 # Minor-free graphs have light spanners

We show that every H-minor-free graph has a light (1+ϵ)-spanner, resolving an open problem of Grigni and Sissokho and proving a conjecture of Grigni and Hung. Our lightness bound is O(σ_H/ϵ^31/ϵ) where σ_H = |V(H)|√( |V(H)|) is the sparsity coefficient of H-minor-free graphs. That is, it has a practical dependency on the size of the minor H. Our result also implies that the polynomial time approximation scheme (PTAS) for the Travelling Salesperson Problem (TSP) in H-minor-free graphs by Demaine, Hajiaghayi and Kawarabayashi is an efficient PTAS whose running time is 2^O_H(1/ϵ^41/ϵ)n^O(1) where O_H ignores dependencies on the size of H. Our techniques significantly deviate from existing lines of research on spanners for H-minor-free graphs, but build upon the work of Chechik and Wulff-Nilsen for spanners of general graphs.

## Authors

##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## 1 Introduction

Peleg and Schäffer  introduced -spanners of graphs as a way to sparsify graphs while approximately preserving pairwise distances between vertices. A -spanner of a graph is a subgraph of such that for all vertices 111We use standard graph terminology, which can be found in Appendix A.. Two parameters of -spanners that are of interest are their sparsity and lightness. The sparsity of is the ratio of the number of edges to the number of vertices of . The lightness of is the ratio of the total weight of the edges of to the weight of an of ; generally, we assume that (and so ). Here, we are concerned with the lightness of -spanners, where , and so we refer to -spanners simply as spanners.

We say that a spanner is light if the lightness does not depend on the number of vertices in the graph. Grigni and Sissokho  showed that -minor-free graphs have spanners of lightness

 O(1ϵσHlogn). (1)

where is the sparsity coefficient of -minor-free graphs; namely that an -minor-free graph of vertices has edges222This bound is tight . . Later Grigni and Hung , in showing that graphs of bounded pathwidth have light spanners, conjectured that -minor-free graphs also have light spanners; that is, that the dependence on can be removed from the lightness above. In this paper, we resolve this conjecture positively, proving:

###### Theorem 1.

Every -minor-free graph has a -spanner of lightness

 O(σHϵ3log1ϵ). (2)

Our algorithm consists of a reduction phase and a greedy phase. In the reduction phase, we adopt a technique of Chechik and Wulff-Nilsen : edges of the graph are subdivided and their weights are rounded and scaled to guarantee that every -edge has unit weight and we include all very low weight edges in the spanner (Appendix C). In the greedy phase, we use the standard greedy algorithm for constructing a spanner to select edges from edges of the graph not included in the reduction phase (Appendix B).

As a result of the reduction phase, our spanner is not the ubiquitous greedy spanner. However, since Filtser and Solomon have shown that greedy spanners are (nearly) optimal in their lightness , our result implies that the greedy spanner for -minor-free graphs is also light.

### 1.1 Implication: Approximating TSP

Light spanners have been used to give PTASes, and in some cases efficient PTASes, for the traveling salesperson problem (TSP) on various classes of graphs. A PTAS, or polynomial-time approximation scheme, is an algorithm which, for a fixed error parameter , finds a solution whose value is within of optimal in polynomial time. A PTAS is efficient if its running time is where is a function of . Rao and Smith  used light spanners of Euclidean graphs to give an EPTAS for Euclidean TSP. Arora, Grigni, Karger, Klein and Woloszyn  used light spanners of planar graphs, given by Althöfer, Das, Dobkin, Joseph and Soares , to design a PTAS for TSP in planar graphs with running time . Klein  improved upon this running time to by modifying the PTAS framework, using the same light spanner. Borradaile, Demaine and Tazari generalized Klein’s EPTAS to bounded genus graphs .

