Improved Kernels for Tracking Path Problems

01/09/2020
by   Pratibha Choudhary, et al.
0

Given a graph G, terminal vertices s and t, and an integer k, the Tracking Paths problem asks whether there exists at most k vertices, which if marked as trackers, would ensure that the sequence of trackers encountered in each s-t path is unique. The problem was first proposed by Banik et al. in <cit.>, where they gave results for some restricted versions of the problem. The problem was further studied in <cit.>, where the authors proved it to be NP-complete and fixed-parameter tractable. They obtained a kernel (i.e. an equivalent instance) with O(k^6) vertices and O(k^7) edges, in polynomial time. We improve on this by giving a quadratic kernel, i.e. O(k^2) vertices and edges. Further, we prove that if G is a d-degenerate graph then there exists a linear kernel. We also show that finding a tracking set of size at most n-k for a graph on n vertices is hard for the parameterized complexity class W[1], when parameterized by k.

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1 Introduction

Graphs serve as a systematic model for modeling and analysis of many real life problems. One of the commonly studied problems in areas like networks and machine learning is tracking of moving objects. Typically a secure environment or setup that needs to be monitored, has one or more source and destination points. The requirement is usually to identify the path traced by object(s) in a network. This can be implemented by placing trackers at a subset of checkpoints.

Coordinated path tracking and framework for multi-target tracking have been discussed in [11] and [10]. Tracking algorithms can be used in designing debugging tools. Another useful application is the problem of leakage detection systems. In these kind of problems it would be resource efficient if a small subset of nodes/checkpoints are sufficient to trace the movements of entities in the network.

While most of the work done on these kind of problems has been heuristic-based, Banik et al. 

[2] considered the problem of target tracking theoretically and modeled it as the following graph theoretic problem. Let be an undirected graph without any self loops or parallel edges and suppose has a unique entry vertex (source) and a unique exit vertex (destination) . A simple path from to is called an - path. The problem is to find a set of vertices be a set of vertices such that for any two distinct - paths, say and , the sequence of vertices in as encountered in is different from the sequence of vertices in as encountered in . Here is called a tracking set for the graph , and the vertices in are called trackers. Banik et.al. [2] proved that the problem of finding a minimum-cardinality tracking set with respect to shortest - paths (Tracking Shortest Paths problem) is NP-hard and APX-hard. The problem was first studied from a parameterized perspective in [4], the parameterized version of the problem being as follows.

Tracking Paths Parameter: Input: An undirected graph with two distinguished vertices and , and a non-negative integer . Question: Is there a tracking set of size at most for ?

In [4], above problem was proven to be NP-complete, and further it also shown to be fixed-parameter tractable by existence of a polynomial kernel. Specifically it was proven that an instance of Tracking Paths can be reduced to an equivalent instance of size in polynomial time, where is the desired size of the tracking set.

Our Results and Methods. In this paper we give a quadratic kernel for Tracking Paths on general graphs, which is a major improvement from the kernel given in [4]. We also give a linear kernel for Tracking Paths on -degenerate graphs. Further we prove that deciding if there exists a tracking set of size at most , where is the number of vertices in the graph, is W[1]-hard.

Given an instance , we give a polynomial time algorithm that either determines that is a NO instance or produces an equivalent instance with vertices and edges. This polynomial time algorithm is called a kernelization algorithm and the reduced instance is called a kernel. For more details about parameterized complexity and kernelization we refer to monographs [7, 5].

The kernelization algorithm works along the following lines. Let be an input instance to Tracking Paths. Two main results used to build the kernel in [4] were every tracking set is also a feedback vertex set (set of vertices whose deletion removes all cycles from graph), if there exist more than paths between a pair of vertices in , then cannot be tracked with at most trackers. However, in this paper, though we start the algorithm with a -approximate solution for feedback vertex set (FVS), we use another newly introduced result. Specifically, we prove that if there exists an induced subgraph in which consists of a tree with all of its leaves adjacent to a particular vertex , then the size of a minimum tracking set for is at least one less than the number of neighbors of in this tree. Then if is an FVS of size at most , we give bounds on different types of vertices in , based on how they share neighbors in . Combining all these bounds we prove the existence of a quadratic kernel for general graphs and linear kernel for -degenerate graphs. A -degenerate graph is a graph in which each subgraph has a vertex whose degree is at most . Observe that for such a graph , the number of edges in is for any subgraph of . Note that graphs with bounded degeneracy include planar graphs. Eppstein et al. studied Tracking Paths for planar graphs in [8], where they show that Tracking Paths remains NP-complete when the graph is planar, and give a 4-approximation algorithm for this setting.

