In streaming algorithms, a graph is presented as a sequence of edges. In the simplest of this model, we have a stream of edge arrivals, where each edge adds to the graph seen so far, or may include a dynamic mixture of arrivals and departures of edges. In either case, the primary objective is to quickly answer some basic questions over the current state of the graph, such as finding a (maximal) matching over the current graph edges, or finding a (minimum) vertex cover, while storing only a small amount of information. In the most restrictive model, we only allow bits of space for storage. However, using methods from communication complexity one can show that most problems do not admit such algorithms. Thus one relaxes this notion and defines what is called a semi-streaming model, which allows bits of space. This model is extremely successful for graph streaming algorithms and a plethora of non-trivial algorithms has been made in this model [AKL16, GVV17, KKSV17]. There is a vast literature on graph streaming and we refer to the survey by McGregor [McG14] for more details.
The theme of this paper is parameterized streaming algorithms. So, before we go into parametrized streaming let us introduce a few basic definitions in parameterized complexity. The goal of parameterized complexity is to find ways of solving NP-hard problems more efficiently than brute force: the aim is to restrict the combinatorial explosion to a parameter that is hopefully much smaller than the input size. Formally, a parameterization of a problem is assigning an integer to each input instance. A parameterized problem is said to be fixed-parameter tractable (FPT) if there is an algorithm that solves the problem in time , where is the size of the input and is an arbitrary computable function depending only on the parameter . There is a long list of NP-hard graph problems that are FPT under various parameterizations: finding a vertex cover of size , finding a cycle of length , finding a maximum independent set in a graph of treewidth at most , etc. For more details, the reader is referred to the monograph [CFK15]. Given the definition of FPT for parameterized problems, it is natural to expect an efficient algorithm for the corresponding parameterized streaming versions to allow bits of space, where is an arbitrary computable function depending on the parameter .
There are several ways to formalize the parameterized streaming question, and in literature certain natural models are considered. The basic case is when the input of a given problem consists of a sequence of edge arrivals only, for which one seeks a parameterized streaming algorithm (PSA). It is more challenging when the input stream is dynamic, and contains both deletions and insertions of edges. In this case one seeks a dynamic parameterized streaming algorithm (DPSA). Notice that when an edge in the matching is deleted, we sometimes need substantial work to repair the solution and have to ensure that the algorithm has enough information to do so, while keeping only a bounded amount of working space. If we are promised that at every timestamp there is a solution of cost , then we seek a promised dynamic parameterized streaming algorithm (PDPSA). These notions were formalized in the following two papers [CCE16, CCE15] and several results for Vertex Cover and Maximum Matching were presented there. Unfortunately, this relaxation to bits of space does not buy us too many new results. Most of the problems for which parameterized streaming algorithms are known are “local problems”. Other local problems like Cluster Vertex Deletion and Triangle Deletion do not have positive results. Also, problems that require some global checking – such as Feedback Vertex Set, Even Cycle Transversal, Odd Cycle Transversal etc. remain elusive. In fact, one can show that, when edges of the graph arrive in an arbitrary order, using reductions from communication complexity all of the above problems will require space even if we allow a constant number of passes over the data stream [CCE16].
The starting point of this paper is the above mentioned lower bounds on basic graph problems. We ask the most natural question – how do we deconstruct these intractability results? When we look deeper we realize that, to the best of our knowledge the only parameter that has been used in parameterized streaming algorithms is the size of the solution that we are seeking. Indeed this is the most well-studied parameter, but there is no reason to only use solution size as a parameter.
[backgroundcolor=yellow!25] In parameterized complexity, when faced with such obstacles, we either study a problem with respect to parameters larger than the solution size or consider some structural parameters. We export this approach to parameterized streaming algorithms. This is our main conceptual contribution, that is, to introduce the concept of structural parameterizations to the study of parameterized streaming algorithms.
Parameters, models, problems and our results
What parameters to use? In parameterized complexity, after solution size and treewidth, arguably the most notable structural parameter is vertex cover size [CFK15, FJP14]. For all the vertex deletion problems that we consider in this paper, a vertex cover is also a solution. Thus, the vertex cover size is always larger than the solution size for all the above problems. We do a thorough study of vertex deletion problems from the view point of parameterized streaming in all known models and show dichotomy when moving across parameters and streaming models. The main conceptual contribution of this paper is to introduce structural parameter in parameterized streaming algorithms.