In fact, it was in pursuit of a PTAS for TSP in -minor-free graphs that Grigni and Sissokho discovered the logarithmic bound on lightness (Equation (1)); however, the logarithmic bound implies only a quasi-polynomial time approximation scheme (QPTAS) for TSP . Demaine, Hajiaghayi and Kawarabayashi  used Grigni and Sissokho’s spanner to give a PTAS for TSP in -minor-free graphs with running time ; that is, not an efficient PTAS. However, Demaine, Hajiaghayi and Kawarabayashi’s PTAS is efficient if the spanner used is light. Thus, the main result of this paper implies an efficient PTAS for TSP in -minor-free graphs.

### 1.2 Techniques

In proving the lightness of spanners in planar graphs  and bounded genus graphs , the embedding of the graph was heavily used. Thus, it is natural to expect that showing minor-free graphs have light spanners would rely on the decomposition theorem of minor-free graphs by Robertson and Seymour , which shows that graphs excluding a fixed minor can be decomposed into the clique-sum of graphs nearly embedded on surfaces of fixed genus. Borradaile and Le  use this decomposition theorem to show that if graphs of bounded treewidth have light spanners, then -minor-free graphs also have light spanners. As graphs of bounded treewidth are generally regarded as easy instances of -minor-free graphs, it may be possible to give a simpler proof of lightness of spanners for -minor-free graphs using this implication.

However, relying on the Robertson and Seymour decomposition theorem generally results in constants which are galactic in the size of the the minor [16, 14]. In this work, we take a different approach which avoids this problem. Our method is inspired from the recent work of Chechik and Wulff-Nilsen  on spanners for general graphs which uses an iterative super-clustering technique [3, 8]. Using the same technique in combination with amortized analysis, we show that -minor-free graphs not only have light spanners, but also that the dependency of the lightness on and is practical (Equation (2)).

At a high level, our proof shares several ideas with the work of Chechik and Wulff-Nilsen  who prove that (general) graphs have -spanners with lightness , removing a factor of from the previous best-known bound and matching Erdős’s girth conjecture  up to a factor. Our work differs from Chechik and Wulff-Nilsen in two major aspects. First, Chechik and Wulff-Nilsen reduce their problem down to a single hard case where the edges of the graph have weight at most for some constant . In our problem, we must partition the edges according to their weight along a logarithmic scale and deal with each class of edges separately. Second, we must employ the fact that -minor-free graphs (and their minors) are sparse in order to get a lightness bound that does not depend on .

### 1.3 Future directions

Since we avoid relying on Robertson and Seymour’s decomposition theorem and derive bounds using only the sparsity of graphs excluding a fixed minor, it is possible this technique could be extended to related spanner-like constructions that are used in the design of PTASes for connectivity problems. Except for TSP, many connectivity problems  have PTASes for bounded genus graphs but are not known to have PTASes for -minor-free graphs – for example, subset TSP and Steiner tree. The PTASes for these problems rely on having a light subgraph that approximates the optimal solution within (and hence is spanner-like). The construction of these subgraphs, though, rely heavily on the embedding of the graph on a surface and since the Robertson and Seymour decomposition gives only a weak notion of embedding for -minor-free graphs, pushing these PTASes beyond surface embedded-graphs does not seem likely. The work of this paper may be regarded as a first step toward designing spanner-like graphs for problems such as subset TSP and Steiner tree that do not rely on the embedding.

## 2 Bounding the lightness of a (1+ϵ)-spanner

As we already indicated, we start with a reduction that allows us to assume that the edges of the of the graph each have unit weight. (For details, see Appendix C.) For simplicity of presentation, we will also assume that the spanner is a greedy -spanner for a sufficiently large constant ; this does not change the asymptotics of our lightness bound.

Herein, we let be the edges of a greedy -spanner of graph with an having edges all of unit weight. We simply refer to as the spanner. The greedy spanner considers the edges in non-decreasing order of weights and adds an edge if is at most the -to- distance in the current spanner (see Appendix B for a review).

We partition the edges of according to their weight as it will be simpler to bound the weight of subsets of . Let be the edges of of weight in the range ; note that and, since has edges and ,

 w(J0)=O(σHn/ϵ)=O(σHϵw(\textscMST)) (3)

Let be the edges of of weight in the range for every and . Let . We will prove that

###### Lemma 2.

There exists a set of spanner edges such that and for every ,

 w(\textscMST∪(Jj∖B))=O(σHϵ3)w(\textscMST).

