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Max Weight Independent Set in graphs with no long claws: An analog of the Gyárfás' path argument

We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw S_t,t,t as an induced subgraph [Chudnovsky, Pilipczuk, Pilipczuk, Thomassé, SODA 2020] and provide a subexponential-time algorithm with improved running time 2^𝒪(√(n)log n) and a quasipolynomial-time approximation scheme with improved running time 2^𝒪(ε^-1log^5 n). The Gyárfás' path argument, a powerful tool that is the main building block for many algorithms in P_t-free graphs, ensures that given an n-vertex P_t-free graph, in polynomial time we can find a set P of at most t-1 vertices, such that every connected component of G-N[P] has at most n/2 vertices. Our main technical contribution is an analog of this result for S_t,t,t-free graphs: given an n-vertex S_t,t,t-free graph, in polynomial time we can find a set P of 𝒪(t log n) vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of G-N[P] such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most n/2 vertices.

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1 Introduction

The complexity of the Maximum Weight Independent Set problem (MWIS

for short), one of the classic combinatorial optimization problems, varies depending on the restrictions imposed on the input graph from polynomial-time solvable (e.g., in bipartite or chordal graphs) through known to admit a quasipolynomial-time algorithm (graphs with bounded longest induced path 

[15]), a polynomial-time approximation scheme and a fixed-parameter algorithm (planar graphs [8]), a quasipolynomial-time approximation scheme (graphs excluding a fixed subdivided claw as an induced subgraph [10, 11]), to being NP-hard and hard to approximate within factor in general graphs [20, 25]. A methodological study of this behavior leads to the following question:

For which structures in the input graph, the assumption of their absence from the input graph makes MWIS easier and by how much?

The “absence of structures” notion can be made precise by specifying the forbidden structure and the containment relation, for example as a minor, topological minor, induced minor, subgraph, or induced subgraph. The last one — induced subgraph relation — is the weakest one, and thus the most expressible. This leads to the study of the complexity of MWIS in various hereditary graph classes, that is, graph classes closed under vertex deletion and thus definable by a (possibly infinite) list of forbidden induced subgraphs.

While a general classification of all hereditary graph classes with regards to the complexity of MWIS

(or other classic graph problems) may be too complex, classifying graph classes with one forbidden induced subgraph looks more feasible. That is, we focus on

-free graphs, graphs excluding a fixed graph as an induced subgraph. Furthermore, the complexity of a given problem (here, MWIS) in -free graphs may indicate the impact of forbidding as an induced subgraph on the complexity of MWIS in more general settings.

As observed by Alekseev [5, 6], the fact that MWIS remains NP-hard and APX-hard in subcubic graphs, together with the observation that subdividing every edge twice in a graph increases the size of the maximum independent set by exactly the number of edges of the original graph, leads to the conclusion that MWIS remains NP-hard and APX-hard in -free graphs unless every connected component of is a path or a tree with three leaves.

In what follows, for integers , by we denote the path on vertices, and by we denote the tree with three leaves within distance , , and from the unique vertex of degree of the tree. Since 1980s, it has been known that MWIS is polynomial-time solvable in -free graphs (because of their strong structural properties) and in -free graphs [22, 24] (because the notion of an augmenting path from the matching problem generalizes to MWIS in -free, i.e., claw-free graphs). For many years, only partial results in subclasses were obtained until the area started to develop rapidly around 2014.

Lokshtanov, Vatshelle, and Villanger [21] adapted the framework of potential maximal cliques [9] to show a polynomial-time algorithm for MWIS in -free graphs; this was later generalized to -free graphs [17] and other related graph classes [3, 4]. More importantly for this work, Bacsó et al. [7] observed that the classic Gyárfás’ path argument, developed to show that for every fixed the class of -free graphs is -bounded [18, 19], also easily gives a subexponential-time algorithm for MWIS in -free graphs. The crucial corollary of the Gyárfás’ path argument lies in the following. Given an -vertex graph , one can in polynomial time find an induced path in such that every connected component of has at most vertices. For -free graphs the said path has at most vertices. Bacsó et al. [7] observed that branching either on the highest degree vertex (if this degree is larger than ) or on the whole set for the path coming from Section 1 (otherwise) gives an algorithm with running time bound exponential in .

