Parameterized Complexity of Secluded Connectivity Problems

The Secluded Path problem models a situation where a sensitive information has to be transmitted between a pair of nodes along a path in a network. The measure of the quality of a selected path is its exposure, which is the total weight of vertices in its closed neighborhood. In order to minimize the risk of intercepting the information, we are interested in selecting a secluded path, i.e. a path with a small exposure. Similarly, the Secluded Steiner Tree problem is to find a tree in a graph connecting a given set of terminals such that the exposure of the tree is minimized. The problems were introduced by Chechik et al. in [ESA 2013]. Among other results, Chechik et al. have shown that Secluded Path is fixed-parameter tractable (FPT) on unweighted graphs being parameterized by the maximum vertex degree of the graph and that Secluded Steiner Tree is FPT parameterized by the treewidth of the graph. In this work, we obtain the following results about parameterized complexity of secluded connectivity problems. We give FPT-algorithms deciding if a graph G with a given cost function contains a secluded path and a secluded Steiner tree of exposure at most k with the cost at most C. We initiate the study of"above guarantee"parameterizations for secluded problems, where the lower bound is given by the size of a Steiner tree. We investigate Secluded Steiner Tree from kernelization perspective and provide several lower and upper bounds when parameters are the treewidth, the size of a vertex cover, maximum vertex degree and the solution size. Finally, we refine the algorithmic result of Chechik et al. by improving the exponential dependence from the treewidth of the input graph.


Introduction
Secluded Path and Secluded Steiner Tree problems were introduced in Chechik et al. in [8]. In the Secluded Path problem, for given vertices and of a graph , the task is to find an , -path with the minimum exposure, i.e. a path such that the number of vertices from plus the number of vertices of adjacent to vertices of is minimized. The name secluded comes from the setting where one wants to transfer a confident information over a path in a network which can be intercepted either while passing through a vertex of the path or from some adjacent vertex. Thus the problem is to select a secluded path minimizing the risk of interception of the information. When instead of connecting two vertices one needs to connect a set of terminals, we arrive naturally to the Secluded Steiner Tree.
More precisely, Secluded Steiner Tree is the following problem. If ( ) = 1 for each ∈ ( ) and = , then we have an instance of Secluded Steiner Tree without costs; respectively, we omit and whenever we consider such instances.
Clearly, it can be assumed that is a tree, and thus the problem can be seen as a variant of the classical Steiner Tree problem. For the special case = 2, we call the problem Secluded Path. Previous work. The study of the secluded connectivity was initiated by Chechik et al. [7,8] who showed that the decision version of Secluded Path without costs is NP-complete. Moreover, for the optimization version of the problem, it is hard to approximate within a factor of (2 log 1− ), is the number of vertices in the input graph, for any > 0 (under an appropriate complexity assumption) [8]. Chechik et al. [8] also provided several approximation and parameterized algorithms for Secluded Path and Secluded Steiner Tree. Interestingly, when there are no costs, Secluded Path is solvable in time Δ Δ · (1) , where Δ is the maximum vertex degree and and thus is FPT being parameterized by Δ. Chechik et al. [8] also showed that when the treewidth of the input graph does not exceed , then the Secluded Steiner Tree problem is solvable in time 2 ( log ) · (1) · log , 1 where is the maximum value of on an input graph . Johnson et al. [19] obtained several approximation results for Secluded Path and showed that the problem with costs is NP-hard for subcubic graphs improving the previous result of Chechik et al. [8] for graphs of maximum degree 4.
The problems related to secluded path and connectivity under different names were considered by several authors. Motivated by secure communications in wireless ad hoc networks, Gao et al. [15] introduced the very similar notion of the thinnest path. The motivation of Gilbers [17], who introduced the problem under the name of the minimum witness path, came from the study of art gallery problems. Our results. In this paper we initiate the systematic study of both problems from the Parameterized Complexity perspective and obtain the following results. In Section 3, we show that Secluded Path and Secluded Steiner Tree are FPT when parameterized by the size of the solution by giving algorithms of running time (3 /3 · ( + ) log ) and (2 · ( + ) 2 log ), where is the maximum value of on an input graph , correspondingly.