Our linear kernel for -degenerate graphs uses the fact that in a -degenerate graph , any subgraph has edges. We also prove that finding a tracking set of size at most is W[1]-hard on a graph with vertices, where the parameter is .

Although a tracking set is also a feedback vertex set, both are fundamentally very different. A graph may have a small FVS but the tracking set may be arbitrarily larger than the FVS. Moreover, Tracking Paths is more demanding as a problem compared to the classic covering problems studied in graph theory. While covering problems aim at hitting a particular type of structure in graphs, Tracking Paths requires distinguishing each - path uniquely using a small set of vertices. In particular, even proving that Tracking Paths is NP is non-trivial. See [4] for details. A combinatorial generalization of Tracking Paths is studied in [3], where the input is a set system, and it is required to find a subset of elements from the universe that have a unique intersection with each set in the family. The problem has been shown to be a dual of the test cover problem. A related problem, Identifying Path Cover has been discussed in [9]. Identifying Path Cover requires finding a set of paths that cover all the vertices in a graph, and also uniquely identify each vertex by inclusion in a distinct set of paths.

Preliminaries

A kernelization algorithm is typically obtained using what are called reduction rules. These rules transform a given parameterized instance in polynomial time to an equivalent instance, and a rule is said to be safe if the resulting graph has a tracking set of size at most if and only if the original instance has one.

Throughout the paper, we assume graphs to be simple i.e. there are no self loops and multi-edges. When considering tracking set for a graph , we assume that the given graph is an - graph, i.e. the graph contains a unique source and a unique destination (both and are known), and we aim to find a tracking set that can distinguish between all simple paths between and . Here and are also referred as the terminal vertices. In this paper, when we refer to tracking set, we mean tracking set for all - paths. If , then unless otherwise stated, represents the set of vertices and , and represents an edge between and . For a vertex , neighborhood of is denoted by . We use to denote degree of vertex . For a vertex and a subgraph , . For a subset of vertices we use to denote . With slight abuse of notation we use to denote . For a graph and a set of vertices , denotes the subgraph induced by the vertex set . If is a singleton, we may use to denote , where . is used to denote the set of integers .

For a path , denotes the vertex set of path , and for a subgraph(or graph) , denotes the vertex set of . For a subgraph(or graph) , denotes the edge set of . Let be a path between vertices and , and be a path between vertices and , such that . By , we denote the path between and , formed by concatenating paths and at . Two paths and are said to be vertex disjoint if their vertex sets do not intersect except possibly at the end points, i.e. , where and are the starting and end points of the paths. By distance we mean length of the shortest path, i.e. the number of edges in that path. For a graph , an FVS is a set of vertices such that is a forest.

1.1 Organization of the Paper

Section 2 analyzes graph structures that have a strict lower bound in terms of the number of trackers required in them if they appear as subgraphs in the input graph. The lemmas in this section form the basis of the reduction rules used in Sections 3 and 4 to get the respective kernels. Section 3 gives an kernel for general graphs, where is the desired size of a tracking set. We start by finding a 2-approximate FVS for the input graph . Next the vertices in are categorized on the basis of how they share neighbors in with other vertices in . Section 4 uses a similar approach to derive an kernel for -degenerate graphs, while using the fact that a -degenerate graph has edges. Section 5 discusses the W[1]-hardness for the problem of finding whether a graph can be tracked with trackers, where is the number of vertices in the graph. Finally Section 6 summarizes the results in the paper with some open problems. Section Preliminaries explaining the terms and notations used in the paper can be found in Appendix.

2 Analyzing structures

We first give some reduction rules and basic results from [4] followed by additional ones. Then we give an important lemma based on tree like structures, which forms the base for vertex counting arguments for kernels in subsequent sections.