Chudnovsky, Pilipczuk, Pilipczuk, and Thomassé [10, 11] added to the mix an observation that a simple branching algorithm is able to get rid of heavy vertices: vertices of the input graph whose neighborhood contains a large fraction of the sought independent set. Once this branching is executed and the graph does not have heavy vertices, the set from Section 1 contains only a small fraction of the sought solution and, if one aims for an approximation algorithm, can be just sacrificed, yielding a quasipolynomial-time approximation scheme (QPTAS) for MWIS in -free graphs. Using this as a starting point and leveraging on the celebrated three-in-a-tree theorem of Chudnovsky and Seymour [13], they developed a much more involved QPTAS and a subexponential algorithm (with running time bound ) for MWIS in -free graphs.

Consider the following simple template for a branching algorithm for MWIS: if the current graph is disconnected, solve independently every connected component; otherwise, pick a vertex (pivot) and branch whether is in the sought independent set (recursing on ) or not (recursing on ). The performance of such an algorithm highly depends on how we choose the pivot . Section 1 suggests that in -free graphs the vertices of may be good choices: there is only a bounded number of them, and the deletion of the whole neighborhood splits into multiplicatively smaller pieces. In a breakthrough result, Gartland and Lokshtanov [15] showed how to choose the pivot and measure the progress of the algorithm, obtaining a quasipolynomial-time algorithm for MWIS in -free graphs. Later, Pilipczuk, Pilipczuk, and Rzążewski [23] provided an arguably simpler measure, leading to an improved (but still quasipolynomial) running time bound. These developments have been subsequently generalized to a larger class of problems beyond MWIS and to -free graphs (graphs without induced cycle of length more than [16].

This progress suggests that MWIS may be actually solvable in polynomial time in -free graphs for all open cases, that is, whenever is a forest whose every connected component has at most three leaves. However, we seem still far from proving it: not only we do not know how to improve the quasipolynomial bounds of [15, 23] to polynomial ones, but also it remains unclear how to merge the approach of [15, 23] with the way how [10, 11] used the three-in-a-tree theorem [13].

In this work, we make a step in this direction, providing an analog of Section 1 for -free graphs. Before we state it, let us briefly discuss what we can hope for in the class of -free graphs.

Consider an example of a graph being the line graph of a clique . The graph is -free, but does not admit any (balanced in any useful sense) separator of the form for a small set . The MWIS problem on translates back to the maximum weight matching problem in the clique ; this problem is polynomial-time solvable, but with very different methods than branching. In particular, we are not aware of any way of solving maximum weight matching in a clique in quasipolynomial time by simple branching. Thus, we expect that an algorithm for MWIS in -free graphs, given such a graph , will discover that it is actually working with the line graph of a clique and apply maximum weight matching techniques to the preimage graph .

Chudnovsky and Seymour, in their project to understand claw-free graphs [12], developed a good way of describing that a graph “looks like a line graph” by the notion of an extended strip decomposition. The formal definition can be found in Section 2. Here, we remark that in an extended strip decomposition of a graph, one can distinguish particles being induced subgraphs of the graph; an algorithm for MWIS can recurse on individual particles, compute the maximum weight independent sets there, and combine the results into a maximum weight independent set in the whole graph using a maximum weight matching algorithm on an auxiliary graph (cf. [10, 11]). Thus, an extended strip decomposition of a graph with particles of multiplicatively smaller size is very useful for recursion; it can be seen as an analog of splitting into connected components of multiplicatively smaller size, as it is in the case of the components of in Section 1.

With the above discussion in mind, we can now state our main technical result. Given an -vertex graph and , one can in polynomial time either:

  • output an induced copy of in , or

  • output a set consisting of at most induced paths in , each of length at most , and a rigid extended strip decomposition of whose every particle has at most vertices.