We consider the "above guarantee" parameterizations of both problems in Section 4. Recall that if 1 , . . . , are vertices of a graph , then a connected subgraph of of minimum size such that 1 , . . . , ∈ ( ) is called a Steiner tree for the terminals 1 , . . . , . If = 2, then a Steiner tree is a shortest ( 1 , 2 )-path. Clearly, if ℓ is the size (the number of vertices) of a Steiner tree, then for any connected subgraph of with ⊆ ( ), | [ ( )]| ≥ ℓ. Recall that the Steiner Tree problem is well known to be NP-complete as it is included in the famous Karp's list of 21 NP-complete problems [20], but in 1971 Dreyfus and Wagner [11] proved that the problem can be solved in time * (3 ), i.e., it is FPT when parameterized by the number of terminals. The currently best FPT-algorithms for Steiner Tree running in time * (2 ) are given by Björklund et al. [2] and Nederlof [24] (the first algorithm demands exponential in space and the latter uses polynomial space). In Section 4 we show that Secluded Path and Secluded Steiner Tree are FPT when the problems are parameterized by + , where = − ℓ. From the other side, we show that the problem is co-W[1]-hard when parameterized by only.
In Section 5, we provide a thorough study of the kernelization of the problem from the structural paramaterization perspective. We consider parameterizations by the treewidth, size of the solution, maximum degree and the size of a vertex cover of the input graph. We show that it is unlikely that Secluded Path (even without costs) parameterized by the solution size, the treewidth and the maximum degree of the input graph, admits a polynomial kernel. In particular, this complements the FPT algorithmic findings of Chechik et al. [8] for graphs of bounded treewdith and of bounded maximum vertex degree. The same holds for the "above guarantee" parameterization instead the solution size as well. On the other hand, we show that Secluded Steiner Tree has a polynomial kernel when parameterized by and the vertex cover number of the input graph. Interestingly, when we parameterize only by the vertex cover number, again, we show that most likely the problem does not admit a polynomial kernel. Finally, we refine the algorithm on graphs of bounded treewidth of Chechik et al. [8] by showing that Secluded Steiner Treewithout costs can be solved by a randomized algorithm in time that single-exponentially depends on treewidth by applying the Count & Color technique of Cygan et al. [9] and further observe that for the general variant of the problem with costs, the same Count & Color technique can be used as well and also a single-exponential deterministic algorithm can be obtained by making use the representative set technique developed by Fomin et al. [13].

Basic definitions and preliminaries
We consider only finite undirected graphs without loops or multiple edges. The vertex set of a graph is denoted by ( ) and the edge set is denoted by ( ). Throughout the paper we typically use and to denote the number of vertices and edges respectively.
For a set of vertices ⊆ ( ), [ ] denotes the subgraph of induced by . For a vertex , we denote by ( ) its (open) neighborhood, that is, the set of vertices which are adjacent to , and for a set ⊆ ( ), A tree decomposition of a graph is a pair (ℬ, ) where is a tree and ℬ = { | ∈ ( )} is a collection of subsets (called bags) of ( ) such that i) ⋃︀ ii) for each edge ∈ ( ), , ∈ for some ∈ ( ), and iii) for each ∈ ( ) the set { | ∈ } induces a connected subtree of .
The width of a tree decomposition ({ | ∈ ( )}, ) is max ∈ ( ) {| | − 1}. The treewidth of a graph (denoted as tw( )) is the minimum width over all tree decompositions of . A set ⊆ ( ) is a vertex cover of if for any edge of , ∈ or ∈ . The vertex cover number ( ) is the size of a minimum vertex cover.
Parameterized complexity is a two dimensional framework for studying the computational complexity of a problem. One dimension is the input size and another one is a parameter . It is said that a problem is fixed parameter tractable (or FPT), if it can be solved in time ( )· (1) for some function . A kernelization for a parameterized problem is a polynomial algorithm that maps each instance ( , ) with the input and the parameter to an instance ( ′ , ′ ) such that i) ( , ) is a yes-instance if and only if ( ′ , ′ ) is a yes-instance of the problem, and ii) the size of ′ is bounded by ( ) for a computable function . The output ( ′ , ′ ) is called a kernel. The function is said to be a size of a kernel. Respectively, a kernel is polynomial if is polynomial. While a parameterized problem is FPT if and only if it has a kernel, it is widely believed that not all FPT problems have polynomial kernels. In particular, Bodlaender et al. [4,5] introduced techniques that allow to show that a parameterized problem has no polynomial kernel unless NP ⊆ coNP /poly. We refer to the book of Downey and Fellows [10], for detailed introductions to parameterized complexity.