Following reduction rules are applied in the given sequence as long as they are applicable.

Reduction Rule 1

[4] If there exists a vertex or an edge that does not participate in any - path then delete it.

In the rest of the paper we assume that each vertex and edge participates in at least one - path.

Reduction Rule 2

If degree of (or ) is and (), then delete (), and label the vertex adjacent to it as ().

Lemma 1

Reduction Rule 2 is safe and can be implemented in polynomial time.

Proof

Consider a graph where (). Note that since all - paths pass through , a minimal tracking set need not contain . Also the sequence of each - path starts (ends) with () followed (preceded) by . Hence we can simply relabel as the source (destination) vertex (), delete () from the graph , and this would not affect a tracking set for the graph. Note that this reduction rule cannot be applied if and are neighbors, since we cant delete either of them from . ∎

Next we give a reduction rule that bounds the number of vertices with degree two in . A similar rule has been given in [4]. However, our rule is slightly tighter in the sense that we retain only one vertex of degree two as compared to two vertices being retained in the corresponding reduction rule given in [4], if multiple vertices of degree two are found connected in series (induced path).

Reduction Rule 3

If there exist such that and , then delete and introduce an edge between and .

Lemma 2

Reduction Rule 3 is safe and can be implemented in polynomial time.

Proof

Observe that if , then , and form a triangle such that and do not participate in any - path (as they form a triangle). Such a case is not possible due to Reduction Rule 1. Next observe that the set of - paths that passes through is the same as the set of - paths that pass through and the set of - paths that pass through . For identifying one or more - paths by their vertex sets, any one amongst , or is sufficient as a tracker. And at most two among are sufficient to help distinguish sequence of - paths that pass through these vertices. Thus a minimal tracking set would not contain all three of and .

Let be the graph obtained after applying the reduction rule. We claim that there exists a tracking set of size at most in if and only if there exists a tracking set of size at most in . First we prove the forward direction. Suppose that there exists a tracking set of size at most in . If , then is also a tracking set for . Let . If , then is a tracking set for . If , then it must have been the case when and both were chosen as trackers in order to distinguish sequence between some - paths. In this case is a tracking set for .

Now we prove the reverse direction. Let be a tracking set of size at most in . We claim that is a tracking set for as well. Suppose not. Then there exist two distinct - paths, and in that contain the same sequence of trackers. If and , then these paths exist in as well, contradicting the assumption that is a tracking set for . If and , again it contradicts the assumption that is a tracking set for . Now consider the case when one of these - paths passes through while the other does not. Without loss of generality, let and . Since all paths that pass through , also pass through , there exists a path in that is obtained by applying the reduction rule. Hence there exist paths and with the same sequence of trackers. This contradicts the assumption that is a tracking set for . Thus the reduction rule is safe.

To apply the rule we can consider all adjacent pairs of vertices, such that both vertices in the pair are of degree . This takes time. Hence the rule is applicable in polynomial time. ∎

Reduction Rule 4

If , then return a trivial YES instance.

Lemma 3

Reduction Rule 4 is safe and can be implemented in polynomial time.

Proof

Observe that if , then consists of only the edge . Since there exists only one - path, no trackers are needed for distinguishing. Hence, the given instance is a YES instance. It can be seen that such a case can be identified in constant time, and thus the rule is applicable in polynomial time. ∎

Next we recall a reduction rule from [8] that removes all those triangles from that contain a vertex of degree two.

Reduction Rule 5

If there exist such that and and , then mark as a tracker, delete from and set .

Next we recall a monotonicity lemma and a corollary from [4], which says that if a subgraph of cannot be tracked with trackers, then cannot be tracked with trackers either.

Lemma 4

[4] Let be a graph and be a subgraph of such that . If is a tracking set for and is a minimum tracking set for , then .

Lemma 5

[4] Any induced subgraph of containing at least one edge will contain a pair of vertices and that satisfy following conditions: (a) there exists a path in from to , say , and another path from to , say , (b) , (c) and .

For a subgraph , and a pair of vertices that satisfy the conditions of Lemma 5, we call vertex as a local source for and vertex as a local destination. Hence it follows from Lemma 5 that any subgraph containing at least one edge has a local source and a local destination. Note that a subgraph may have multiple local source-destination pairs.

