A Generalization of Nemhauser and Trotter’s Local Optimization Theorem

A Generalization of Nemhauser and Trotter’s
Local Optimization Theorem

M.R. Fellows michael.fellows@newcastle.edu.au J. Guo H. Moser  and  R. Niedermeier (guo,moser,niedermr)@minet.uni-jena.de
Abstract.

The Nemhauser-Trotter local optimization theorem applies to the NP-hard Vertex Cover problem and has applications in approximation as well as parameterized algorithmics. We present a framework that generalizes Nemhauser and Trotter’s result to vertex deletion and graph packing problems, introducing novel algorithmic strategies based on purely combinatorial arguments (not referring to linear programming as the Nemhauser-Trotter result originally did).

We exhibit our framework using a generalization of Vertex Cover, called Bounded-Degree Deletion, that has promise to become an important tool in the analysis of gene and other biological networks. For some fixed , Bounded-Degree Deletion asks to delete as few vertices as possible from a graph in order to transform it into a graph with maximum vertex degree at most . Vertex Cover is the special case of . Our generalization of the Nemhauser-Trotter theorem implies that Bounded-Degree Deletion has a problem kernel with a linear number of vertices for every constant . We also outline an application of our extremal combinatorial approach to the problem of packing stars with a bounded number of leaves. Finally, charting the border between (parameterized) tractability and intractability for Bounded-Degree Deletion, we provide a W[2]-hardness result for Bounded-Degree Deletion in case of unbounded -values.

Key words and phrases:
Algorithms, computational complexity, NP-hard problems, W[2]-completeness, graph problems, combinatorial optimization, fixed-parameter tractability, kernelization
The first author was supported by the Australian Research Council. Work done while staying in Jena as a recipient of the Humboldt Research Award of the Alexander von Humboldt Foundation, Bonn, Germany. The second author was supported by the DFG, Emmy Noether research group PIAF, NI 369/4, and project DARE, GU 1023/1. The third author was supported by the DFG, projects ITKO, NI 369/5, and AREG, NI 369/9.
copyright: ©:

Michael R. Fellows \@ifemptypcru

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pcru Jiong Guo Hannes Moser Rolf Niedermeier \@ifemptyjena

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jena


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section1[Introduction]Introduction Nemhauser and Trotter [20] proved a famous theorem in combinatorial optimization. In terms of the NP-hard Vertex Cover111Vertex Cover is the following problem: Given an undirected graph, find a minimum-cardinality set  of vertices such that each edge has at least one endpoint in . problem, it can be formulated as follows:

NT-Theorem [20, 4]. For an undirected graph  one can compute in polynomial time two disjoint vertex subsets  and , such that the following three properties hold:

  1. If  is a vertex cover of the induced subgraph , then  is a vertex cover of .

  2. There is a minimum-cardinality vertex cover  of  with .

  3. Every vertex cover of the induced subgraph  has size at least .

In other words, the NT-Theorem provides a polynomial-time data reduction for Vertex Cover. That is, for vertices in  it can already be decided in polynomial time to put them into the solution set and vertices in  can be ignored for finding a solution. The NT-Theorem is very useful for approximating Vertex Cover. The point is that the search for an approximate solution can be restricted to the induced subgraph . The NT-Theorem directly delivers a factor- approximation for Vertex Cover by choosing  as the vertex cover. Chen et al. [7] first observed that the NT-Theorem directly yields a -vertex problem kernel for Vertex Cover, where the parameter  denotes the size of the solution set. Indeed, this is in a sense an “ultimate” kernelization result in parameterized complexity analysis [10, 11, 21] because there is good reason to believe that there is a matching lower bound  for the kernel size unless PNP [16].

Since its publication numerous authors have referred to the importance of the NT-Theorem from the viewpoint of polynomial-time approximation algorithms (e.g., [4, 17]) as well as from the viewpoint of parameterized algorithmics (e.g., [1, 7, 9]). The relevance of the NT-Theorem comes from both its practical usefulness in solving the Vertex Cover problem as well as its theoretical depth having led to numerous further studies and follow-up work [1, 4, 9]. In this work, our main contribution is to provide a more general and more widely applicable version of the NT-Theorem. The corresponding algorithmic strategies and proof techniques, however, are not achieved by a generalization of known proofs of the NT-Theorem but are completely different and are based on extremal combinatorial arguments. Vertex Cover can be formulated as the problem of finding a minimum-cardinality set of vertices whose deletion makes a graph edge-free, that is, the remaining vertices have degree . Our main result is to prove a generalization of the NT-Theorem that helps in finding a minimum-cardinality set of vertices whose deletion leaves a graph of maximum degree  for arbitrary but fixed . Clearly, is the special case of Vertex Cover.