We use randomized algorithms for our problems. Recall that a Monte Carlo algorithm is a randomized algorithm whose running time is deterministic, but whose output may be incorrect with a certain (typically small) probability. A Monte-Carlo algorithm is true-biased (false-biased respectively) if it always returns a correct answer when it returns a yes-answer (a no-answer respectively).

FPT-algorithms for the problems parameterized by the solution size
In this section we consider Secluded Path and Secluded Steiner Tree problems parameterized by the size of the solution, i.e., by . We also show how these parameterized algorithms can be used to design faster exact exponential algorithms. We start with Secluded Path.
where is the maximum value of on an input graph .
Proof. Let us observe first that if there is an optimal secluded path, then there is an optimal secluded induced path-shortcutting a path cannot increase the size of its neighbourhood. We give an algorithm that enumerates all induced paths from to such that | [ ( )]| ≤ in time (3 /3 · ) for a graph with vertices. Then picking up a secluded path of minimum cost will complete the proof. The algorithm is based on the standard branching ideas. If | [ ]| > the algorithm reports that no such path exist and stops. If | [ ]| ≤ and = the algorithm outputs the path consisting of the single vertex . Otherwise a path from to must go through one of the neighbors of . Since we are looking for an induced path it must never return to a vertex from [ ]. This allows us to branch as follows. For each ∈ ( ), we check recursively whether the graph This way we get the following recurrence on the number of nodes ( ) in the corresponding recursion tree. If = , then there is only one path from to , and ( ) ≤ 1. If ̸ = , then ( ) ≤ · ( − ), where = | ( )|. This is a well known recurrence implying that ( ) = (3 /3 ) (see, e.g., the analysis of the algorithm enumerating all maximal independent sets in Chapter 1 of [12]). Note that we spend only a linear time ( ) in each vertex of the recursion tree. Since the length of each path can be computed in time ( log ), we can find a path of minimum cost it time (3 /3 log ). Therefore, the total running time is (3 /3 · log ).
For Secluded Steiner Tree we prove the following theorem.
Theorem 2. Secluded Steiner Tree can be solved in time (2 · ( + ) 2 log ), where is the maximum value of on an input graph .
The following proposition from [14] will be useful for us.  Parameterized algorithms for Secluded Path and Secluded Steiner Tree combined with a brute-force procedure imply the following exact exponential algorithms for the problems. )︀ . Thus for all integers between 0.77923· and , we enumerate sets of size − , while for all integers between 1 and 0.77923 · we use Theorem 2 to find if there is a solution of size at most . The running time of this algorithm is (2 0.77923 · log ) = (1.7088 · log ).

FPT-algorithms for the problems parameterized above the guaranteed value
In this section we show that Secluded Path and Secluded Steiner Tree are FPT when the problems are parameterized by + where = −ℓ and ℓ is the size of a Steiner tree for .
where ℓ is the length of a shortest ( , )-path for { , } = and is the maximum value of on an input graph .
Proof. The proof of this theorem is very similar to the proof of Theorem 1. For an integer ℎ, we enumerate in the graph all induced paths from to of length at most ℎ − 1 such that | ( ( ))| ≤ − ℎ. The only difference with Theorem 1 is that this time we bound the running time of the algorithm as a function of − ℎ.
If | ( )| > − ℎ the algorithm reports that no such path exist and stops. If | [ ]| ≤ −ℎ and = the algorithm outputs the path consisting of the single vertex . Otherwise, we branch by checking recursively for each ∈ ( ), whether the graph . This way we get the following recurrence on the number of nodes ( − ℎ) in the corresponding recursion tree. If = , then there is only one path from to , and ( − ℎ) ≤ 1. If ̸ = , then We need some structural properties of solutions of Secluded Steiner Tree. We start with an auxiliary lemma bounding the number of vertices of degree at least three in as well as the number of their neighbors. Proof. Let ℬ be the set of blocks of . Consider bipartite graph with the bipartition ( ( ), ℬ) of the vertex such that ∈ ( ) and ∈ ℬ are adjacent if and only if is a vertex of . Notice that is a tree. Recall that the vertex dissolution operation for a vertex of degree 2 deletes together with incident edges and replaces them by the edge joining the neighbors of . Denote by ′ the tree obtained from by consequent dissolving all vertices of of degree 2 that are not in . Denote by the set of leaves of . By the minimality of , ⊆ . Let 1 = | | ≤ , and let 2 be the number of degree 2 vertices and 3 be the number of vertices of degree at least 3 in . Clearly, either is included in at least 3 blocks of , or is in a block of size at least 3. In the second case, is adjacent to a vertex ∈ ℬ of with degree at least 3. It implies that The following lemma provides a bound on the number of vertices of a tree that have neighbors outside the tree.