Combining Section 1 with previously known algorithmic techniques, we derive two algorithms for MWIS in -free graphs. Actually, our algorithms work in a slightly more general setting. For integers , by we denote the graph with connected components, each isomorphic to . Recall that by the observation of Alekseev [5, 6] the only graphs , for which we can hope for tractability results for MWIS in -free graphs, are forests whose every component has at most three leaves. We observe that each such is contained in , for some and depending on . Thus algorithms for -free graphs, for every and , cover all potential positive cases.

First, we observe that the statement of Section 1 seamlessly combines with the method how [7] obtained a subexponential-time algorithm for MWIS in -free graphs. As a result, we obtain a subexponential-time algorithm for MWIS in -free graphs with improved running time as compared to [10, 11].

Let be constants. Given an -vertex -free graph with weights on vertices, one can in time exponential in compute an independent set in of maximum possible weight.

Second, we observe that the statement of Section 1 again seamlessly combines with the method how [10, 11] obtained a QPTAS for MWIS in -free graphs, obtaining an arguably simpler QPTAS for MWIS in -free graphs with improved running time (compared to [10, 11]). Let be constants. Given an -vertex -free graph with weights on vertices, and a real , one can in time exponential in compute an independent set in that is within a factor of of the maximum possible weight.

After preliminaries in Section 2, we prove Section 1 in Section 3. Proofs of Theorems 1 and 1 are provided in Section 4. Finally, we discuss future steps in Section 5.

2 Preliminaries

Notation.

For a family of sets, by we denote . If the base of a logarithmic function is not specified, we mean the logarithm of base 2, i.e., . For a function and subset , we denote .

Let be a graph. For , by we denote the subgraph of induced by , i.e., . If the graph is clear from the context, we will often identify induced subgraphs with their vertex sets. The sets are complete to each other if for every and the edge is present in . Note that this, in particular, implies that and are disjoint. We say that two sets touch if or there is an edge with one end in and another in .

For a vertex , by we denote the set of neighbors of , and by we denote the set . For a set , we also define , and . If it does not lead to confusion, we omit the subscript and write simply and .

By , we denote the set of all triangles in . Similarly to writing , we will write to indicate that .

Extended strip decompositions.

Now let us define a certain graph decomposition which will play an important role in the paper. An extended strip decomposition of a graph is a pair that consists of:

  • a simple graph ,

  • a set for every ,

  • a set for every , and its subsets ,

  • a set for every ,

which satisfy the following properties (see also see Figure 1):

  1. is a partition of ,

  2. for every and every distinct , the set is complete to ,

  3. every is contained in one of the sets for , or is as follows:

    • for some and , or

    • for some , or

    • and for some .

Figure 1: A graph and as extended strip decomposition of a graph . Sets corresponding to vertices, edges, and the triangle of are marked green, blue, and orange, respectively. The edges between distinct sets are drawn thick if they must exist, and thin if they may exist.

Note that for an extended strip decomposition of a graph , the number of vertices of can be much larger than the number of vertices of . However, in such case many sets are empty and thus is “unnecessarily complicated.” An extended strip decomposition is rigid if (i) for every it holds that , and (ii) for every such that is an isolated vertex it holds that .

[] Let be a rigid extended strip decomposition of an -vertex graph . Then and .

Proof.

Recall that since is rigid, for every we have that , and for every isolated vertex of we have .

Let and denote, respectively, the sets of vertices of with degree 0 and more than 0. As the family consists of pairwise disjoint nonempty subsets of , we conclude that and therefore .

Note that by the handshaking lemma we have , and so by the previous argument. ∎

We say that a vertex is peripheral in if there is a degree-one vertex of , such that , where is the (unique) neighbor of in . For a set , we say that is an extended strip decomposition of if has degree-one vertices and each vertex of is peripheral in .

The following theorem by Chudnovsky and Seymour [13] is a slight strengthening of their celebrated solution of the famous three-in-a-tree problem. We will use it as a black-box to build extended strip decompositions.