Streaming models The models that we consider are: (1) Edge Arrival (Ea) model; (2) Dynamic Edge Arrival (Dea) model; (3) Vertex Arrival (Va) model (In a step, a vertex is exposed along with all the edges between and already exposed neighbors of .); and (4) Adjacency List (Al) model (In a step, a vertex is exposed along with all edges incident on ). The formal definitions are in Section 2.
What problems to study? We study the streaming complexity of parameterized versions of -Subgraph deletion, -Minor deletion and Cluster Vertex Deletion (CVD). These problems are one of the most well studied ones in parametertized complexity [Cai96, CM15, CM16, FJP14, FLM16, FLMS12, KLP16, KP14, Mar10, RSV04, Tho10] and have led to development of the field. The parameters we consider in this paper are (i) the solution size and (ii) the size of the vertex cover of the input graph . In -Subgraph deletion, -Minor deletion and CVD, the objective is to decide whether there exists of size at most such that has no graphs in as a subgraph, has no graphs in as a minor and has no induced , respectively. -Subgraph deletion, -Minor deletion and CVD are interesting due to the following reasons. Feedback Vertex set (FVS), Even Cycle Transversal (ECT), Odd Cycle Transversal (OCT) and Triangle Deletion (TD) are special cases of -Subgraph deletion when , , and , respectively. FVS is also a special case of -Minor deletion when . CVD is different as we are looking for induced structures.
Our results. Let a graph and a non-negative integer be the inputs to the graph problems we consider. Notice that for -Subgraph deletion, -Minor deletion and CVD, . Interestingly, the parameter also has different effects on the above mentioned problems in the different streaming models. We show that structural parameters help to obtain efficient parameterized streaming algorithms for some of the problems, while no such effect is observed for other problems. This throws up the more general and deeper question in parameterized streaming complexity of classification of problems based on the different graph streaming models and different parameterization. We believe that our results and concepts will be instrumental in opening up the avenue for such studies in future.
In particular, we obtain a range of streaming algorithms as well as lower bounds on streaming complexity for the problems we consider. Informally, for a streaming model and a parameterized problem , if there is a -pass randomized streaming algorithm for that uses space then we say that is -streamable. Similarly, if there is no -pass algorithm using bits555It is usual in streaming that the lower bound results are in bits, and the upper bound results are in words. of storage then is said to be -hard. For formal definitions please refer to Section 2. When we omit , it means we are considering one pass of the input stream. The highlight of our results are captured by the -Subgraph deletion, -Minor deletion and CVD problems.
Consider -Subgraph deletion in the Al model. Parameterized by solution size , -Subgraph deletion is -hard. However, when parameterized by vertex cover , -Subgraph deletion is -streamable. Here is the maximum degree of any graph in .
The above Theorem is in contrast to results shown in [CCE16]. First, we would like to point out that to the best of our knowledge this is the first set of results on hardness in the Al model. The results in [CCE16] showed that -Subgraph deletion is -hard. A hardness result in the Al model implies one in the Ea model (Refer to Section 2). Thus, our result (Proof in Theorem 5.1) implies a stronger lower bound for -Subgraph deletion particularly in the Ea model. On the positive side, we show that -Subgraph deletion parameterized by the vertex cover size , is -streamable (Proof in Theorem 4.4).
Our hardness results are obtained from reductions from well-known problems in communication complexity. The problems we reduced from are , and (Please refer to Section 5.1 for details). In order to obtain the algorithm, one of the main technical contributions of this paper is the introduction of the Common Neighbor problem which plays a crucial role in designing streaming algorithms in this paper. We show that -Subgraph deletion and many of the other considered problems, like -Minor deletion parameterized by vertex cover size , have a unifying structure that can be solved via Common Neighbor, when the edges of the graph are arriving in the Al model. In Common Neighbor, the objective is to obtain a subgraph of the input graph such that the subgraph contains a maximal matching of . Also, for each pair of vertices 666 denotes the set of all vertices present in the matching , the edge is present in if and only if , and enough 777By enough, we mean in this case. common neighbors of all subsets of at most vertices of are retained in . Using structural properties of such a subgraph, called the common neighbor subgraph, we show that it is enough to solve -Subgraph deletion on the common neighbor subgraph. Similar algorithmic and lower bound results can be obtained for -Minor deletion. The following theorem can be proven using Theorem 4.7 in Section 4 and Theorem 5.1 in Section 5.