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paragraph4[Motivation.]Motivation. Since the NP-hard Bounded-Degree Deletion problem—given a graph and two positive integers  and , find at most  vertices whose deletion leaves a graph of maximum vertex degree —stands in the center of our considerations, some more explanations about its relevance follow. Bounded-Degree Deletion (or its dual problem) already appears in some theoretical work, e.g., [6, 18, 22], but so far it has received considerably less attention than Vertex Cover, one of the best studied problems in combinatorial optimization [17]. To advocate and justify more research on Bounded-Degree Deletion, we describe an application in computational biology. In the analysis of genetic networks based on micro-array data, recently a clique-centric approach has shown great success [3, 8]. Roughly speaking, finding cliques or near-cliques (called paracliques [8]) has been a central tool. Since finding cliques is computationally hard (also with respect to approximation), Chesler et al. [8, page 241] state that “cliques are identified through a transformation to the complementary dual Vertex Cover problem and the use of highly parallel algorithms based on the notion of fixed-parameter tractability.” More specifically, in these Vertex Cover-based algorithms polynomial-time data reduction (such as the NT-Theorem) plays a decisive role [19] (also see [1]) for efficient solvability of the given real-world data. However, since biological and other real-world data typically contain errors, the demand for finding cliques (that is, fully connected subgraphs) often seems overly restrictive and somewhat relaxed notations of cliques are more appropriate. For instance, Chesler et al. [8] introduced paracliques, which are achieved by greedily extending the found cliques by vertices that are connected to almost all (para)clique vertices. An elegant mathematical concept of “relaxed cliques” is that of -plexes222Introduced in 1978 by Seidman and Foster [24] in the context of social network analysis. Recently, this concept has again found increased interest [2, 18]. where one demands that each -plex vertex does not need to be connected to all other vertices in the -plex but to all but . Thus, cliques are -plexes. The corresponding problem to find maximum-cardinality -plexes in a graph is basically as computationally hard as clique detection is [2, 18]. However, as Vertex Cover is the dual problem for clique detection, Bounded-Degree Deletion is the dual problem for -plex detection: An -vertex graph has an -plex of size  iff its complement graph has a solution set for Bounded-Degree Deletion with  of size , and the solution sets can directly be computed from each other. The Vertex Cover polynomial-time data reduction algorithm has played an important role in the practical success story of analyzing real-world genetic and other biological networks [3, 8]. Our new polynomial-time data reduction algorithms for Bounded-Degree Deletion have the potential to play a similar role.


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paragraph4[Our results.]Our results. Our main theorem can be formulated as follows.

BDD-DR-Theorem (Theorem 2). For an undirected -vertex and -edge graph , we can compute two disjoint vertex subsets  and  in  time, such that the following three properties hold:

  1. If  is a solution set for Bounded-Degree Deletion of the induced subgraph , then  is a solution set for Bounded-Degree Deletion of .

  2. There is a minimum-cardinality solution set  for Bounded-Degree Deletion of  with .

  3. Every solution set for Bounded-Degree Deletion of the induced subgraph  has size at least

In terms of parameterized algorithmics, this gives a -vertex problem kernel for Bounded-Degree Deletion, which is linear in  for constant -values, thus joining a number of other recent “linear kernelization results” [5, 12, 14, 15]. Our general result specializes to a -vertex problem kernel for Vertex Cover (the NT-Theorem provides a size- problem kernel), but applies to a larger class of problems. For instance, a slightly modified version of the BDD-DR-Theorem (with essentially the same proof) yields a -vertex problem kernel for the problem of packing at least  vertex-disjoint length- paths of an input graph, giving the same bound as shown in work focussing on this problem [23].333Very recently, Wang et al. [25] improved the -bound to a -bound. We claim that our kernelization based on the BDD-DR-Theorem method can be easily adapted to also deliver the -bound. For the problem, where, given an undirected graph, one seeks a set of at least  vertex-disjoint stars444A star is a tree where all of the vertices but one are leaves. of the same constant size, we show that a kernel with a linear number of vertices can be achieved, improving the best previous quadratic kernelization [23]. We emphasize that our data reduction technique is based on extremal combinatorial arguments; the resulting combinatorial kernelization algorithm has practical potential and implementation work is underway. Note that for  our algorithm computes the same type of structure as in the “crown decomposition” kernelization for Vertex Cover (see, for example, [1]). However, for  the structure returned by our algorithm is much more complicated; in particular, unlike for Vertex Cover crown decompositions, in the BDD-DR-Theorem the set  is not necessarily a separator and the set  does not necessarily form an independent set.