[Chudnovsky, Seymour [13, Section 6]] Let be an -vertex graph and consider with . There is an algorithm that runs in time and returns one of the following:

  • an induced subtree of containing at least three elements of ,

  • a rigid extended strip decomposition of .

Let us point out that actually, an extended strip decomposition produced by Figure 1 satisfies more structural properties, but of our purpose, we will only use the fact that it is rigid.

Particles of extended strip decompositions.

Let be an extended strip decomposition of a graph . We introduce some special subsets of called particles, divided into five types.

vertex particle:
edge interior particle:
half-edge particle:
full edge particle:
triangle particle:

Observe that the number of all particles of is at most . However, the number of nonempty particles is linear in the number of vertices of .

[] Let be an extended strip decomposition of an -vertex graph. Then the number of nonempty particles of is bounded by .

Proof.

Let , respectively, the subsets consisting of those elements of , , or , for which . Observe that each gives rise to one nonempty particle , and each gives rise to at most four nonempty particles: . Moreover, since are pairwise disjoint subsets of , we have that . Hence, the number of nonempty particles is bounded by

A vertex particle is trivial if is an isolated vertex in . Similarly, an extended strip decomposition is trivial if is an edgeless graph. The following observation follows immediately from the definitions of an extended strip decomposition and particles.

Let be an extended strip decomposition of a graph . For each the following hold:

  1. ,

  2. for any and we have .

We conclude this section by recalling an important property of particles of extended strip decompositions, observed by Chudnovsky et al. [10].

[Chudnovsky et al. [10, Lemma 6.8]] Let be an extended strip decomposition of . Suppose are three induced paths in that do not touch each other, and moreover each of has an endvertex that is peripheral in . Then in there is no particle that touches each of .

3 Main result

In this section, we prove our main result, i.e., Section 1. Let us first give an overview of our approach. We present a recursive algorithm that, for a given graph , will return one of the outcomes of Section 1. Let be the number of vertices in the input graph; the value of will not change throughout the recursive steps of the algorithm. We start with finding a Gyárfás path navigating towards the largest component in . That is, by Section 1, we find such that each connected component of is of size at most . Finding such a small connected component is a great outcome as we can readily include it as a small trivial vertex particle of an extended strip decomposition we are constructing. We say that a particle is small if its size is at most , and an extended strip decomposition is refined if all its particles are small. Observe that if , we immediately get the desired refined extended strip decomposition of . Otherwise, we proceed to the main part of the algorithm. At each step, we will remove some vertices from , and will measure the progress of our algorithm in the number of the remaining vertices of .

Formally, we create a set of pairwise not touching induced paths such that and . At each step of recursion we obtain a set with that represents for the next step. Hence, in recursive steps, drops below . In the base case of the recursion, when , we return the refined trivial extended strip decomposition ensured by maintaining the property that has connected components of size at most throughout the recursive steps. In each step of recursion, we further split the induced path(s) in so we are able to use Figure 1 to obtain an extended strip decomposition . If is already refined, then we are done. Otherwise, it contains a particle that is not small. We use Section 2 to select at most two paths touching . Then it is easy to separate with the respective touching paths from the rest of the graph. The graph induced by and the touching paths form a smaller instance, i.e., an instance where drops by a factor of . We need to ensure that at every recursive step, we include only a constant number of paths of length into (i.e., the set of paths in the second outcome of Section 1). We now prove the core recursive formulation of the algorithm formally.

[Recursion] Given a graph and a set of at most two induced paths (vertex disjoint non-adjacent), and a refined extended strip decomposition of . In polynomial time, we can output one of the following:

  • an induced copy of in , or

  • , , and a refined extended strip decomposition , so that and the longest path in has at most vertices.

Proof.

If the longest path of has at most vertices, return where each path in may be further split in at most three paths on at most vertices, and . Hence, we output the extended strip decomposition we were given by the assumptions of the lemma.