Let be a connected graph and ⊆ ( ), = | |. Let ℓ be the size of a Steiner tree for and be a positive integer. Suppose that is an inclusion minimal subgraph of such that is a tree spanning and Proof. Denote by the set of leaves of and by the set of vertices of degree at least 3 in . Clearly, ⊆ . We select a leaf of as the root of . The selection of a root defines a parent-child relation on .
Proof of the claim. Since all leaves of including are in , we have that ∈ ( ′ ). To prove the claim, we show that for each vertex ∈ ( ′ ), there is a ( , )-path in ′ . Every vertex ∈ has a neighbor ( ) in ′ . Hence, it is sufficient to prove the existence of ( , )-paths for ∈ ( )∖ ′ . The proof is by induction on the distance between and in . If = , then we have a trivial ( , )-path. Assume that ̸ = . Let be the parent of in . If ∈ ( ′ ), then by the inductive hypothesis, there is a ( , )-path in ′ and it implies the existence of a ( , )-path. Suppose that / ∈ ( ′ ), i.e., ∈ ′ . Since ( ) = 2, there is ∈ such that = ( , ). The distance in between and ( ) is less than the distance between and . Therefore, by the inductive hypothesis, there is a ( , ( ))-path in ′ . It remains to observe that because ( ) , ∈ ( ′ ), ′ has a ( , )-path as well. This concludes the proof to the claim.
Denote by the set of the children of the vertices of ∪ in . Observe Let ′ be the tree obtained from by consequent dissolving all the vertices of degree 2 that are not in .
Now we are ready to prove the main result of the section.
Theorem 5. Secluded Steiner Tree can be solved in time 2 ( + ) · · log by a true-biased Monte-Carlo algorithm and in time 2 ( + ) · log · log by a deterministic algorithm for graphs with vertices and edges, where = − ℓ and ℓ is the size of a Steiner tree for and is the maximum value of on an input graph .
Proof. We construct an FPT-algorithm for Secluded Steiner Tree parameterized by + . The algorithm is based on the random separation techniques introduced by Cai, Chan, and Chan [6] (see also [1]). We first describe a randomized algorithm and then explain how it can be derandomized.
Let ℐ = ( , , , , ) be an instance of Secluded Steiner Tree, ℓ be the size of a Steiner tree for = { 1 , . . . , } and = − ℓ. Without loss of generality we assume that ≥ 2 and ≥ 1 as for = 1 or = 0, the problem is trivial. We also can assume that is connected.
Description of the algorithm In each iteration of the algorithm we color the vertices of independently and uniformly at random by two colors. In other words, we partition ( ) into two sets and . We say that the vertices of are red, and the vertices of are blue. Our algorithm can recolor some blue vertices red, i.e., the sets and can be modified. Our aim is to find a connected subgraph of with Step 1. Step 2. If there is ∈ such that / ∈ or ( ) ∩ = ∅, then return that ℐ is no-instance and stop.
Step 3. Find a component of [ ] with 1 ∈ ( ). If there is a pendant vertex / ∈ of that is adjacent in to the unique vertex ∈ , then find a component of [ ] that contains , recolor its vertices red and then return to Step 1. Otherwise, return that ( , , ) is no-instance and stop.
We repeat at most 2 ( + ) iterations. If on some iteration we obtain a yes-answer, then we return it and the corresponding solution. Otherwise, if on every iteration we get a no-answer, we return a no-answer. ∩ ( ) ⊆ and ∖ ( ) ⊆ . The probability that for a random coloring, the vertices of are colored incorrectly, i.e., ∩ ( ) ∩ ̸ = ∅ or ( ∖ ( )) ∩ ̸ = ∅, is at most 1 − 2 − . Hence, if we consider 2 random colorings, then the probability that the vertices of are colored incorrectly for all the colorings is at most (1 − 2 − ) 2 , and with probability at least 1 − (1 − 2 − ) 2 for at least one coloring we will have ∩ ( ) ⊆ and

Correctness of the algorithm
Thus if ℐ is a yes-instance, after 2 random colorings of , we have that at least one of the colorings is successful with a constant success probability = 1 − 1/ .