Due to Lemma 4 and Lemma 5, we have the following two corollaries.

Corollary 1

[4] If a subgraph of that contains both and cannot be tracked by trackers, then cannot be tracked by trackers either.

Corollary 2

If there exists a subgraph of , and there exists a pair of vertices , such that is a local source for and is a local destination for , and all paths between and in cannot be tracked by at most trackers, then cannot be tracked by at most trackers.

In rest of the paper the phrase ‘subgraph cannot be tracked by trackers’ implies that the paths between a local source and destination in a subgraph cannot be tracked with trackers. Next corollary forms a starting point for the kernelization algorithms.

Corollary 3

[4] The size of a minimum tracking set for is at least the size of a minimum FVS for .

For rest of the paper, we assume that the graph in context has already been preprocessed using Reduction Rules 1 to 5.

2.1 Vertex Disjoint Paths

Here we give a bound on the number of vertex disjoint paths that can exist between a pair of vertices in a graph , given that can be tracked with at most trackers. While in [4] it is proven that there can exist at most vertex disjoint paths between a pair of vertices in , we improve the bound to . The new bound allows easy analysis and computation in future lemmas.

Lemma 6

If there exists two vertices such that there exists more than vertex disjoint paths between and , and the graph induced by these paths along with and has as a local source and as a local destination, then cannot be tracked with at most trackers.

Proof

For a contradiction assume that there exist vertex disjoint paths between a pair of vertices and in , and can be tracked with at most trackers. Let be the subgraph induced by . Since is a local source for and is a local destination for , there exists a path that starts at and ends at and does not contain any vertex from , and there exists a path that starts at and ends at and does not contain any vertex from . See Figure 2. Consider a pair of paths . Let and do not contain any trackers (except for and ). Now consider the - paths and . Note that and contain the same sequence of trackers. Since these paths differ only in the vertices on paths and , at least one vertex (except and ) either on or has to be a tracker. Thus as long as there are two paths in without any trackers, there will be two - paths with same sequence of trackers. Hence, at least paths in need a tracker on them. Due to Corollary 2, we know that if cannot be tracked with at most trackers then cannot be tracked with at most trackers. This contradicts the initial assumption, and hence completes the proof. ∎

Figure 1: Vertex disjoint paths between a local source and destination
Figure 2: Vertex disjoint paths between a pair of vertices
Lemma 7

If there exists two vertices such that there exists more than vertex disjoint paths between and , then cannot be tracked with at most trackers.

Proof

For a contradiction assume that there exist vertex disjoint paths between a pair of vertices and in , and can be tracked with at most trackers. Let be the subgraph induced by . Due to Lemma 5, there exists a local source, say , and a local destination, say , in . We consider various cases possible based on position of and in .

  • When and , or and . Both of these cases are symmetric to each other, and have been proven in Lemma 6.

  • When . First we consider the case when and lie on different paths in . See Figure 2. We denote the path between and (subpath of ) by , the path between and (subpath of ) by , the path between and (subpath of ) by , and the path between and (subpath of ) by . We denote the paths in by . Any - path in that passes through will be one among the following types:

    1. , where

    2. , where

    Consider the first two cases. Let be the graph induced by . Observe that and are local source and destination for , since there exists a path from to , and a path from to , and these paths intersect with only at and . Since there are paths between and in , due to Lemma 6, these require at least trackers in . If each of the paths in has a tracker, the paths indicated in cases 3,4 have the same sequence of trackers, and this contradicts the assumption that has a tracking set of size . Else, without loss of generality, let be the path in that is left without a tracker.

    Cases 3,4 denote two vertex disjoint paths between and along and . Hence, due to Lemma 6, there must be a tracker on either or . We consider following cases:

    • There exists a tracker in : Paths and contain the same set of trackers.

    • There exists a tracker in : Paths and contain the same set of trackers.

    • There exists a tracker in : Paths and contain the same set of trackers.

    • There exists a tracker in : Paths and contain the same set of trackers.

    All the above cases contradict the assumption that can be tracked with at most trackers.