Otherwise, let be the longest path in . Let and be the -th and the -th vertex of , respectively. The removal of and from divides the path into three induced non-touching subpaths , , and , each of length at least . Let be the remaining path of , should it exist. We define if exists, or , otherwise. Consult Figure 2 to see an overview of the definitions described in this paragraph. For each path we define as the set comprising:

  • first vertices of (or all vertices of if ), and

  • the separating vertex of directly preceding if .

It can be easily seen that the set of vertices forms an induced path of length at most . We finally define shells of paths in . Given a path , we set if and otherwise. Intuitively, if , the shell of takes the whole neighborhood as we do not have a use for a short path in the next stage of our algorithm. For a long enough path , the shell of intersects all short paths connecting the first vertex of with the rest of the graph. Thus, each path from the first vertex of to any vertex of outside of will have length at least . To ease the notation, we define , , and .

Figure 2: Definitions of and in case of .

Now, we use the algorithm from Figure 1 on being the set of the first vertices of paths in and the graph defined as . If Figure 1 produced an induced tree with three leaves among , we return it as an induced , since those must have been induced branches at least vertices long in . Hence, we obtained an extended strip decomposition of . If the obtained decomposition is refined, we return , , and the extended strip decomposition .

Therefore, the obtained extended strip decomposition of contains a particle which is not small, i.e., is composed of at least vertices. As is peripheral, we know that no three paths in touch one particle by Section 2. Therefore, we take the set of at most two paths, say and , touching (for convenience, let or be an empty set if it does not exist). We now compute the maximum proportion of put to . If both , then this fraction is at most as by the definition , for . If one is and the other comes from

, then we estimate

for . Hence, we know that . We define to use Section 3 on a smaller instance. Now, we need to verify that the assumption of the lemma holds. We claim the following:

has a refined extended strip decomposition.

As is an induced subgraph of and has a refined extended strip decomposition, we know that has a refined extended strip decomposition. First, recall that , which is disjoint with . Analogously is disjoint with . Also, if then is disjoint with as well. Hence, . Also, recall that the only paths among that touch are in . Hence, observe that .

Therefore, we can apply Section 3 inductively on and , obtaining and , and a refined extended strip decomposition  of . We need to combine the extended strip decomposition obtained from the recursion with the extended strip decomposition we obtained earlier.

We can always suppose that particle is of type for some edge , unless is of type for an isolated vertex . That is because is the superset of all possible particle types. As Figure 1 gives us that both and are nonempty, we can select and (possibly ). By Section 2, the set

separates from the rest of . Set . In the case of such that is an independent vertex, we set and and still such is separated from the rest of by . We return:

  • ,

  • ,

  • an extended strip decomposition of , where is with an additional isolated vertex , and is with an additional trivial vertex particle containing all vertices of .

We compute that as we added at most six new paths into . Observe that the described algorithm runs in polynomial time as we just computed that the depth of recurrence is logarithmic in and each recursive call takes polynomial time in the size of .

Proof of Section 1.

Using Section 1 we find a Gyárfás path . We get the desired outcome by Section 3 on with . The extended strip decomposition needed by the lemma’s assumption is trivial. That is, each connected component of is represented by a vertex particle of small size. We conclude the proof of Section 1 by the following calculation:

Note that for any extended strip decomposition we can easily add the assumption that sets for any edge . As suppose ; then we can update by adding to and removing from . Moreover, we can simply remove any empty trivial vertex particle form and the corresponding isolated vertex from . Therefore, we may suppose that the obtained extended strip decomposition is rigid. ∎

In the following simple corollary we apply Section 1 to -free graphs, for some .

[] Let be constants. Let be an -free graph on vertices. Then in polynomial time we can find a set consisting of at most vertices and a rigid extended strip decomposition of whose every particle has at most vertices.

Proof.

Induction on . If , then we obtain the result immediately by Section 1. Thus let us assume that and the theorem holds for -free graphs.

We exhaustively check if there is some with , such that ; we can do it in time . If such does not exist, then we can immediately apply Section 1, and the proof is complete. Thus suppose that exists.