Assume that for a random red-blue coloring of , ∩ ( ) ⊆ and ∖ ( ) ⊆ . We show that in this case the algorithm finds a tree ′ with We claim that for every connected component of [ ], either ( ) ⊆ ( ) or ( ) ∩ ( ) = ∅. To obtain a contradiction, assume that there are , ∈ ( ) such that ∈ ( ) and / ∈ ( ). Indeed, is connected, and thus contains an ( , ) path . Since goes from ( ) to ̸ ∈ ( ), path should contain a vertex ∈ ( ) = . But is colored blue, which is a contradiction to the assumption that is in the red component . Let us assume that the algorithm does not stop at Step 1. For the right coloring, because ⊆ and ( ) ⊆ ′ , for every ∈ , we have that ∈ . Moreover, because ≥ 2, at least one neighbor of in is in . Thus the only reason why the algorithm stops at Step 2 is due to the wrong coloring. Consider the case when the algorithm does not stop after Step 2.
Suppose that is a component of [ ] with 1 ∈ ( ). Because the algorithm did not stop in Step 2, such a component exists and has at least 2 vertices. Recall that ( ) ⊆ ( ). Because we proceed in Step 1, we conclude that ∖ ( ) ̸ = ∅. Then there is a vertex ∈ ( ) which has a neighbor in such that ∈ . If ∈ , then ∈ ′ , but this contradicts the assumption ′ ⊆ . Hence, / ∈ . Suppose that ( ) ≥ 2. In this case ( ) ≥ 3 and ∈ ′ ; a contradiction. Therefore, is a pendant vertex of .
Let / ∈ be an arbitrary pendant vertex of . If has no neighbors in , then is a leaf of that does not belong to but this contradicts the inclusion minimality of . Assume that is adjacent to at least two distinct vertices of . Because is an inclusion minimal tree spanning , vertex has at least two neighbors in and has a neighbor ∈ in . Let ∈ ( ( ) ∩ ) ∖ { }. If ∈ ( ), then ( ) ≥ 3 and, therefore, ∈ and , ∈ ′ ; a contradiction with ′ ⊆ . Hence, / ∈ ( ). Moreover, is the unique neighbor of in that belongs to . Then ∈ and ∈ { 1 ( ), 2 ( )}; a contradiction with ⊆ . We obtain that is adjacent in to the unique vertex ∈ . Let ′ be the component of [ ] that contains . Since is an inclusion minimal tree that spans , has at least two neighbors in . It implies that ∈ ( ), therefore ( ′ ) ⊆ ( ). We recolor the vertices of ′ red in Step 3. For the new coloring the vertices of are blue and the vertices of ∖ are red. Therefore, we keep the crucial property of the considered coloring but we increase the size of the component of [ ] containing 1 .
To conclude the correctness proof, it remains to observe that in Step 3 we increase the number of vertices in the component of [ ] that contains 1 . Hence, after at most iterations, we obtain a component in [ ] that includes and return a solution in Step 1.
It is straightforward to verify that each of Steps 1-3 can be done in time ( log ). Because the number of iterations is at most , we obtain that the total running time is 2 ( + ) · log . This algorithm can be derandomized by standard techniques (see [1,6]). The random colorings can be replaced by the colorings induced by universal sets. Let and be positive integers, ≤ . An ( , )-universal set is a collection of binary vectors of length such that for each index subset of size , each of the 2 possible combinations of values appears in some vector of the set. It is known that an ( , )-universal set can be constructed in FPT-time with the parameter . The best construction is due to Naor, Schulman and Srinivasan [23]. They obtained an ( , )-universal set of size 2 · (log ) log , and proved that the elements of the sets can be listed in time that is linear in the size of the set. In our case is the number of vertices of and = 20 + 7 − 27.
We complement Theorem 5 by showing that it is unlikely that Secluded Steiner Tree is FPT if parameterized by only. To show it, we use the standard reduction from the Set Cover problem (see, e.g., [20]). Notice that we prove that Secluded Steiner Tree is co-W[1]-hard, i.e., we show that it is W[1]-hard to decide whether we have a no-answer. Since | ( )| ≤ + + 1, we obtain that ′ ≤ . It remains to note that 1 , . . . , ′ cover and, therefore, ( , 1 , . . . , , ) is a yes-instance of Set Cover.

Structural parameterizations of Secluded Steiner Tree
In this section we consider different algorithmic and complexity results concerning different structural parameterizations of secluded connectivity problems. We consider parameterizations by the treewidth, size of the solution, maximum degree and the size of a vertex cover of the input graph. (See Appendix for definitions of these parameters.) We show that it is unlikely that Secluded Path without costs parameterized by , the treewidth and the maximum degree of the input graph has a polynomial kernel. We obtain the same result for the cases when the problem is parameterized by − ℓ, the treewidth and the maximum degree of the input graph, where ℓ is the length of the shortest path between terminals.