Combined with Equation (3), Lemma 2 gives us

 w(S)=w(B)+⌈log1ϵ⌉∑j=0w(Jj∖B)=O(σHϵ3log1ϵ)w(\textscMST)

which, combined with the reduction to unit-weight -edges, proves Theorem 1 (noting that the stretch condition of is satisfied since is a greedy spanner of ).

In the remainder, we prove Lemma 2 for a fixed . Let for this fixed and some . Let ; then, the weight of the edges in are in the range . Let . We refer to the indices of the edge partition as levels.

### 2.1 Proof overview

To prove Lemma 2, we use an amortized analysis, initially assigning each edge of a credit of . For each level, we partition the vertices of the spanner into clusters where each cluster is defined by a subgraph of the graph formed by the edges in levels 0 through . (Note that not every edge of level 0 through may belong to a cluster; some edges may go between clusters.) Level clusters are a refinements of level clusters. We prove (by induction over the levels), that the clusters for each level satisfy the following diameter-credit invariants:

DC1

A cluster in level of diameter has at least credits.

DC2

A cluster in level has diameter at most for some constant (specified later).

We achieve the diameter-credit invariants for the base case (level ) as follows. Although a simpler proof could be given, the following method we use will be revisited in later, more complex, constructions. Recall that and that, in a greedy spanner, the shortest path between endpoints of any edge is the edge itself. If the diameter of is , edges in the spanner have length at most . Thus, it is trivial to bound the weight of all the spanner edges across all levels using the sparsity of -minor-free graphs. Assuming a higher diameter, let be a maximal collection of vertex-disjoint subtrees of , each having diameter (chosen, for example, greedily). Delete from . What is left is a set of trees , each of diameter . For each tree , let be the union of with any neighboring trees in (connected to by a single edge of ). By construction, has diameter at most (giving DC2). is assigned the credits of all the edges in the cluster each of which have credit (giving DC1).

We build the clusters for level from the clusters of level in a series of four phases (Section 3). We call the clusters of level -clusters, since the diameter of clusters in level are an -fraction of the diameters of clusters in level . A cluster in level is induced by a group of -clusters.

We try to group the -clusters so that the diameter of the group is smaller than the sum of the diameters of the -clusters in the group (Phases 1 to 3). This diameter reduction will give us an excess of credit beyond what is needed to maintain DC1 which allows us to pay for the edges of . We will use the sparsity of -minor free graphs to argue that each -cluster needs to pay for, on average, a constant number of edges of . In Phase 4, we further grow existing clusters via edges and unpaid edges of .

Showing that the clusters for level satisfy invariant DC2 will be seen directly from the construction. However, satisfying invariant DC1 is trickier. Consider a path witnessing the diameter of a level- cluster . Let be the graph obtained from by contracting -clusters; we call the cluster-diameter path. The edges of are a subset of . If does not contain an edge of , the credits from the -clusters and edges of are sufficient for satisfying invariant DC1 for . However, since edges of are not initialized with any credit, when contains an edge of , we must use credits of the -clusters of outside to satisfy DC1 as well as pay for . Finally, we need to pay for edges of that go between clusters. We do so in two ways. First, some edges of will be paid for by this level by using credit leftover after satisfying DC1. Second, the remaining edges will be paid for at the end of the entire process (over all levels); we show that there are few such edges over all levels (the edges of Lemma 2).

In our proof below, the fixed constant required in DC2 is roughly 100 and is sufficiently smaller than . For simplicity of presentation, we make no attempt to optimize . We note that a -spanner is also a -spanner for any constant and the asymptotic dependency of the lightness on remains unchanged. That is, requiring that is sufficiently small is not a limitation on the range of the parameter .