We observe that the graph is -free. Denote . By the inductive assumption, in time we can obtain a set of size at most and a rigid extended strip decomposition of whose every particle is of size at most .

We set . Now and satisfy the statement of the theorem, as and . The total running time is polynomial in as the depth of the recursion is . ∎

4 Algorithmic applications

In this section we will show how to combine Section 1 with the approach of Chudnovsky et al. [10, 11] in order to obtain a QPTAS and a subexponential-time algorithm for MWIS in -free graphs, i.e., we prove Theorems 1 and 1.

Both algorithms follow the same general outline; let us sketch it before we get into the details of each particular case. Each algorithm is a recursive procedure, which consists of two phases. In the first one, we deal with the vertices of that are heavy, which means that their neighborhood is “large”, where the exact meaning of “large” depends on the particular algorithm.

Once there are no heavy vertices, i.e., the neighborhood of each vertex is “small”, we proceed to the second phase. We call Section 3 for the current instance , obtaining a small-sized set and a rigid extended strip decomposition of , whose every particle is of small size. The crux is that since we are in the second phase, all vertices in are not heavy, and since is of small size, the whole set is “small”. We treat separately in a way that depends on the particular algorithm.

Next, for each particle of , we call the algorithm recursively for , obtaining (a good approximation of) a maximum-weight independent set in . Finally, we combine the obtained results to derive (a good approximation of) a maximum-weight independent set in . This last step is based on the idea of Chudnovsky et al. [10, 11] to reduce the problem to finding a maximum-weight matching in a graph obtained by a simple modification of . Since the size of is linear in (by Figure 1), this problem can be solved in time polynomial in using, e.g., the classic algorithm of Edmonds [14]. The last step is encapsulated in the following lemma, whose exact statement comes from Abrishami et al. [1].

[Chudnovsky et al. [10, 11]] Let be a real number. Let be an -vertex graph equipped with a weight function . Suppose that is given along with an extended strip decomposition , where has vertices.

Let be a fixed independent set in . Furthermore, assume that for each particle of we are given an independent set in such that . Then in time polynomial in we can compute an independent set in such that .

Let us stress out that the algorithm from Section 4 does not need to know the value of or the independent set .

The main difference between our approach and the one of Chudnovsky et al. [11] is that we use Section 1 and its consequence, i.e., Section 3. The previous algorithms used a similar statement but with a worse (and much more involved) guarantee on the size of and each particle. Furthermore, the way we obtain our set is significantly simpler.

4.1 Proof of Section 1

Before we proceed to the proof, let us first explain the meaning of “small”, and how to deal with in this particular case. Here the neighborhood of a vertex is “small” if it has few vertices (more specifically, at most ). In the first phase, we deal with heavy vertices (i.e., of large degree) with simple branching: we guess whether is included in our optimum solution or not. Since the degree of is large, in the first branch, we obtain significant progress, which is enough to obtain a subexponential running time.

In the second phase, since is the neighborhood of vertices, each of degree , the total size of is . Thus we can afford to exhaustively guess the intersection of our optimum solution with .

Proof of Section 1.

Let be constants and let be an instance of MWIS, where is -free and has vertices. We observe that if is small, i.e., bounded by a constant. Then we can solve the problem by brute force. Thus we assume that , where is a constant (depending on and ) whose exact value follows from the reasoning below.

First, consider the case that there exists such that . We branch on including in the final solution: we either delete from , or we delete and add to the solution returned by the recursive call. Then we output the one of these two solutions that has a larger weight. The correctness of this step of the algorithm is straightforward.

Hence, we can assume that for every it holds that . By Section 3, since is -free, we obtain a set of size (here we use that is large), and a rigid extended strip decomposition of whose every particle has at most vertices.

We exhaustively guess an independent set ; think of it as an intersection of the intended optimum solution with . Consider the graph . We modify by removing the vertices from from the sets . Let us call the obtained strip decomposition ; note that it might not be rigid. We call the algorithm recursively for the subgraph for every nonempty particle of . Let be the solution. If , then . By the inductive assumption is a maximum-weight independent set in . Then we use Section 4 for to combine the solutions into a maximum-weight independent set of . Finally, we return the independent set whose weight is maximum over all choices of . Note that the correctness of this step is guaranteed by the exhaustive guessing of and Section 4.