    Next we consider the case when both and lie on the same path in . Without loss of generality assume that and both lie on . Here note that there exists one path between and that is a strict subpath of , and the remaining paths between and pass through , via vertices and . Observe that and are a local source and destination for the subgraph induced by . Since there are paths between and in the subgraph , due to Lemma 6, at least trackers are required in . If there are trackers in , it contradicts the assumption that can be tracked with at most trackers. If there are trackers in , without loss of generality, let be the path without any tracker. Now observe that there are two paths between and , the one that does not pass through and (subpath of ) and the one that passes through and , through , that do not have any trackers on them. Due to Lemma 6, at least one tracker is required on one of these paths. Hence we have a contradiction to the assumption that the graph can be tracked with at most trackers.

  • When and , or, and . Consider , and . This case can be argued similar to the second case, except that now . Similarly, if , and , the case is similar to the second case, except that now .

Note that the case when , is not possible after application of Reduction Rule 1. Correctness of the proof follows from Corollary 1. Hence the lemma holds. ∎

Next we give a reduction rule based on Lemma 7.

Reduction Rule 6

Let be a subgraph of , consisting of a pair of vertices adjacent to vertices each of degree two. If is a local source for and is a local destination for , then arbitrarily mark of the vertices of degree two as trackers and delete them. If return a NO instance, else set .

Lemma 8

Reduction Rule 6 is safe and can be implemented in polynomial time.

Proof

Let be the set of vertices of degree two that are adjacent to and and let be the set of vertices that were marked as trackers and deleted. Let be the newly created graph after the deletion of . We claim that has a tracking set of size if and only if has a tracking set of size . Suppose has a tracking set of size . If we add the vertices of back to along with their edges, there exists vertex disjoint paths between and . Since and are the local source and destination, due to Lemma 7 at least trackers are required on the vertices of . We mark all the vertices in as trackers. Now all paths between and are tracked. Since all other paths were already being tracked by in , is a tracking set of size for .

In the other direction let be a tracking set of size in . Let . We claim that has a tracking set of size . Suppose not. Then there exists two - paths, say and , in that have the same sequence of trackers. Observe that if both and do not intersect with edges and , then is not a tracking set of size in . This implies that and are also two paths with same sequence of trackers in . Note that the trackers on vertices in cannot be used to distinguish between and , as that would leave some untracked paths between and . Thus is not a tracking set for , which is a contradiction. This completes the proof of safeness of Reduction Rule 6.

In order to implement Reduction Rule  6, we consider each pair of vertices , and compare all their neighbors, to check for common neighbors of degree two. This can be done in time. Hence the rule can be applied in polynomial time. ∎

2.2 Tree-sink structure

In this section we discuss a specific graph structure, namely the tree-sink structure, and prove a lower bound for the number of trackers required if such a structure exists in an - graph. A tree-sink structure in a graph , is a subgraph such that , where is a tree with at least two vertices, and all of its leaves are adjacent to . Here is the tree while is the sink of the tree-sink structure. Note that may or may not be adjacent to the non-leaf vertices of . We prove that if the sink is adjacent to more than vertices in , then cannot be tracked with at most trackers.

We start with a simple case when the graph itself is a tree-sink structure and either or is the sink.

Figure 3: Tree-sink structure: Solid lines are edges of tree, and dashed lines are edges between the sink and vertices of the tree.
Lemma 9

Let be an - graph that forms a tree-sink structure with as the sink and . If , then trackers are required in , and these trackers have to be in , where is the tree induced by .

Proof

Without loss of generalization we assume that . We root at the source vertex . Consider that graph has already been preprocessed using Reduction Rules 1, 2, 3, and 4.

We prove the lemma by induction on the value of . Observe that due to Reduction Rule 1, 2, 3, and 4, is not possible. Thus the base case for induction is when . Note that in this case is either a triangle or a four cycle. See Figure 4.

Figure 4: Possible cases when the sink has two neighbors in the tree induced by .

Consider the case when is a triangle. Due to Reduction Rule 5, the vertex is marked as a tracker and deleted. Consider the case when is a four cycle. Observe that there exist two vertices, say , of degree two each, adjacent to and . Due to Reduction Rule 6, one among and is marked as a tracker and deleted. Note that in both the cases, after application of the corresponding reduction rules, comprises only of the edge . Due to Reduction Rule 4, this is a trivial YES instance. Hence, when , exactly one tracker is required in . This proves that the claim holds for the base case.