Consider -Minor deletion in the Al model. Parameterized by solution size , -Minor deletion is -hard. However, when parameterized by vertex cover , -Minor deletion is -streamable. Here is the maximum degree of any graph in .
The result on CVD is stated in the following Theorem.
Parameterized by solution size , CVD is -hard. However, when parameterized by vertex cover CVD is -streamable.
The CVD problem behaves very differently from the above two problems. We show that the problem is -hard (Theorem 5.3). In contrast, in [CCE16] the -hardness for the problem was shown, and we are able to extend this result to the Va model (Refer to Section 2 for relations between the models considered). Surprisingly, when we parameterize by , CVD is -streamable (Theorem 3.1). In fact, this implies -streamability for . To design our algorithm, we build on the sampling technique for Vertex Cover [CCE16] to solve CVD in Dea model. Our analysis of the sampling technique exploits the structure of a cluster graph.
Though we have mentioned the main algorithmic and lower bound result in the above theorems, we have a list of other algorithmic and lower bound results in the different streaming models. The full list of results are summed up in Table 1. To understand the full strength of our contribution, we request the reader to go to Section 2 to see the relations between different streaming models and the notion of hardness and streamability.
|Parameter||Al model||Va model||Ea/Dea model|
Related Work. Problems in class P have been extensively studied in streaming complexity in the last decade [McG14]. Recently, there has been a lot of interest in studying streaming complexity of NP-hard problems like Hitting Set, Set Cover, Max Cut and Max CSP [GVV17, KKSV17, AKL16]. Fafianie and Kratsch [FK14] were the first to study parameterized streaming complexity of NP-hard problems like -Hitting Set and Edge Dominating Set in graphs. Chitnis et al. [CCE16, CCE15, CCHM15] developed a sampling technique to design efficient parameterized streaming algorithms for promised variants of Vertex Cover, -Hitting Set problem, -Matching etc. They also proved lower bounds for problems like -Free Deletion, -Editing, Cluster Vertex Deletion etc. [CCE16].
Organisation of the paper. Section 2 contains preliminary definitions. Our algorithm for CVD is described in Section 3. The algorithms for Common Neighbor, -Subgraph deletion and -Minor deletion are given in Section 4. The lower bound results are in Section 5. Appendix A has all formal problem definitions.
2 Preliminaries, Model and Relationship Between Models
In this section we state formally the models of streaming algorithms we use in this paper, relationship between them and some preliminary notations that we make use of.
A promising prospect to deal with problems on large graphs is the study of streaming algorithms, where a compact sketch of the subgraph whose edges have been streamed/revealed so far, is stored and computations are done on this sketch. Algorithms that can access the sequence of edges of the input graph, times in the same order, are defined as -pass streaming algorithms. For simplicity, we refer to 1-pass streaming algorithms as streaming algorithms. The space used by a (-pass) streaming algorithm, is defined as the streaming complexity of the algorithm. The algorithmic model to deal with streaming graphs is determined by the way the graph is revealed. Streaming algorithms for graph problems are usually studied in the following models [CDK19, McG14, MVV16]. For the upcoming discussion, and will denote the vertex and edge set, respectively of the graph having vertices.
[noitemsep, wide=0pt, leftmargin=]
Edge Arrival (Ea) model: The stream consists of edges of in an arbitrary order.
Dynamic Edge Arrival (Dea) model: Each element of the input stream is a pair , where and describes whether is being inserted into or deleted from the current graph.
Vertex Arrival (Va) model: The vertices of are exposed in an arbitrary order. After a vertex is exposed, all the edges between and neighbors of that have already been exposed, are revealed. This set of edges are revealed one by one in an arbitrary order.
Adjacency List (Al) model: The vertices of are exposed in an arbitrary order. When a vertex is exposed, all the edges that are incident to , are revealed one by one in an arbitrary order. Note that in this model each edge is exposed twice, once for each exposure of an endpoint.
Streamability and Hardness.
Let be a parameterized graph problem that takes as input a graph on vertices and a parameter . Let be a computable function. For a model , whenever we say that an algorithm solves with complexity in model , we mean is a randomized algorithm that for any input instance of in model
gives the correct output with probabilityand has streaming complexity .