## 3 Achieving diameter-credit invariants

In this section, we construct clusters for level that satisfy DC2 using the induction hypothesis that -clusters (clusters of level ) satisfy the diameter-credit invariants (DC1 and DC2). Since , we let , and drop the subscript in the remainder. For DC2, we need to group -clusters into clusters of diameter . Let be the collection of -clusters and be the set of clusters that we construct for level . Initially, . We define a cluster graph whose vertices are the -clusters and edges are the edges of . can be obtained from the subgraph of formed by the edges of the -clusters and by contracting each -cluster to a single vertex. Recall each -cluster is a subgraph of the graph formed by the edges in levels 0 through .

###### Observation 3.

is a simple graph.

###### Proof.

Since when is sufficiently smaller than , there are no self-loops in . Suppose that there are parallel edges and where and . Let , w.l.o.g.. Then, the path consisting of the shortest -to- path in , edge and the shortest -to- path in has length at most by DC2. Since and , has length at most . Therefore, if our spanner is a greedy -spanner, would not be added to the spanner. ∎

We call an -cluster high-degree if its degree in the cluster graph is at least , and low-degree otherwise. For each -cluster , we use to denote the cluster in that contains . To both maintaining diameter-credit invariants and buying edges of , we use credits of -clusters in and edges connecting -clusters in . We save credits of a subset of -clusters of and -edges connecting -clusters in for maintaining invariant DC1. We then reserve credits of another subset of -clusters to pay for edges of of incident to -clusters in . We let other -clusters in release their credits to pay for their incident edges of ; we call such -clusters releasing -clusters. We designate an -cluster in to be its center and let the center collect the credits of -clusters in . The credits collected by the center are used to pay for edges of incident to non-releasing -clusters.

### 3.1 Phase 1: High-degree ϵ-clusters

In this phase, we group high-degree -clusters. The goal is to ensure that any edge of not incident to a low-degree -cluster has both endpoints in the new clusters formed (possibly in distinct clusters). Then we can use sparsity of the subgraph of induced by the -clusters that were clustered to argue that the clusters can pay for all such edges; this is possible since this subgraph is a minor of . The remaining edges that have not been paid for are all incident to low-degree -clusters which we deal with in later phases.

With all -clusters initially unmarked, we apply Step 1 until it no longer applies and then apply Step 2 to all remaining high-degree -clusters at once and breaking ties arbitrarily:

1. If there is a high-degree -cluster such that all of its neighbor -clusters in are unmarked, we group , edges in incident to and its neighboring -cluster into a new cluster . We then mark all -clusters in . We call the center -cluster of .

2. After Step 1, any unmarked high-degree -cluster, say , must have at least one marked neighboring -cluster, say . We add and the edge of between and to and mark . Figure 1: A cluster C(X) formed in Phase 1 is enclosed in the dotted blue curve. The set S(X) consists of five gray ϵ-clusters inside the dotted blue closed curve and S(X) consists of green-shaded ϵ-clusters. Remaining hollow ϵ-clusters are releasing. Cluster C(X) will be augmented further in Phase 4 and augmenting ϵ-clusters are outside the dotted blue curve. Solid blue edges are in Ei and thin black edges are in MST. The diameter path D is highlighted by the dashed red curve and ϵ-clusters in D are gray-shaded.

In the following, the upper bound is used to guarantee DC2 and the lower bound will be used to guarantee DC1.

###### Claim 4.

The diameter of each cluster added in Phase 1 is at least and at most .

###### Proof.

Since the clusters formed are trees each containing at least two edges of and since each edge of has weight at least , the resulting clusters have diameter at least .

Consider an -cluster that is the center of a cluster in Step 1 that is augmented to in Step 2 (where, possibly ). The upper bound on the diameter of comes from observing that any two vertices in are connected via at most 5 -clusters and via at most 4 edges of (each -cluster that is clustered in Step 2 is the neighbor of a marked -cluster from Step 1). Since -clusters have diameter at most and edges of have weight at most , the diameter of is at most . ∎

Let be a cluster in Phase 1 with the center . Let be the set of ’s neighbors in the cluster graph . By construction, is a tree of -clusters. Thus, at most five -clusters in would be in the cluster-diameter path while at most three of them are in . We use the credit of and of two -clusters in for maintaining DC1. Let this set of three -clusters be . Since is high-degree and , has at least -clusters. Let be any subset of -clusters in . The center collects the credits of -clusters in . We let other -clusters in release their own credits; we call such -clusters releasing -clusters. By diameter-credit invariants for level , each -cluster has at least credits. Thus, we have:

###### Observation 5.

The center of collects at least credits.