Next, for induction hypothesis, we assume that the claim holds for , i.e. if the sink is adjacent to vertices, then trackers are required in . Consider the case when . Note that here . Due to Reduction Rule 1, all leaves in are adjacent to , being the tree induced by . Consider a leaf vertex, say , that is at maximum distance from . Since , due to Reduction Rule 3, the degree of its parent node, say , is at least . Thus either has another child node, or is adjacent to . We analyze both the possibilities:

  • Case I: has another child node, say . Since is at maximum distance possible from , is a leaf node in . Observe that the graph induced by , , and has as a local source and as a local destination, and . Due to Reduction Rule 6, either or will be marked a tracker and deleted. This reduces the value of from to , while using one tracker.

  • Case II: is adjacent to . Observe that , and form and triangle and . Due to Reduction Rule 5, will be marked as a tracker and deleted. This reduces the value of from to , while using one tracker.

In both the above cases, after application of reduction rule, . Due to induction hypothesis, we know that when , then trackers are required in . Since we already used a tracker in both the above cases, the total number of trackers required when , is . Since the sink is itself, all the trackers need to be in . This completes the proof. ∎

Next we give a corollary which makes the above lemma more usable for the sake of our future arguments.

Corollary 4

Let be a graph and be a subgraph of such that induces a tree-sink structure with as its sink. If , and is either a local source or a local destination for , then the size of a tracking set for is at least . Further these trackers need to be in .

Proof

Consider the subgraph . Without loss of generality, we assume that is a local destination for . Let be a local source corresponding to the local destination . Due to Lemma 9, we have that trackers are required in to track all paths between and . From Corollary 2, if in a subgraph all paths between a local source and destination cannot be tracked with trackers then the graph cannot be tracked with trackers. Hence if , then cannot be tracked with at most trackers. Thus the size of a tracking set for is at least . It follows from Lemma 9 that these trackers need to be in . ∎

The next lemma generalizes the result in Corollary 4. We prove that regardless of where and lie in graph , if forms a tree-sink structure, then the size of the tracking set for is at least the number of neighbors of the sink in the tree minus one.

Lemma 10

If an - graph forms a tree-sink structure such that is the sink and induces a tree and , then the size of a tracking set for is at least , and at least trackers are required in .

Proof

Let be the tree induced by . The case when has been proven in Lemma 9. Consider the case when . We start by rooting the tree at . Now create a graph by removing the edge between and its parent vertex, say , in . Observe that in , there exists a tree, say that can be considered rooted at , consisting of all those vertices in that are not descendants of in , with all its leaves adjacent to the vertex . There exists another tree, say , rooted at , consisting of all of its descendants in , with all of its leaves adjacent to . See Figure 5. We denote the graph induced by by , and the graph induced by by . Let be the number of leaves in , and be the number of leaves in . Note that .

Figure 5: Removing edge from creates two tree-sink structures in , with trees (shown with solid lines) and (shown in dashed lines) and sink .

Note that is a local destination for . Hence by Corollary 4, since has many leaves, the size of a tracking set for is at least , and all these trackers must be in .

Note that is a local source for . Hence by Corollary 4, since has many leaves, the size of a tracking set for is at least , and all these trackers must be in .

If there exists at least trackers in , then the lemma holds. Else there exist trackers in and trackers in . Hence, there exists exactly one path in , say , from to that does not contain any trackers, and exactly one path in , say , from to that does not contain any trackers. Consider the path . Note that if contains a total of trackers, then is not a tracker and hence does not contain any trackers. Recall the edge that was initially removed between and its parent, , in . Consider the path in from to , say . We consider the following two scenarios.

  • is a subpath of the path . Consider the paths , and . Observe that both these paths have no trackers. Hence one more tracker is needed, either in or in order to distinguish them in .

  • is not a subpath of the path . If does not have a tracker, both the paths and do not contain any trackers. If has a tracker, let be the tracker that is closest to . Since is the minimum number of trackers required in , there exists a path from to (and not passing through ) in that does not contain any trackers. Lets denote this path by . Let be the path from to that is a subpath of . Now observe that paths and have the same set of trackers. Hence in both the cases discussed one more tracker is required in .