A parameterized graph problem , that takes an -vertex graph and a parameter as input, is -pass hard in the Edge Arrival model, or in short is -hard, if there does not exist any -pass streaming algorithm of streaming complexity bits that can solve in model .
Analogously, -hard, -hard and -hard are defined.
A graph problem , that takes an -vertex graph and a parameter as input, is -pass streamable in Edge Arrival model, or in short is -streamable if there exists a -pass streaming algorithm of streaming complexity words 888It is usual in streaming that the lower bound results are in bits, and the upper bound results are in words. that can solve in Edge Arrival model.
-streamable, -streamable and -streamable are defined analogously. For simplicity, we refer to -hard and -streamable as -hard and -streamable, respectively, where .
Let be two streaming models, be a computable function, and .
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If for any parameterized graph problem , -hardness of implies -hardness of , then we say .
If for any parameterized graph problem , -streamability of implies -streamability of , then we say .
; ; ; .
This observation has the following implication. If we prove a lower (upper) bound result for some problem in model , then it also holds in any model such that (). For example, if we prove a lower bound result in Al or Va model, it also holds in Ea and Dea model; if we prove an upper bound result in Dea model, it also holds in Ea, Va and Al model. In general, there is no direct connection between Al and Va. In Al and Va, the vertices are exposed in an arbitrary order. However, we can say the following when the vertices arrive in a fixed (known) order.
Let () be the restricted version of Al (Va), where the vertices are exposed in a fixed (known) order. Then and .
Now, we remark the implication of the relation between different models discussed in this section to our results mentioned in Table 1.
In Table 1, the lower bound results in Va and Al hold even if we know the sequence in which vertices are exposed, and the upper bound results hold even if the vertices arrive in an arbitrary order. In general, the lower bound in the Al model for some problem does not imply the lower bound in the Va model for . However, our lower bound proofs in the Al model hold even if we know the order in which vertices are exposed. So, the lower bounds for FVS, ECT, OCT in the Al model imply the lower bound in the Va model. By Observations 2.4 and 2.5, we will be done by showing a subset of the algorithmic and lower bound results mentioned in the Table 1.
The set is denoted as . Without loss of generality, we assume that the number of vertices in the graph is , which is a power of . Given an integer and , denotes the bit in the bit expansion of . The union of two graphs and with , is , where and . For , is the subgraph of induced by . The degree of a vertex , is denoted by . The maximum and average degrees of the vertices in are denoted as and , respectively. For a family of graphs , . A graph is a subgraph of a graph if and be the set of edges that can be formed only between vertices of . A graph is said to be a minor of a graph if can be obtained from by deleting edges and vertices and by contracting edges. The neighborhood of a vertex is denoted by . For , denotes the set of vertices in that are neighbors of every vertex in . A vertex is said to be a common neighbor of in . The size of any minimum vertex cover in is denoted by . A cycle on the sequence of vertices is denoted as . For a matching in , the vertices in the matching are denoted by . denotes a cycle of length . denotes a path having vertices. A graph is said to a cluster graph if is a disjoint union of cliques, that is, no three vertices of can form an induced .
3 Cvd in the Dea model
In this Section, we show that CVD parameterized by vertex cover size , is -streamable. By Observation 2.4, this implies -streamability for all . The sketch of the algorithm for CVD parameterized by vertex cover size in the Dea model is in Algorithm 1. The algorithm is inspired by the streaming algorithm for Vertex Cover [CCE16]. Before discussing the algorithm, let us discuss some terms.
A family of hash functions of the form is said to be pairwise independent hash family if for a pair and a randomly chosen from the family, . Such a hash function can be stored efficiently by using bits [MR95].
-sampler [CF14]: Given a dynamic graph stream, an -sampler does the following: with probability at least , where is a positive constant, it produces an edge uniformly at random from the set of edges that have been inserted so far but not deleted. If no such edge exists, -sampler reports Null. The total space used by the sampler is .
CVD, parameterized by vertex cover size , is -streamable.