Exploring the borders of parameterized tractability of Bounded-Degree Deletion for arbitrary values of the degree value , we show the following.

Theorem 1.

For unbounded  (given as part of the input), Bounded-Degree Deletion is -complete with respect to the parameter  denoting the number of vertices to delete.

In other words, there is no hope for fixed-parameter tractability with respect to the parameter  in the case of unbounded -values. Due to the lack of space the proof of Theorem 1 and several proofs of lemmas needed to show Theorem 2 are omitted.


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section1[Preliminaries]Preliminaries

A bdd--set for a graph  is a vertex subset whose removal from  yields a graph in which each vertex has degree at most . The central problem of this paper is

Bounded-Degree Deletion

Input:

An undirected graph , and integers  and .

Question:

Does there exist a bdd--set  of size at most  for ?

In this paper, for a graph  and a vertex set , let  be the subgraph of  induced by  and . The open neighborhood of a vertex  or a vertex set  in a graph  is denoted as  and , respectively. The closed neighborhood is denoted as  and . We write  and  to denote the vertex and edge set of , respectively. A packing  of a graph  is a set of pairwise vertex-disjoint subgraphs of . A graph has maximum degree  when every vertex in the graph has degree at most . A graph property is called hereditary if every induced subgraph of a graph with this property has the property as well.

Parameterized algorithmics [10, 11, 21] is an approach to finding optimal solutions for NP-hard problems. A common method in parameterized algorithmics is to provide polynomial-time executable data reduction rules that lead to a problem kernel [13]. This is the most important concept for this paper. Given a parameterized problem instance , a data reduction rule replaces  by an instance  in polynomial time such that , and  is a Yes-instance if and only if  is a Yes-instance. A parameterized problem is said to have a problem kernel, or, equivalently, kernelization, if, after the exhaustive application of the data reduction rules, the resulting reduced instance has size  for a function  depending only on . Roughly speaking, the kernel size  plays a similar role in the subject of problem kernelization as the approximation factor plays for approximation algorithms.


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section1[A Local Optimization Algorithm for Bounded-Degree Deletion]A Local Optimization Algorithm for Bounded-Degree Deletion The main result of this section is the following generalization of the Nemhauser-Trotter-Theorem [20] for Bounded-Degree Deletion with constant .

Theorem 2 (BDD-DR-Theorem).

For an -vertex and -edge graph , we can compute two disjoint vertex subsets  and  in  time, such that the following three properties hold:

  1. If  is a bdd--set of , then  is a bdd--set of .

  2. There is a minimum-cardinality bdd--set  of  with .

  3. Every bdd--set of  has size at least .

This first two properties are called the local optimality conditions. The remainder of this section is dedicated to the proof of this theorem. More specifically, we present an algorithm called compute_AB (see Figure 1) which outputs two sets  and  fulfilling the three properties given in Theorem 2. The core of this algorithm is the procedure find_extremal (see Figure 2) running in  time. This procedure returns two disjoint vertex subsets  and  that, among others, satisfy the local optimality conditions. The procedure is iteratively called by compute_AB. The overall output sets  and  then are the union of the outputs of all applications of find_extremal. Actually, find_extremal searches for  satisfying the following two conditions:

  1. Each vertex in  has degree at most  in , and

  2. is a minimum-cardinality bdd--set for .

It is not hard to see that these two conditions are stronger than the local optimality conditions of Theorem 2:

Lemma 1.

Let  and  be two vertex subsets satisfying conditions C1 and C2. Then, the following is true:

  1. If  is a bdd--set of , then  is a bdd--set of .

  2. There is a minimum-cardinality bdd--set  of  with .

Lemma 1 will be used in the proof of Theorem 2—it helps to make the description of the underlying algorithm and the corresponding correctness proofs more accessible. As a direct application of Theorem 2, we get the following corollary.

Corollary 1.

Bounded-Degree Deletion with constant  admits a problem kernel with at most  vertices, which is computable in  time.

We use the following easy-to-verify forbidden subgraph characterization of bounded-degree graphs: A graph  has maximum degree  if and only if there is no “-star” in .

Definition 0.1.

For , the graph  is called an -star. The vertex  is called the center of the star. The vertices  are the leaves of the star. A -star is an -star with .