Let be the set of edges of that have both endpoints in marked -clusters.

###### Claim 6.

If , we can buy edges of using credits deposited in the centers and credit of releasing -clusters.

###### Proof.

Since the subgraph of induced by marked -clusters and edges of is -minor-free, each marked -cluster, on average, is incident to at most edges of . Thus, each -cluster must be responsible for buying edges of .

Consider a cluster . The total credits of each releasing -clusters is at least , which is when . For non-releasing -clusters, we use credits from their center to pay for incident edges of . Recall that non-releasing -clusters are in and:

 |R(X)∪S(X)|≤5+18ϵ (4)

Thus, non-releasing -cluster are responsible for paying at most edges of and credits suffice if . ∎

By Claim 6, each center -cluster has at least credits remaining after paying for . We note that clusters in Phase 1 could be augmented further in Phase 4. We will use these remaining credits at the centers to pay for edges of in Phase 4.

### 3.2 Phase 2: Low-degree, branching ϵ-clusters

Let be a maximal forest whose nodes are the -clusters that remain unmarked after Phase 1 and whose edges are edges between pairs of such -clusters.

Let be the diameter of a path in , which is the diameter of the subgraph of formed by edges inside -clusters and MST edges connecting -clusters of . We define the effective diameter to be the sum of the diameters of the -clusters in . Since the edges of have unit weight (since they are edges), the true diameter of a path in is bounded by the effective diameter of plus the number of edges in the path. Since each -cluster has diameter at least (by construction of the base case), we have:

###### Observation 7.

.

We define the effective diameter of a tree (in ) to be the maximum effective diameter over all paths of the tree. Let be a tree in that is not a path and such that . Let be a branching vertex of , i.e., a vertex of of degree is at least , and let be a minimal subtree of that contains and ’s neighbors and such that . We add to and delete from ; this process is repeated until no such tree exists in . We refer to as the center -cluster of . Figure 2: A cluster C(X) formed in Phase 2 is enclosed in the dotted blue curve. C(X) will be augmented further in Phase 4 and augmenting ϵ-clusters are outside the dotted blue curve. Edges connecting ϵ-clusters are MST edges.
###### Claim 8.

The diameter of each cluster added in Phase 2 is at most .

###### Proof.

Since is minimal, its effective diameter is at most . The claim follows from Observation 7. ∎

Let be a set of -clusters. We define a subset of as follows:

 ⌊X⌋2g/ϵ={Xif |X|≤2g/ϵany subset of 2g/ϵ of X% otherwise

By definition, we have:

 |⌊X⌋2g/ϵ|≤2gϵ (5)

Let where is the diameter path of . We save credits of -clusters in for maintaining DC1 and we use credits of -clusters in to buy edges of incident to -clusters in . Since is branching, at least one neighbor -cluster of , say , is not in . Let . The center collects credits of clusters in ; other -clusters in release their credits.

Let be the set of unpaid edges of incident to -clusters grouped in Phase 2.

###### Claim 9.

If , we can buy edges of using credits from the center -clusters and half the credit from releasing -clusters.

###### Proof.

Consider a cluster formed in Phase 2. Recall -clusters in Phase 2 are low-degree. Thus, each -cluster in is incident to at most edges of . We need to argue that each -cluster has at least credits to pay for edges of . By invariant DC1 for level , half credits of releasing -clusters are at least , which is when .

Since , the center collects at least credits by invariant DC1 for level . Recall non-releasing -clusters are all in . Thus, by Equation 5, the total number of edges of incident to -clusters in is at most:

 (2gϵ+1)20ϵ=O(gϵ2)

Since , credits of the center is at least which suffices to buy all edges of incident to -clusters in . ∎

We use remaining half the credit of releasing -clusters to achieve invariant DC1. More details will be given later when we show diameter-credit invariants of .