Let be the input graph of the streaming algorithm and by assumption . Let be a set of pairwise independent hash functions such that each chosen uniformly and independently at random from a pairwise independent family of hash functions, where , and are suitable constants. For each hash function and pair , let be the subgraph of induced by the vertex set . For the hash function and for each pair , we initiate an sampler for the dynamic stream restricted to the subgraph . Therefore, there is a set of -samplers corresponding to the hash function . Now, we describe what our algorithm does when an edge is either inserted or deleted. A pseudocode of our algorithm for CVD is given in Algorithm 1. When an edge arrives in the stream, that is is inserted or deleted, we give the respective input to , where . At the end of the stream, for each , we construct a subgraph by taking the outputs of all the -samplers corresponding to the hash function . Let . We run the classical FPT algorithm for CVD on the subgraph and solution size bound [CFK15], and report YES to CVD if and only if we get YES as answer from the above FPT algorithm on . If we output YES , then we also give the solution on as our solution to .
The correctness of the algorithm needs an existential structural result on (Claim 3.2) and the fact that if there exists a set whose deletion turns into a cluster graph, then the same deleted from will turn it into a cluster graph with high probability (Claim 3.3).
There exists a partition of into such that the subgraph induced in by each , is a clique with at least vertices, and the subgraph induced by is the empty graph.
Proof of Claim 3.2.
We start with a partition which may not have the properties of the claim and modify it iteratively such that the final partition does have all the properties of the Claim. Let us start with a partition that does not satisfy the given condition. First, if there exists a part having one vertex , we create a new partition by adding to . Next, if there exists a part having at least two vertices and the subgraph induced by is not a clique, then we partition into smaller parts such that each smaller part is either a clique having at least two vertices or a singleton vertex. We create a new partition by replacing with the smaller cliques of size at least and adding all the singleton vertices to . Now, let be the new partition of obtained after all the above modifications. In , each part except is a clique of at least two vertices. If the subgraph induced by has no edges, satisfies the properties in the Claim and we are done. Otherwise, there exists such that . In this case, we create a new part with , and remove both and from . Note that in the above iterative description, each vertex goes to a new part at most times - (i) it can move at most once from a part to a smaller part that is a clique on at least vertices and such a vertex will remain in the same part in all steps afterwards, or it can move at most once from a to , and (ii) a vertex can move at most once from to become a part of a clique with at least vertices and such a vertex will remain in the same part in all steps after that. Therefore, this process is finite and there is a final partition that we obtain in the end. This final partition has all the properties of the claim. ∎
Let be such that is a cluster graph. Then is a cluster graph with high probability.
Consider a partition of into as mentioned in Claim 3.2. Note that our algorithm does not need to find such a partition. The existence of will be used only for the analysis purpose. Let . Note that since , each can have at most vertices, and it must be true that . In fact, we can obtain the following stronger bound that . The total number of vertices in is at most . Since , the total number of vertices in is at most .
A vertex , is said to be of high degree if , and low degree, otherwise. Let be the set of all high degree vertices and be the set of low degree vertices in . Let be the set of edges in having both the endpoints in . It can be shown [CCE16] that
Fact-1: , ;
Fact-2: , and for each , with probability at least .
Note that Fact-2 makes our algorithmic result for CVD probabilistic.
Let denote a minimum set of vertices such that is a cluster graph. Our parametric assumption says that . Now consider the fact that a graph is a cluster graph if and only if it does not have any induced . First, we show that the high degree vertices in surely need to be deleted to make it a cluster graph, i.e., . Let us consider a vertex . As the subgraph induced by has no edges and , each vertex in is of degree at most . So, must be in some in the partition . As , using , must have at least many vertices from as its neighbors in . Thus, there are at least edge disjoint induced ’s that are formed with and its neighbors in . If , then more than neighbors of that are in must be present in . It will contradict the fact that . Similarly, we can also argue that as by Fact-2.
Next, we show that an induced is present in if and only if it is present in . Removal of from removes all the induced ’s in having at least one vertex in . Any induced in (or ) must have all of its vertices as low degree vertices. Now, using Fact-2, note that all the edges, in , between low degree vertices are in . In other words, an induced is present in if and only if it is present in . Thus for a set , if is a cluster graph then is also a cluster graph.