Due to this forbidden subgraph characterization of bounded-degree graphs, we can also derive a linear kernelization for the -Star Packing problem. In this problem, given an undirected graph, one seeks for at least  vertex-disjoint -stars for a constant . With a slight modification of the proof of Theorem 2, we get the following corollary.

Corollary 2.

-Star Packing admits a problem kernel with at most  vertices, which is computable in  time.

For , the best known kernelization result was a  kernel [23]. Note that the special case of -Star Packing with  is also called -Packing, a problem well-studied in the literature, see [23, 25]. Corollary 2 gives a -vertex problem kernel. The best-known bound is  [25]. However, the improvement from the formerly best bound  [23] is achieved by improving a properly defined witness structure by local modifications. This trick also works with our approach, that is, we can show that the NT-like approach also yields a -vertex problem kernel for -Star Packing.


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subsection2[The Algorithm]The Algorithm We start with an informal description of the algorithm. As stated in the introduction of this section, the central part is Algorithm compute_AB shown in Figure 1.

  Algorithm: compute_AB 

Input: An undirected graph .

Output: Vertex subsets  and  satisfying the three properties of Theorem 2.

  1. Compute a witness  and the corresponding residual  for 

  2. If  then return 

  3. find_extremal .

  4. goto line 2

 

Figure 1. Pseudo-code of the main algorithm for computing  and .

Using the characterization of bounded-degree graphs by forbidding large stars, in line 2 compute_AB starts with computing two vertex sets  and : First, with a straightforward greedy algorithm, compute a maximal -star packing  of , that is, a set of vertex-disjoint -stars that cannot be extended by adding another -star. Let  be the set of vertices of the star packing. Since the number of stars in the packing is a lower bound for the size of a minimum bdd--set,  is a factor- approximate bdd--set. Greedily remove vertices from  such that  is still a bdd--set, and finally set . We call  the witness and  the corresponding residual.

  Procedure: find_extremal 

Input: An undirected graph , witness , and residual .

Output: Vertex subsets  and  satisfying the local optimality conditions.

  1. bipartite graph with  and  as its two vertex subsets and

  2. Initialize empty set of forbidden vertices

  3. start with  and while  do Loop while not all vertices in  are forbidden

  4.   Determine forbidden vertices in 

  5.   star-packing

  6.   Vertices in  that are not forbidden and not in 

  7.   start with  and repeat Start search for  satisfying C2

  8.  

  9.  

  10.  

  11.   until 

  12.  

  13.   if  then also satisfy C1

  14.   return 

  15.   Determine forbidden vertices in  for next iteration

  16.  

  17. end while

  18. Recompute forbidden vertices in  (as in line 4)

  19. return 

  Procedure: star-packing 

Input: A bipartite graph  with two vertex subsets  and .

Output: A maximum-edge packing of stars that have their centers in  and have at most  leaves in .

See Lemma 2, the straightforward implementation details using matching techniques are omitted.

Figure 2. Pseudo-code of the procedure computing the intermediary vertex subset pair .

If the residual  is too big (condition in line 3), the sets  and  are passed in line 4 to the procedure find_extremal in Figure 2 which computes two sets  and  satisfying conditions C1 and C2. Computing  and  represents the first step to find a subset pair satisfying condition C1: Since there is no vertex that has degree more than  in  (due to the fact that  is a bdd--set), the search is limited to those subset pairs where  is a subset of the witness  and  is a subset of .

Algorithm compute_AB calls find_extremal iteratively until the sets  and , which are constructed by the union of the outputs of all applications of find_extremal (see line 5), satisfy the third property in Theorem 2. In the following, we intuitively describe the basic ideas behind find_extremal.

To construct the set  from , we compute again a star packing  with the centers of the stars being from  and the leaves being from . We relax, on the one hand, the requirement that the stars in the packing have exactly  leaves, that is, the packing  might contain -stars. On the other hand,  should have a maximum number of edges. The rough idea behind the requirement for a maximum number of edges is to maximize the number of -stars in  in the course of the algorithm. Moreover, we can observe that, by setting  equal to the center set of the -stars in  and  equal to the leaf set of the -stars in  is a minimum bdd--set of  (condition C2). We call such a packing a maximum-edge -center -star packing. For computing , the algorithm constructs an auxiliary bipartite graph  with  as one vertex subset and  as the other. The edge set of  consists of the edges in  with exactly one endpoint in . See line 1 of Figure 2. Obviously, a maximum-edge -center -star packing of  corresponds one-to-one with a maximum-edge packing of stars in  that have their centers in  and have at most  leaves in the other vertex subset. Then, the star packing  can be computed by using techniques for computing maximum matchings in  (in the following, let star-packing(,,,) denote an algorithm that computes a maximum-edge -center -star packing  on the bipartite graph ).