### 3.3 Phase 3: Grouping ϵ-clusters in high-diameter paths

In this phase, we consider components of that are paths with high effective diameter. To that end, we partition the components of into HD-components (equiv. HD-paths), those with (high) effective diameter at least (which are all paths) and LD-components, those with (low) effective diameter less that (which may be paths or trees).

#### Phase 3a: Edges of Ei within an HD-path

Consider an HD-path that has an edge with endpoints in -clusters and of such that the two disjoint affices ending at and both have effective diameter at least . We choose such that there is no other edge with the same property on the -to- subpath of (By Observation 3, there is no edge of parallel to ). Let be the -to- subpath of . By the stretch guarantee of the spanner, . Let and be minimal subpaths of the disjoint affices of that end at and , respectively, such that the effective diameters of and are at least . and exist by the way we choose .

##### Case 1: ediam(PX,Y)≤2ℓ

We construct a new cluster consisting of (the -clusters and edges of) , , and edge (see Figure 3(a)). We refer to, w.l.o.g, as the center -cluster of the new cluster.

###### Claim 10.

The diameter of each cluster added in Case 1 of Phase 3a is at least and at most .

###### Proof.

Since the new cluster contains edge of and, in spanner , the shortest path between endpoints of any edge is the edge itself, we get the lower bound of the claim. The effective diameters of and are each at most since they are minimal. By Observation 7, we get that the diameter is at most:

 2(ediam(PX)+ediam(PY)+ediam(PX,Y))≤4(2+ϵg)ℓ+4ℓ=(12+4ϵg)ℓ

###### Claim 11.

Let be any two vertices of in a cluster added in Case 1 of Phase 3a. Let be the shortest -to- path in as a subgraph of . Let be obtained from by contracting -clusters into a single vertex. Then, is a simple path.

###### Proof.

By construction, the only cycle of -clusters in is (see Figure 3(a)). Therefore, if is not simple, and must enter and leave at some -cluster . In this case, could be short-cut through , reducing the weight of the path by at least and increasing its weight by at most . This contradicts the shortness of for sufficiently smaller than (). ∎ Figure 3: (a) A cluster of C in Case 1 of Phase 3a and (b) a cluster of C in Case 2 of Phase 3a. Thin edges are edges of MST, solid blue edges are edges of Ei and vertices are ϵ-clusters. Edges and vertices inside the dashed red curves are grouped into a new cluster.

Since is the only cycle of -clusters, by Claim 11, -clusters in form a simple subpath of where is the diameter path of . We have:

.

###### Proof.

For otherwise, could be shortcut through at a cost of

 ≤ diam(X)+diam(Y)+w(e)cost % of shortcut−(diam(PX,Y)−diam(X)−diam(Y))lower bound on diameter ≤ +w(e)+4gϵℓ−(1+sϵ)w(e)(by the stretch % condition for e) ≤ 4gϵℓ−sϵℓ/2(since w(e)≥ℓ/2)

This change in cost is negative for . ∎

Let and . The center collects the credits of -clusters in and edges outside connecting -clusters of . We let other -clusters in release their credits.

###### Claim 13.

If does not contain , then has at least credits. Otherwise, has at least credits.

###### Proof.

If contains at least -clusters, then . Thus, by invariant DC1 for level , the total credit of -clusters in is at least:

 2gϵ⋅cϵℓ2=gcℓ≥cℓ+cϵℓ2 (for g≥2 and ϵ<1)≥cw(e)+cϵℓ2(since w(e)≤ℓ)

Thus, we can assume that contains less than -clusters. In this case, . Since by Observation 12, does not contain, w.l.o.g., . Thus, contains at least one -cluster and the claim holds for the case that .

Suppose that contains and an internal -clusters of , then w.l.o.g., does not contain . has credit . Since and when is sufficiently small (), the claim holds.