Putting everything together, if is such that is a cluster graph, then is also a cluster graph. ∎
Coming back to the proof of Theorem 3.1, we are using hash functions, and each hash function requires a storage of bits. There are -samplers for each hash function and each -sampler needs bits of storage. Thus, the total space used by our algorithm is . ∎
4 Deterministic algorithms in the Al model
In this Section, we show that -Subgraph deletion is -streamable when the vertex cover of the input graph is parameterized by . This will imply that FVS, ECT, OCT and TD parameterized by vertex cover size , are -streamable. This complements the results in Theorems 5.1 and 5.2 (in Section 5) that show that the problems parameterized by vertex cover size are -hard (see also Table 1). Note that by Observation 2.4, this also implies that the problems parameterized by vertex cover size are -hard when . Finally, we design an algorithm for -Minor deletion that is inspired by the algorithm for -Subgraph deletion.
For the algorithm for -Subgraph deletion, we define an auxiliary problem Common Neighbor and a streaming algorithm for it. This works as a subroutine for our algorithm for -Subgraph deletion.
4.1 Common Neighbor problem
For a graph and a parameter , will be called a common neighbor subgraph for if
such that has no isolated vertex;
contains the edges
of a maximal matching of along with the edges where both the endpoints are from ,
such that for each subset , , , that is, contains edges to at most common neighbors of in .
In simple words, a common neighbor subgraph of contains the subgraph of induced by as a subgraph of for some maximal matching in . Also, for each subset of at most vertices in , contains edges to sufficient common neighbors of in . The parameters and are referred to as the degree parameter and common neighbor parameter, respectively.
The Common Neighbor problem is formally defined as follows. It takes as input a graph with , degree parameter and common neighbor parameter and produces a common neighbor subgraph of as the output. Common Neighbor parameterized by vertex cover size , has the following result.
Common Neighbor, with a commmon neighbor parameter and parameterized by vertex cover size , is -streamable.
We start our algorithm by initializing and construct a matching in that is maximal under inclusion; See Algorithm 2. As , . Recall that we are considering the Al model here. Let and be the maximal matchings just before and after the exposure of the vertex (including the processing of the edges adjacent to ), respectively. Note that, by construction these partial matchings and are also maximal matchings in the subgraph exposed so far. The following Lemma will be useful for the proof.
Let for some . Then , that is, is exposed, after all the vertices in are declared as vertices of .
Observe that if there exists such that , then after is exposed, there exists such that is present in . This implies , which is a contradiction to . ∎
Now, we describe what our algorithm does when a vertex is exposed. A complete pseudocode of our algorithm for Common Neighbor is given in Algorithm 2. When a vertex is exposed in the stream, we try to extend the maximal matching . Also, we store all the edges of the form such that , in a temporary memory . As , we are storing at most many edges in . Now, there are the following possibilities.
If , that is, either or the matching is extended by one of the edges stored in , then we add all the edges stored in to .
Otherwise, for each such that and , we check whether the number of common neighbors of the vertices present in , that are already stored, is less than . If yes, we add all the edges of the form such that to ; else, we do nothing. Now, we reset to .
As , . We are storing at most common neighbors for each with and the number of edges having both the endpoints in is at most , the total amount of space used is at most . ∎
We call our algorithm described in the proof of Lemma 4.1 and given in Algorithm 2, as . The following structural Lemma of the common neighbor subgraph of , obtained by algorithm is important for the design and analysis of streaming algorithms for -Subgraph deletion. The proof of this structural result is similar to that in [FJP14].
Let be a graph with and let be a connected graph with . Let be the common neighbor subgraph of with degree parameter and common neighbor parameter , obtained by running the algorithm . Then the following holds in : For any subset , where , is a subgraph of if and only if is a subgraph of , such that and are isomorphic.
Let the common neighbor subgraph , obtained by algorithm , contain a maximal matching of . First, observe that since , the size of a subgraph in is at most . Now let us consider a subset such that . First, suppose that is a subgraph of and is isomorphic to . Then since is a subgraph of , is also a subgraph of . Therefore, and we are done.
Conversely, suppose is a subgraph of that is not a subgraph in . We show that there is a subgraph of such that is isomorphic to . Consider an arbitrary ordering ; note that . We describe an iterative subroutine that converts the subgraph to through steps, or equivalently, through a sequence of isomorphic subgraphs in such that and .
Let us discuss the consequence of such an iterative routine. Just before the starting of step , we have the subgraph such that is isomorphic to and the set of edges in is a subset of . In step , we convert the subgraph into such that is isomorphic to . Just after the step , we have the subgraph such that is isomorphic to and the set of edges in is a subset of . In particular, in the end is a subgraph both in and .