The most involved part of find_extremal in Figure 2 is to guarantee that the output subsets in line 4 fulfill condition C1. To this end, one uses an iterative approach to compute the star packing . Roughly speaking, in each iteration, if the subsets  and  do not fulfill condition C1, then exclude from further iterations the vertices from  that themselves or whose neighbors violate this condition. See lines 2 to 15 of Figure 2 for more details of the iterative computation. Herein, for , the sets  and , where  is initialized with the empty set, and  is computed using , store the vertices excluded from computing . To find the vertices that themselves cause the violation of the condition, that is, vertices in  that have neighbors in , one uses an augmenting path computation in lines 7 to 11 to get in line 12 subsets  and  such that the vertices in  do not themselves violate the condition. Roughly speaking, the existence of an edge  from some vertex in  to some vertex in  would imply that the -star packing is not maximum (witnessed by an augmenting path beginning with —in principle, this idea is also used for finding crown decompositions, cf. [1]). The vertices whose neighbors cause the violation of condition C1 are all vertices in  with neighbors in  that themselves have neighbors in . These neighbors in  and the corresponding vertices in  are excluded in line 4 and line 18. We will see that the number of all excluded vertices is , thus, in total, we do not exclude too many vertices with this iterative method. The formal proof of correctness is given in the following subsection.


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subsection2[Running Time and Correctness]Running Time and Correctness Now, we show that compute_AB in Figure 1 computes in the claimed time two vertex subsets  and  that fulfill the three properties given in Theorem 2.


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subsubsection3[Running Time of find_extremal.]Running Time of find_extremal. We begin with the proof of the running time of the procedure find_extremal in Figure 2, which uses the following lemmas.

Lemma 2.

Procedure star-packing in Figure 2 runs in  time.

The next lemma is also used for the correctness proof; in particular, it guarantees the termination of the algorithm.

Lemma 3.

If the condition in line 13 of Figure 2 is false for a , then .

Proof.

In lines 4 and 5 of Figure 2, all vertices in  and their neighbors  are excluded from the star packing  in the th iteration of the outer loop. Moreover, the vertices in  are excluded from the set  (line 6). Therefore, a vertex in  cannot be added to  in line 12. Thus  (set to  in line 15) contains . Moreover, this containment is proper, as otherwise the condition in line 13 would be true.    

Lemma 4.

Procedure find_extremal runs in  time.


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subsubsection3[Correctness of find_extremal.]Correctness of find_extremal. The correctness proof for find_extremal in Figure 2 is more involved than its running time analysis. The following lemmas provide some properties of  which are needed.

Lemma 5.

For each  the following properties hold after the execution of line 12 in Figure 2:

  1. every vertex in  is a center vertex of a -star in , and

  2. the leaves of every star in  with center in  are vertices in .

Proof.

(Sketch) To prove (1), first of all, we show that  implies , since, otherwise, we could get a -augmenting path from some element in  to . A -augmenting path is a path where the edges in  and the edges not in  alternate, and the first and the last edge are not in . This -augmenting path can be constructed in an inductive way by simulating the construction of  in lines 6 to 11 of Figure 2. From this -augmenting path, we can then construct a -center -star packing that has more edges than , contradicting that  has maximum cardinality. Second, every vertex in  is a center of a star due to the definition of  and Procedure star-packing. Finally, if a vertex  is the center of a star with less than  leaves, then again we get a -augmenting path from some element in  to .

The second statement follows easily from Procedure star-packing and the pseudo-code in lines 6 to 12.    

Lemma 6.

For each  there is no edge in  between  and .

Proof.

The vertices in  and the vertices in  are excluded from the computation of  and are not contained in  (lines 4 to 6 in Figure 2). Thus, and therefore there are no edges in  between  and .    

The next lemma shows that the output of find_extremal fulfills the local optimality conditions.

Lemma 7.

Procedure find_extremal returns two disjoint vertex subsets fulfilling conditions C1 and C2.

Proof.