Thus the total number of required trackers in is at least , i.e. . Since the sink can be a tracker as well, a tree-sink structure requires at least trackers in the vertex set of the tree. ∎

Lemma 10 along with Corollary 1 gives us the following corollary.

Corollary 5

In a graph , if there exists a subgraph and a vertex , such that forms a tree-sink structure with as a sink, and then the size of a tracking set for is at least . Further at least trackers are required to be in .

3 Quadratic Kernel for General Graphs

In this section we show that an instance of Tracking Paths, can be reduced to an equivalent instance such that if is an YES instance, then , and . We start by applying Reduction Rules 1, 2, 3, 4 and 5. If the instance is not termed a NO instance by any of the reduction rules, we proceed with following. Recall from Corollary 3, that the size of a minimum tracking set for is at least the size of a minimum FVS for . We start by finding a 2-approximate Feedback Vertex set , using [1]. From Corollary 3, we have the following reduction rule.

Reduction Rule 7

[4] Apply the algorithm of [1] to find a 2-approximate solution, for Feedback Vertex set. If , then return that the given instance is a NO instance.

Observe that is a forest. Now we try to bound the number of vertices and edges in for the case when all - paths in can be tracked with at most trackers. Unless specified, we do not assume that the tree in context is a tree in . When referring to a tree-sink structure, by ‘tree’ we mean the tree that forms the tree-sink structure.

We give some counting arguments to bound the vertices in and the edges incident on these vertices. We first categorize the vertices in as following:

  • : vertices that share a neighbor in with another vertex in the same tree as they belong.

  • : vertices that share a neighbor in with another vertex from some other tree in .

  • : vertices that do not share any of their neighbors in with any other vertices in .

  • : vertices that do not have any neighbors in .

denotes the set of edges between the set of vertices and , where . Note that some vertices in may belong to more than one of the above mentioned categories. While giving the counting arguments, we may allow this possible over counting since it does not change the asymptotic value of the bound on . Note that since each vertex in can have at most one vertex from adjacent to it, and , it follows that . The total number of vertices in will be less than or equal to . Now we explore each of the above categories in detail. Henceforth when we use the term neighbor(s) of a vertex for a vertex in , we assume that we refer to the neighbor(s) of the vertex in .

3.1 Vertices that share a neighbor in with another vertex in the same tree

We first give a lemma that bounds the number of trees that can form a tree-sink structure with a common sink. This in turn helps us bound the number of trees in whose vertices can form tree-sink structures with vertices in as sinks.

Lemma 11

Let there be a vertex such that is a sink for tree-sink structures, then the numbers of trackers required is at least .

Figure 6: Two tree-sink structures with a common sink.
Proof

Suppose is a sink for trees. Let be the graph induced by along with all the trees that form tree-sink structures with as the sink. Due to Lemma 5, there exists a local source and a local destination in . Note that if were either the local source or the local destination, then due to Corollary 4 each of the trees requires a tracker in their vertex set, and hence the lemma holds. Suppose not. Let denote the tree and denote the graph induced by the vertex set of along with the vertex , for . Then due to Lemma 5, each graph has at least one pair of local source and destination vertices. Consider induced graphs and . See Figure 6. Note that for there exists a path from to via , that intersects with only at the sink , thus making a local destination for . Hence due to Corollary 4, at least one tracker is needed in . Next consider . Note that there exists a path from to the sink , via , that intersects at only at , thus making a local source for . Hence by Corollary 4, at least one tracker is needed in . Since these arguments can be extended for any induced graph , it holds that at least one tracker is required in the vertex set of each of the trees. ∎

Next we give two lemmas to bound the vertices in and edges in .

Lemma 12

For a vertex , the number of vertices in that form tree-sink structures with as a sink is at most in a YES instance.

Proof

Let and be the set of vertices that form tree-sink structures with as a sink. Let be the number of trees that form tree-sink structures with as sink, each with number of vertices adjacent to , where . Note that .