Now consider the instance just before step . We show how we select the subgraph from . Let . Note that . By the definition of the maximal matching in , it must be the case that . From the construction of the common neighbor subgraph , if both and are in , then . So, exactly one of and is present in . Without loss of generality, let . Observe that is a common neighbor of in . Because of the maximality of , each vertex in is present in . Now, as , is not a common neighbor of in . From the construction of the common neighbor subgraph, contains common neighbors of all the vertices present in . Of these common neighbors, at most common neighbors can be vertices in . Thus, there is a vertex that is a common neighbor of all the vertices present in in such that is a subgraph that is isomorphic to . Moreover, . Thus, this leads to the fact that there is a subgraph in that is isomorphic to the subgraph in . ∎
4.2 Streambality results for -Subgraph deletion and -Minor deletion
Our result on Common Neighbor leads us to the following streamability result for -Subgraph deletion and -Minor deletion. We first discuss the result on -Subgraph deletion, which is stated in the following theorem.
-Subgraph deletion parameterized by vertex cover size is -streamable, where .
Let be an input for -Subgraph deletion, where is the input graph, is the size of the solution of -Subgraph deletion, and the parameter is at least .
Now, we describe the streaming algorithm for -Subgraph deletion. First, we run the Common Neighbor streaming algorithm described in Lemma 4.1 (and given in Algorithm 2) with degree parameter and common neighbor parameter , and let the common neighbor subgraph obtained be . We run a traditional FPT algorithm for -Subgraph deletion [CFK15] on and output YES if and only if the output on is YES.
Let us argue the correctness of this algorithm. By Lemma 4.3, for any subset , where , is a subgraph of if and only if , such that is isomorphic to , is a subgraph of . In particular, let be a -sized vertex set of . As mentioned before, . Thus, by Lemma 4.3, is a solution of -Subgraph deletion in if and only if is a solution of -Subgraph deletion in . Therefore, we are done with the correctness of the streaming algorithm for -Subgraph deletion.
The streaming complexity of -Subgraph deletion is same as the streaming complexity for the algorithm from Lemma 4.1 with degree parameter and common neighbor parameter . Therefore, the streaming complexity of -Subgraph deletion is . ∎
FVS, ECT, OCT and TD parameterized by vertex cover size are -streamable due to deterministic algorithms.
4.3 Algorithm for -Minor deletion
Finally, we describe a streaming algorithm for -Minor deletion that works similar to that of -Subgraph deletion due to the following proposition and the result is stated in Theorem 4.7.
Proposition 4.6 ([Fjp14]).
Let be a graph with as a minor and . Then there exists a subgraph of that has as a minor such that and
-Minor deletion parameterized by vertex cover size are -streamable, where .
Let be an input for -Minor deletion, where is the input graph, is the size of the solution of -Minor deletion we are looking for, and the parameter is such that . Note that, .
Now, we describe the streaming algorithm for -Minor deletion. First, we run the Common Neighbor streaming algorithm described in Lemma 4.1 with degree parameter and common neighbor parameter , and let the common neighbor subgraph obtained be . We run a traditional FPT algorithm for -Minor deletion [CFK15] and output YES if and only if the output on is YES.
Let us argue the correctness of this algorithm, that is, we prove the following for any . contains as a minor if and only if contains as a minor such that and are isomorphic, where is of size at most . For the only if part, suppose contains as a minor. Then since is a subgraph of , contains as a minor. For the if part, let contains as a minor. By Proposition 4.6, conatins a subgraph such that contains as a minor and . Now, Lemma 4.3 implies that also contains a subgraph that is isomorphic to . Hence, contains as a monor such that is isomorphic to .
The streaming complexity of the streaming algorithm for -Minor deletion is same as the streaming complexity for the algorithm from Lemma 4.1 with degree parameter and common neighbor parameter . Therefore, the streaming complexity for -Minor deletion is . ∎
5 The Lower Bounds
Before we prove the lower bound results presented in Table 1, note that a lower bound on Feedback Vertex Set is also a lower bound for -Subgraph deletion (deletion of cycles as subgraphs) and -Minor deletion (deletion of 3-cycles as minors). Thus, we will be done by proving the following theorems; Observations 2.4 and 2.5 imply the other hardness results.
Feedback Vertex Set