Clearly, the output consists of two disjoints sets. The algorithm returns in lines 14 or 19 of Figure 2. If it returns in line 19, then the output  is empty and  contains only vertices that have a distance at least  to the vertices in : The condition in line 3 implies  and, therefore,  contains all vertices in  that have distance at most  to the vertices in . Since  is a bdd--set of , all vertices in  and their neighbors in  have a degree at most . This implies that both conditions hold for the output returned in this line. It remains to consider the output returned in line 14.

To show that condition C1 holds, recall that  has maximum degree  and that . Therefore, if for a vertex  in  we have , then  has degree at most  in . Thus, to show that each vertex in  has degree at most  in , it suffices to prove that . We show separately that  and that .

The assignment in line 8 and the until-condition in line 11 directly give . Due to Lemma 6 there is no edge in  between  and , where  (the if-condition in line 13, which has to be satisfied for the procedure to return in line 14). From this it follows that the vertices in  have no vertex in  as neighbor and, thus, . Therefore, .

By Properties 1 and 2 of Lemma 5, there are exactly  many vertex-disjoint -stars in . Moreover, there is no -star in , since  is a bdd--set of . Thus, is a minimum-cardinality bdd--set of .    


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subsubsection3[Running Time and Correctness of compute_AB]Running Time and Correctness of compute_AB To prove the running time and correctness of compute_AB, we have to show that the output of find_extremal contains sufficiently many vertices of . To this end, the following lemma plays a decisive role.

Lemma 8.

For all , the set  in line 4 and line 18 of Figure 2 has size at most .

Proof.

The proof is by induction on . The claim trivially holds for , since . Assume that the claim is true for . Since  (Lemma 3), we have

We first bound the size of . Since  was set to  at the end of the th iteration of the outer loop (line 15), the vertices in  were not excluded from computing the packing  (line 5) of the th iteration. Moreover,  for the star packing  computed in the th iteration, since, otherwise, the set  in line 6 would contain a vertex  in  and, then, line 8 would include  into , which would contradict the fact that  (line 15). Due to property 2 in Lemma 5 the leaves of every star in  with center in  are vertices in  and, thus, the vertices in  are leaves of stars in  with centers in . Since each star has at most  leaves, the set  has size at most . The remaining part is easy to bound: since all the vertices in  have degree at most , we get

With the induction hypothesis, we get that

 

Lemma 9.

Procedure find_extremal always finds two sets  and  such that .

Proof.

If find_extremal terminates, then  for the graph  resulting by removing  from . Since  and , we have  and , and by Lemma 8 it follows immediately that .    

Therefore, if , then find_extremal always returns two sets  and  such that  is not empty.

Lemma 10.

Algorithm compute_AB runs in  time.

Lemma 11.

The sets  and  computed by compute_AB fulfill the three properties given in Theorem 2.

Proof.

Since every  output by find_extremal in line 4 of compute_AB in Figure 1 fulfills conditions C1 and C2 (Lemma 7), the pair  output in line 3 of compute_AB fulfills conditions C1 and C2, and, therefore, also the local optimality conditions (Lemma 1). It remains to show that  fulfills the size condition.

Let  and  be the last computed witness and residual, respectively. Since the condition in line 3 is true, we know that . Recall that  is a factor- approximate bdd--set for . Thus, every bdd--set of  has size at least . Since the output sets  and  fulfill the local optimality conditions and the bounded-degree property is hereditary, every bdd--set of  has size at least

The inequality (*) follows from the fact that  is small, that is,  (note that ).    

With Lemmas 10 and 11, the proof of Theorem 2 is completed.

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section1[Conclusion]Conclusion Our main result is to generalize the Nemhauser-Trotter-Theorem, which applies to the Bounded-Degree Deletion problem with (that is, Vertex Cover), to the general case with arbitrary . In particular, in this way we contribute problem kernels with a number of vertices linear in the solution size  for all constant values of  for Bounded-Degree Deletion. To this end, we developed a new algorithmic strategy that is based on extremal combinatorial arguments. The original NT-Theorem [20] has been proven using linear programming relaxations—we see no way how this could have been generalized to Bounded-Degree Deletion. By way of contrast, we presented a purely combinatorial data reduction algorithm which is also completely different from known combinatorial data reduction algorithms for Vertex Cover (see [1, 4, 9]). Finally, Baldwin et al. [3, page 175] remarked that, with respect to practical applicability in the case of Vertex Cover kernelization, combinatorial data reduction algorithms are more powerful than “slower methods that rely on linear programming relaxation”. Hence, we expect that benefits similar to those derived from Vertex Cover kernelization for biological network analysis (see the motivation part of our introductory discussion) may be provided by Bounded-Degree Deletion kernelization.

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