From Corollary 5 it is known that if a tree-sink structure is formed such that the sink is adjacent to vertices of the tree, then at least trackers are required in the tree vertices. Hence, each of the trees forming tree-sink structures with as sink, require trackers in their vertex set, . Note that a tracker in one tree of a tree-sink structure cannot act as a tracker for a tree-sink structure with a disjoint tree. Since the total budget for trackers is , . From Lemma 11, it follows that can be a sink for at most tree-sink structures. Thus . It follows that . ∎

Lemma 13

The number of vertices in that share neighbors in with vertices from the same tree is at most and the number of edges between these vertices and is at most in a YES instance.

Proof

When two or more vertices from a tree in share a common neighbor, say , they form a tree-sink structure, the tree being the minimal connected subtree containing all neighbors of in that tree, and being the sink. Due to Lemma 12 it is known that for a vertex , at most vertices from form tree-sink structures with as a sink. Since , the total number of vertices in is at most i.e. . As we considered only single edges between the sink and its neighbors in the trees of the tree-sink structures, . ∎

Reduction Rule 8

If the number of vertices that share neighbors in with vertices from the same tree are more than , then we return a NO instance.

Lemma 14

Reduction Rule 8 is safe and can be applied in polynomial time.

Proof

Safeness of the reduction rule follows from Lemma 13. To apply the rule, for each vertex in , we consider the subgraph induced by . There can be at most trees in , and each tree can have at most vertices. For each tree we check if at least two vertices are adjacent to . For tree that have at least two vertices adjacent to , we count the number of such vertices. This can be done in time. Since , the total time taken will be . ∎

3.2 Vertices that share a neighbor with a vertex from another tree in

Note that these vertices may or may not share a neighbor with a vertex in the same tree. As mentioned before we allow the possible over counting of vertices of here as the final bound calculated is still .

Observe that if a vertex belongs to a tree such that , then belongs to as well. Thus in such a case we need not count in . Excluding such vertices, we can assume that for each vertex it holds that , where is the tree to which belongs. This implies that either has at least two neighbors in , or there exists a vertex in that is adjacent to a vertex in , where is adjacent to a vertex in another tree in . Since we need an upper bound on , we assume the second case, i.e. for each vertex in there exists another vertex in the same tree and has a different neighbor in .

Let where . If and then at least two vertices among share a neighbor in (either or ) and thus belong to . Hence we can assume that a pair of vertices in is adjacent to at most two vertices from from each tree in .

Lemma 15

The number of vertices in that belong to vertex disjoint paths without any trackers, between a pair of vertices in are at most in a YES instance.

Proof

Let be a pair of vertices such that there exist three vertex disjoint paths (comprising of vertices from between them. Let be the subgraph induced by and along with the three vertex disjoint paths between them. If and are trackers, and are not a local source-destination pair for , then it is possible that no trackers are required on the three paths between them. See Figure 7.

Figure 7: Three paths without any trackers between , when are trackers but do not form a local source-destination pair. Square vertices belong to .

Observe that all such pairs of need to be disjoint, else the condition that they are not local source-destination pairs for the subgraph induced by vertex disjoint paths passing through them, will be violated. Since each such pair of requires two trackers, there can be at most such pairs of vertices in . Further, each such pair, can account for vertices in that form vertex disjoint paths between the pair. Hence, the number of vertices in that belong to vertex disjoint paths without any trackers, between a pair of vertices in are at most . ∎

Next we give a lemma to bound the vertices in .

Lemma 16

The number of vertices in that share a neighbor with a vertex from another tree is at most and the number of edges between these vertices and is at most in a YES instance.

(a) Pairs of vertices in share their neighbors with pair of vertices from another tree. (b) No pair of vertices in share their neighbors with pair of vertices from another tree.
Figure 8: Vertices sharing neighbors with vertices from other trees.
Proof

We subdivide into following two sub-categories:

  • Let be the set of vertices such that each pair of vertices from a tree share their neighbors with a pair of vertices from another tree. See Figure 8(a). Consider a pair of vertices . Observe that if pairs of vertices from different trees are incident to and , they form vertex disjoint paths between and passing through the trees to which they belong. Let be the set of vertices that form vertex disjoint paths without any trackers, between pairs of vertices of . From Lemma 15, .

    Next we consider those pairs of vertices in that are adjacent to at least four pairs of vertices of