Running time.

Let denote the running time of our algorithm for -vertex instances. We prove that . If , then the claim clearly holds. So let us assume that .

In the first case we call the algorithm for two instances, one of size and one of size at most . Hence,

Here we skip the description how this recursion is solved, as it it pretty standard. For a formal proof we refer the reader to Bacsó et al. [7, Lemma 1].

It remains to analyze the running time of the step in which the maximum degree of vertices in is bounded by . Section 3 asserts that we obtain and the rigid extended strip decomposition of in time polynomial in . There are ways of choosing the set . In polynomial time we modify into .

Observe that while might not be rigid, it was obtained from a rigid extended strip decomposition by deleting some vertices from the sets . In particular, both decompositions have the same sets of particles, and every nonempty particle of is also a nonempty particle of . Thus by Section 2 we call the algorithm recursively for at most nonempty particles, each of size at most . By Figure 1, the total number of particles of is polynomial in . Finally, having computed a maximum independent set contained in each particle, by Section 4, we can compute the final solution in time polynomial in . Hence, there are constants , where , such that total running time of this step is bounded by:

(1)

and so is the total complexity of the algorithm. ∎

4.2 Proof of Section 1

Again let us start with explaining the algorithm-specific details of the outline presented at the start of Section 4.

We will use the notion of -heavy vertices from [10, 11]. Consider a graph , a weight function , and an independent set . Let be a real. We say that a vertex is -heavy (with respect to ) if . A set is good for if and contains all vertices that are -heavy with respect to .

[Chudnovsky et al. [10, 11]] Let be an -vertex graph for , be a weight function, be an independent set, and be a real. Then there exists a set of size at most which is good for .

Now the vertex is heavy if it is -heavy for some carefully chosen parameter . This means that a neighborhood of a vertex is “large” if it contains a significant () fraction of the weight of . In the first phase, we exhaustively guess the set that is good for a fixed optimum solution . Note that is of small size and since , we know that contains no vertices from and thus can be safely removed from the graph.

Since is good for , we know that contains no heavy vertices, and for this graph we call Section 3. Now, as is a neighborhood of few non-heavy vertices, we know that the total weight of is small and thus can be sacrificed, as we aim for an approximation.

Proof of Section 1.

Let be constants and let be an instance of MWIS, where is -free and has vertices. Let be fixed. Fix a maximum-weight independent set in with respect to . We describe a procedure that finds in an independent set of weight at least .

Let be the minimum power of two greater than or equal to the size of our initial instance. Note that . The value of will not change throughout the execution of the algorithm.

The algorithm itself is a recursive procedure. The arguments of each call are a graph , which is an induced subgraph of , the weight function on obtained by restricting the domain of , and an integer , which can be intuitively understood as the depth of the current call in the recursion tree. Since it does not lead to confusion, we will always denote the weight function by . We will keep the invariant that for each call it holds that . The initial call, corresponding to the root of the recursion tree, is for .

Consider a call for the instance . If , where is a constant (depending on and ) that follows from the reasoning below, then we can solve the problem by brute force. Thus let us assume that . In particular, .

We set

(2)

It is straightforward to verify that for we have . On the other hand, if , then is of constant size and thus is not computed for such .

Let be the family of all independent sets in of size at most . For each we proceed as follows. If , then we compute a maximum-weight independent set in by brute force. Otherwise, we use Section 3, to obtain a set and a rigid extended strip decomposition of such that each particle of is of size at most . By Section 3, we obtain

(3)

Let . We modify into an extended strip decomposition of as follows. For each , we add to an isolated vertex , and set .111Another possible way of dealing with the set would be to add it directly in the computed solution. However, we decided to restore to the graph, so that these vertices are handled by Section 4 and do not require any special treatment. Let us call this extended strip decomposition