If contains but no internal -clusters of , then

 diam(PX,Y∖{X,Y})≥diam(PX,Y)−diam(X)−diam(Y)≥(1+sϵ)w(e)−diam(X)−diam(Y)   (by the stretch condition)≥we+sϵℓ/2−2gϵℓ   (by bounds % on w(e) and DC2)≥we+ϵℓ/2   (for s≥8g+1, as % previously required) (6)

The credit of the edges and -clusters of is at least:

 c⋅(\textscMST(PX,Y∖{X,Y})+ediam(PX,Y∖{X,Y}))≥c⋅diam(PX,Y∖{X,Y})≥c(we+ϵℓ/2)\qed (7)

Let be the set of unpaid edges of incident to -clusters of clusters in Case 1 of Phase 3a.

###### Claim 14.

If , we can buy edges of using credits from each center and credits of releasing -clusters.

###### Proof.

Consider a cluster in Phase 3. Similar to Claim 9, releasing -clusters can pay for their incident edges in when . By construction, non-releasing clusters of are in . Since and by Equation (5) and since clusters now we are considering have low degree, there are at most

 4gϵ⋅20ϵ=O(gϵ2)

edges of incident to non-releasing -clusters. Thus, if , . That implies credits of suffice to pay for all edges of incident to non-releasing -clusters. ∎

##### Case 2: ediam(PX,Y)>2ℓ

Refer to Figure 3(b). Let and be minimal affices of such that each has effective diameter at least . We construct a new cluster consisting of (the -clusters and edges of) , , and and edge . We refer to as the center of the new cluster.

We apply Case 1 to all edges of satisfying the condition of Case 1 until no such edges exist. We then apply Case 2 to all remaining edges of satisfying the conditions of Case 2. After each new cluster is created (by Case 1 or 2), we delete the -clusters in the new cluster from , reassign the resulting components of to the sets of HD- and LD-components. At the end, any edge of with both endpoints in the same HD-path have both endpoints in two disjoint affixes of effective diameter less than .

We bound the diameter and credit of the centers of clusters in Case 2 of Phase 3a in Phase 3b.

#### Phase 3b: Edges of Ei between HD-paths

Let be an edge of that connects -cluster of HD-path to -cluster of different HD-path such that none affix of effective diameter less than of contains and none affix of effective diameter less than of contains . Such edge is said to have both endpoints far from endpoint -clusters of and .

Let and be minimal edge-disjoint subpaths of that end at and each having effective diameter at least . ( and exist by the way we choose edge .) Similarly, define and . We construct a new cluster consisting of (the -clusters and edges of) and edge (see Figure 4). We refer to as the center of the new cluster. We then delete the -clusters in the new cluster from and , reassign the resulting components of and to the sets of HD- and LD-components. We continue to create such new clusters until there are no edges of connecting HD-paths with far endpoints.

We now bound the diameter and credits of the center of a cluster, say , that is formed in Case 2 of Phase 3a or in Phase 3b. By construction in both cases, consists of two paths and connected by an edge .

###### Claim 15.

The diameter of each cluster in Case 2 of Phase 3a and Phase 3b is at least and at most .

###### Proof.

The lower bound follows from the same argument as in the proof of Claim 10. Since the effective diameters of and are smaller than the effective diameters of and , the diameter of the new cluster is bounded by the sum of the diameters of and and . The upper bound follows from the upper bounds on these diameters as given in the proof of Claim 10. ∎ Figure 4: A cluster of C in Phase 4a. Thin edges are edges of MST, solid blue edges are edges of Ei and vertices are ϵ-clusters. Edges and vertices inside the dashed red curves are grouped into a new cluster.

We show how to pay for unpaid edges of incident to -clusters in Case 2 of Phase 3a and Phase 3b. W.l.o.g, we refer to as the center -cluster of . Let and where is the cluster-diameter path of . We save credits of -clusters in for maintaining invariant DC1. The center collects credits of -clusters in . We let other -clusters in to release their credits.

###### Claim 16.

The center of a cluster in Case 2 of Phase 3a or in Phase 3b has at least credits.