Theorem 7. Secluded Path without costs on graphs of treewidth at most and maximum degree at most Δ admits no polynomial kernel unless NP ⊆ coNP /poly when parameterized by + + Δ or ( − ℓ) + + Δ, where ℓ is the length of the shortest path between terminals.
The proof uses the cross-composition technique introduced by Bodlaender, Jansen and Kratsch [5]. We need the following additional definitions (see [5]).
Let Σ be a finite alphabet. An equivalence relation ℛ on the set of strings Σ * is called a polynomial equivalence relation if the following two conditions hold: i) there is an algorithm that given two strings , ∈ Σ * decides whether and belong to the same equivalence class in time polynomial in | | + | |, ii) for any finite set ⊆ Σ * , the equivalence relation ℛ partitions the elements of into a number of classes that is polynomially bounded in the size of the largest element of .
Let ⊆ Σ * be a language, let ℛ be a polynomial equivalence relation on Σ * , and let ⊆ Σ * × N be a parameterized problem. An OR-crosscomposition of into (with respect to ℛ) is an algorithm that, given instances 1 , 2 , . . . , ∈ Σ * of belonging to the same equivalence class of ℛ, takes time polynomial in ∑︀ =1 | | and outputs an instance ( , ) ∈ Σ * × N such that: i) the parameter value is polynomially bounded in max{| 1 |, . . . , | |}+ log , ii) the instance ( , ) is a yes-instance for if and only if at least one instance is a yes-instance for for ∈ {1, . . . , }.
It is said that OR-cross-composes into if a cross-composition algorithm exists for a suitable relation ℛ.

Theorem 8 ([5]). If an NP-hard language
OR-cross-composes into the parameterized problem , then does not admit a polynomial kernelization unless NP ⊆ coNP /poly. of Theorem 7. First, we prove the claim for the case when the problem is parameterized by + + Δ.
We construct an OR-composition of Secluded Path without costs to the parameterized version of Secluded Path. Recall that Secluded Path without costs was shown to be NP-complete by Chechik et al, [7,8]. We assume that two instances ( , . . , } be equivalent instances of Secluded Path, | ( )| = ≥ 3. Without loss of generality we assume that = 2 for a positive integer ; otherwise, we add minimum number of copies of ( 1 , { 1 1 , 1 2 }, ) to achieve this property. We construct the graph as follows.
ii) Construct a rooted binary tree 1 of height , denote the root by 1 and identify = 2 leaves of the tree with the vertices of 1 1 , . . . , 1 of 1 , . . . , .
iii) Construct a rooted binary tree 2 of height , denote the root by 2 and identify = 2 leaves of the tree with the vertices of 1 2 , . . . , 2 of 1 , . . . , . We set ′ = + 4 and consider the instance ( , The proof for the case when the problem is parameterized by ( −ℓ)+ +Δ uses the same OR-composition. The difference is that now we assume that two instances ( , { 1 , 2 }, ) and ( ′ , { ′ 1 , ′ 2 }, ′ ) are equivalent if | ( )| = | ( ′ )|, = ′ and 1 , 2 and ′ 1 , ′ 2 are at the same distance in and ′ respectively. Let ℓ be the distance between 1 and 2 in for ∈ {1, . . . , }. Then the length of a shortest ( 1 , 2 )-path in Observe that Theorem 7 immediately implies that Secluded Path without costs has no polynomial kernel unless NP ⊆ coNP /poly when parameterized by or − ℓ. The next natural question is if parameterization by a stronger parameter can lead to a polynomial kernel. Let us note that the treewidth of a graph is always at most the minimum size of its vertex cover. The following theorem provides lower bounds for parameterization by the minimum size of a vertex cover.
Theorem 9. Secluded Path without costs on graphs with the vertex cover number at most has no polynomial kernel unless NP ⊆ coNP /poly when parameterized by .
Proof. We show that the 3-Satisfiability problem OR-cross composes into Secluded Path without costs. Recall that 3-Satisfiability asks for given boolean variables 1 , . . . , and clauses 1 , . . . , with 3 literals each, whether the formula = 1 ∧ . . . ∧ can be satisfied. It is well-known that 3-Satisfiability is NP-complete [16]. We assume that two instances of 3-Satisfiability are equivalent if they have the same number of variables and the same number of clauses.
