size . Path-width of grids is in , therefore . If we add set to every node of a path decomposition of we get proper path decomposition of . Since , path-width of is in . The edge subdivision does not increase path-width. Moreover, replacing edges by buttes does not increase it either (up to multiplication constant) because butte has the constant path-width (Observation 5.1). Therefore, for some constant and .

And thus Theorem 1.2 easily follows from Observations 5.2 and 5.5 and Theorems 5.3 and 5.4.

6 Polynomial kernel is questionable

In this section, we prove that for the Minimum Length Bounded Cut problem it is unlikely to admit a polynomial kernel when parametrized by the length and the path-width (tree-width) of the input graph. We will prove this fact by the use of a AND-composition framework—that is by designing an AND-composition algorithm from the unparameterized gapped version of the Minimum Length Bounded Cut problem into itself.

PROBLEM: Decision version of Minimum Length Bounded Cut
Instance: Graph vertices positive integers
Question: Is there an length bounded cut consisting of exactly edges

Baier et al. [2] proved that this problem is -complete even if we restrict the input instances such that

  • the desired length is constant,

  • either there is an bounded cut of size exactly or every bounded cut has at least edges.

Thus, it is possible to define the polynomial relation as follows. We will consider instances of the decision problem with constant Two instances are equivalent if and It is clear that is a polynomial equivalence relation.

AND-composition

We will take graphs the input instances that are equivalent under the relation for the Gapped Minimum Length Bounded Cut problem with the constant As a composition we will take the disjoint union of graphs and unify sources and sinks of the resulting graph and denote the graph as

Now it is easy to see that the path-width of the graph of our construction is at most as we can form bags of the path decomposition by whole graphs add the source and the sink into every bag and connect them into a path. This is indeed a correct path decomposition as the only two vertices in common are in every bag.

As we have taken the parameter to be constant, this finishes arguments about parameters. And thus also the proof of Theorem 1.3.

7 Conclusions

There is another standard generalization of the length bounded cut problem – where we add to each edge also its length. If this length is integral it is possible to extend and use our techniques (we only subdivide edges longer than – this doesn’t raise the tree-width of the graph on the input). On the other hand, if we allow fractional numbers, it is uncertain how to deal with such a generalization.

Acknowledgments

Authors thank to Jiří Fiala, Petr Kolman and Lukáš Folwarczný for fruitful discussions about the problem. We would like to mention that part of this research was done during Summer REU 2014 at Rutgers University.

References

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