###### Proof.

If , has -clusters which have at least total credits by invariant DC1 for level . Since when , the claim holds. Thus, we assume that which implies . By construction, contains -clusters of at most two of four paths . Since each path has effective diameter at least , the -clusters of each path in have total diameter at least . By invariant DC1 for level , each path in has at least credits that implies the claim. ∎

Let be the set of unpaid edges of incident to -clusters of clusters in Case 2 of Phase 3a and clusters in Phase 3b.

###### Claim 17.

If , we can buy edges of using credits of the centers of clusters in Case 2 of Phase 3a and Phase 3b and credits of releasing -clusters.

###### Proof.

Similar to the proof of Claim 9, releasing -clusters of can buy their incident edges in when . By construction, non-releasing -clusters are in . Since and , there are at most edges of incident to non-releasing -clusters. When is sufficiently small (), . Thus, if , and hence, credits suffice to pay for all edges of incident to non-releasing -clusters of . ∎

### 3.4 Phase 4: Remaining HD-paths and LD-components

We assume that after Phase 3. The case when will be handled at the end of this section.

#### Phase 4a: LD-components

Consider a LD-component , that has effective diameter less than . By construction, must have an edge to a cluster, say , in formed in a previous phase. We include and an edge connecting and to . Let be the set of unpaid edges of that incident to -clusters merged into new clusters in this phase. We use credit of the center and -clusters in this phase to pay for . More details will be given in Phase 4b.

#### Phase 4b: Remaining HD-paths

Let be a HD-path. By construction, there is at least one edge connecting to an existing cluster in . Let be one of them. Greedily break into subpaths such that each subpath has effective diameter at least and at most . We call a subpath of a long subpath if it contains at least -clusters and short subpath otherwise. We process subpaths of in two steps. In Step 1, we process affixes of , long subpaths of and the subpath of containing an endpoint -cluster of . In Step 2, we process remaining subpaths of .

##### Step 1

If a subpath of contain an -cluster that is incident to , we merge to the cluster in that contains another endpoint -cluster of . We call the augmenting subpath of . We form a new cluster from each long subpath of and each affix of . It could be that one of two affixes of is augmenting. We repeatedly apply Step 1 for all HD-paths. The remaining cluster paths which are short subpaths of HD-paths would be handled in Step 2. We then pay for every unpaid edges of incident to -clusters in this step. We call a cluster a long cluster if it is a long subpath of and a short cluster if it is a short subpath of .

Let be the set of unpaid edges of incident to -clusters of long clusters. We show below that each long cluster can both maintain diameter-credit invariant and pay for its incident edges in using credits of its -clusters.

Let be the set of unpaid edges of incident to remaining -clusters involved in this step; those belong to augmenting subpaths and short affices of HD-paths. We can pay for edges of incident to -clusters in augmenting subpaths using the similar argument in previous phases. However, we must be careful when paying for other edges of that are incident to -clusters in short affices of . Since short affices of spend all credits of their children -clusters to maintain invariant DC1, we need to use credits of -clusters in to pay for edges of incident to short affices of .

##### Step 2

Let be a short subpath of . If edges of incident to -clusters of are all paid, we let become a new cluster. Suppose that -clusters in are incident to at least one unpaid edge of , say . We have:

###### Observation 18.

Edge must be incident to an -cluster merged in Phase 1.

###### Proof.

Recall that edges of incident to -clusters of clusters initially formed in previous phases except Phase 1 are in ; thus, they are all paid. By construction, edges of between two -clusters in the same cluster initially formed in Phase 1 are in which are also paid. Since is not an affix of , there is no unpaid edge between two -clusters of since otherwise would become a new cluster in Phase 3a; that implies the observation. ∎

We merge -clusters, edges of and to the cluster in that contains another endpoint of . This completes the clustering process. Let be the set of remaining unpaid edges of incident to -clusters involved in Step 2.

We now analyze clusters of which are formed or modified in Phase 4.

###### Claim 19.

Let be a short cluster. Then, and credits of -clusters and edges connecting -clusters in suffice to maintain invariant DC1 for .

Since