Theorem 10. The Secluded Steiner Tree problem admits a kernel with at most 2 ( + 1) vertices on graphs with the vertex cover number at most .
Proof. Let ( , , ) be an instance of Secluded Steiner Tree. We assume that | | ≥ 2, as otherwise the problem is trivial. Our kernelization algorithm uses the following steps.
Step 1. If is disconnected, then return a no-answer and stop if there are distinct components of that contain terminals, and construct the instance It is straightforward to see that our first step is safe to apply, i.e., it either returns a correct answer or creates an equivalent instance of our problem. From now we assume that is connected.
Step 2. Find a set of vertices by taking end-vertices of the edges of a maximal matching in . If | | > 2 , then return a no-answer and stop.
It is well-known (see e.g. [16]) that is a vertex cover and | | gives a factor-2 approximation of the vertex cover number. In particular, if | | > 2 , then has no vertex cover of size at most . To see it, assume that ∈ ( ) ∩ ′ . Since | | ≥ 2, has no isolated vertices and, therefore, has a neighbor in , but then ∈ ∖ ; a contradiction. It proves that Step 3 is safe. Now we give an upper bound for the size of ′ . If ′ = ∅, then ( )∖ ⊆ ( ) and | ( ′ )| ≤ 2 ( + 1). If ′ ̸ = ∅, then ∖ ̸ = ∅ and, therefore, It is straightforward to see that Steps 1-4 can be done in polynomial time and it concludes the proof.
Recall that Chechik et al. [8] showed that if the treewidth of the input graph does not exceed , then the Secluded Steiner Tree problem is solvable in time 2 ( log ) · (1) · log , where is the maximum value of on an input graph . We observe that the running time could be improved by applying modern techniques for dynamic programming over tree decompositions proposed by Cygan et al. [9], Bodlaender et al. [3] and Fomin et al. [13]. Essentially, the algorithms for Secluded Steiner Tree are constructed along the same lines as the algorithms for Steiner Tree described in [9,3,13]. Hence, for simplicity, we only sketch the randomized algorithm based on the Cut&Count technique introduced by Cygan et al. [9] for Secluded Steiner Tree without costs in this conference version of our paper.
Theorem 11. There is a true-biased Monte Carlo algorithm solving the Secluded Steiner Tree without costs in time 4 · (1) , given a tree decomposition of width at most .
We need some additional definitions and auxiliary results. Let (ℬ, ) be a tree decomposition of a graph , ℬ = { | ∈ ( )}. We distinguish one vertex of which is said to be a root of . This introduces natural parent-child and ancestor-descendant relations in the tree . We say that a rooted tree decomposition (ℬ, ) is an extended nice tree decomposition if the following conditions are satisfied:

•
= ∅ and ℓ = ∅ for every leaf ℓ of . In other words, all the leaves as well as the root contain empty bags.
• For every edge ∈ ( ), there is the unique bag assigned to such that , ∈ ; we say that this bag is labeled by .
• Every non-leaf node of is of one of the following three types: -Introduce vertex node: a node ℎ with exactly one child ℎ ′ such that ℎ = ℎ ′ ∪ { } for some vertex / ∈ ℎ ′ ; we say that is introduced at ℎ.
-Introduce edge node: a node ℎ labeled with an edge ∈ ( ) such that , ∈ ℎ , and with exactly one child ℎ ′ such that ℎ = ℎ ′ . We say that edge is introduced at ℎ.
• All the edges incident to a vertex ∈ ( ) are introduced immediately after is introduced.
Using the same arguments as in [21], it is straightforward to show that for a given tree decomposition ( , ) of a graph of width , an extended nice tree decomposition of of width at most such that the total size of the obtained tree is ( 2 | ( )|) can be constructed in linear time.
Assume that each vertex ∈ is assigned an integer weight ( ). To use dynamic programming we relax the restriction that [ ] is connected.
Namely, we view as a union of two disjoint sets 0 and 1 between them. Let ℛ , be the set of all disjoint triples ( 0 , 1 , ) such that Note that any pair ( , ) satisfying (2) such that [ ] consists of connected components, contributes exactly 2 triples to ℛ , (just because each of the connected components can go to either 0 or 1 ). Hence if we compute |ℛ , | modulo 4 all pairs ( , ) with disconnected will cancel out.
Recall that we are given a tree decomposition of of width . Without loss of generality assume that the given tree decomposition is an extended nice decomposition. For a vertex ℎ ∈ ( ), let ℎ ⊆ ( ) be its bag, ℎ ⊆ ( ) and ℎ ⊆ ( ) be all the vertices end edges of respectively that are introduced in the subtree of rooted at ℎ, and ℎ be a graph on the vertex set ℎ containing all the edges introduced in that subtree.
By a coloring of a bag ℎ we mean a mapping : ℎ → {0 0 , 0 1 , 1 0 , 1 1 } assigning four different colors to the vertices of the bag.
• Red, represented by 1 0 . The meaning is that all red vertices have to be contained in 0 .
• Blue, represented by 1 1 . The meaning is that all blue vertices have to be conatained in 1 .
• Green, represented by 0 1 . The meaning is that all green vertices have to be contained in .
• White, represented by 0 0 . The meaning is that all white vertices do not appear in 0 ∪ 1 ∪ .
Given a coloring of a bag ℎ , we say that a triple ( 0 , 1 , ) of pairwise disjoint subsets of ℎ is nice with respect to and if • there are no edges between vertices from 0 and 1 in : • any neighbor of a vertex from 0 ∪ 1 lies in 0 ∪ 1 ∪ : if ∈ 0 ∪ 1 and ∈ ℎ then ∈ 0 ∪ 1 ∪ .
We are now ready to define a state of our dynamic programming algorithm: [ℎ, , , ] is the number modulo 4 of nice triples of size and weight with respect to ℎ and . Clearly, the number of states is ( · 4 · 3 ) (since is at most and is at most 4 2 ). Below we show how to compute all the states by going through the given tree decomposition from the leaves to the root.
Leaf node. If ℎ if a leaf node then ℎ = ∅. Then the only possible coloring is just the empty coloring and the only nice triple with respect to ℎ and this empty coloring is (∅, ∅, ∅). Hence for all , , Introduce vertex node. Let ℎ be an introduce node and ℎ ′ be its child such that ℎ = ℎ ′ ∪ { } for some ̸ ∈ ℎ ′ . Note that is an isolated vertex in ℎ . If is not a terminal vertex (i.e., ̸ ∈ ) it can be colored using any of our four colors. While if is a terminal vertex it should be colored either red or blue. We arrive at the following formula where each case is applied only if none of the previous cases is applicable: Introduce edge node. Let ℎ be an introduce edge with a child ′ such that ℎ = ℎ ′ ∪ { } for some , ∈ ℎ ′ and be a coloring of ℎ . Clearly any triple that is nice with respect to ℎ and is also nice with respect to ℎ ′ and . Hence all we need to do is to check whether all constraints are satisfied for the new edge . I.e., this edge should not join a blue vertex with a red one or a blue/red vertex with a white one. Formally, Join node. Let be a join node with children ℎ 1 and ℎ 2 such that ℎ = ℎ 1 = ℎ 2 . Let be a coloring of ℎ (and hence also a coloring of ℎ 1 and ℎ 2 ). Note that there is a natural one-to-one correspondence between nice triples for ℎ, and and pairs on nice triples for ℎ 1 , and ℎ 2 , . Namely, a nice triple ( 0 , 1 , ) for , defines a nice triple ( 1 0 , 1 1 , 1 ) for ℎ 1 , and a nice triple ( 2 0 , 2 1 , 2 ) for ℎ 2 , as follows ( = 1, 2): And vice versa, two nice triples ( 1 0 , 1 1 , 1 ) and ( 2 0 , 2 1 , 2 ) define a nice triple ( 0 , 1 , ) as follows: It is straightforward to check that the properties (4)-(6) are satisfied for both these maps. This allows us to use the following formula for computing the current state. Let ( ) = | −1 (0 1 ) ∪ −1 (1 0 ) ∪ −1 (1 1 )| and ( ) = ( −1 (1 0 ) ∪ −1 (1 1 )). Then To conclude, it remains to note that each node in the given tree decomposition is processed in time 4 · (1) .
The algorithm based on the Cut&Count technique can be generalized for Secluded Steiner Tree with costs in the same way as the algorithm for Steiner Tree in [9]. This way we can obtain the algorithm that runs in time 4 · ( + ) (1) where is the maximal cost of vertices. One can obtain a deterministic algorithm and improve the dependence on using the representative set technique for dynamic programming over tree decompositions introduced by Fomin et al. [13]. Again by the same approach as for Steiner Tree, it is possible to solve Secluded Steiner Tree deterministically in time ((2 + 2 +1 ) · ( + log ) (1) ) (here is the matrix multiplication constant).