Rounding the Solution

Given a instance $`(G,L,c)`$, the overall algorithm begins by solving the for $`\gamma = \ceil{\tfrac{28M}{\eps^2}}`$. It then computes a solution for $`L^h`$ for the heavily-covered edges $`E^h`$, contracts $`L^h`$ to obtain a new graph $`\overline{G}`$ and decomposes $`\overline{G}`$ and the solution $`x`$ into $`\tfrac{10M}{\eps^2}`$-simple pairs $`(G^i,x^i), i \in [k]`$, as described in Section [sec:decomposition]. In the process, it also computes a set of links $`L^s`$ covering $`E^s`$ as defined in Lemma [lemma:decomposition].

Afterwards, we round each $`(G^i,x^i)`$ individually using the best of the following lemmas – the first one performs well for pairs where cross-links carry much of the cost and is based on the integrality of the odd-cut for instances with only cross- and up-links, while the second performs well for pairs where in-links carry much of the cost and is due to ; it relies on the bundle constraints. The output of the algorithm is then the union of $`L^h`$ and $`L^s`$ as computed by Lemma [lemma:decomposition], with the rounded solutions for all $`(G^i,x^i)`$.

\begin{equation}
  c(S) \leq 2c^\tp \xin + c^\tp \xcr.
\end{equation}

The idea of the rounding algorithm from  is as follows. Given a $`\tfrac{10M}{\eps^2}`$-simple pair $`(H,x)`$ with a $`\beta`$-center $`r`$, we replace all cross-links $`\ell =uv`$ by two up-links $`ur`$ and $`rv`$. After that, we split $`H`$ at $`r`$ into trees $`H_1,\dots, H_m`$ such that $`r`$ is part of all trees. Since we no longer have cross-links, we can split $`x`$ into feasible solutions $`x^1, \dots, x^m`$ for the bundle for $`H_1,\dots,H_m`$ with $`\supp(x^i) \cap \supp(x^j) = \emptyset`$ if $`i \neq j`$. Every sub-tree $`H_1, \dots, H_m`$ of $`H`$ has at most $`\tfrac{10M}{\eps^2}+1`$ leaves (the root causes the additional $`+1`$); it can then be shown that every $`H_j`$ was a $`\ceil{\tfrac{28M}{\eps^2}}`$-bundle in the original graph, allowing us to use a bundle constraint to bound the rounding cost.

\begin{equation}
  c(S) \leq c^\tp \xin + 2c^\tp \xcr + |V^h \cap V[H]|
\end{equation}

For completeness, we give a proof of this lemma in Appendix [appendix:rounding]. Together, these two lemmas allow us to proof the following theorem.

\begin{aligned}
  c(S) &\leq \min(2c^\tp \xin + c^\tp \xcr,c^\tp \xin + 2c^\tp \xcr + |V^h \cap V[H]|)\\
         &\leq \frac{1}{2}\left(2c^\tp \xin + c^\tp \xcr + c^\tp \xin + 2c^\tp \xcr + |V^h \cap V[H]|\right)\\
             &\leq \frac{3}{2} c^\tp \xin + \frac{3}{2} c^\tp \xcr + \frac{1}{2} |V^h \cap V[H]|.\\
 \end{aligned}

\begin{aligned}
  \sum_{i \in [k]} c(S_i) \leq |V^h| + \sum_{i \in [k]} \frac{3}{2}c^\tp x^i = |V^h| + \frac{3}{2} c^\tp x.
 \end{aligned}

Decomposition of solutions

In this section, we will discuss how we use solutions of the to solve a instance approximately. We use the approach of , with two major differences:

• We use the with $`\gamma = \ceil{\tfrac{28M}{\eps^2}}`$ instead of the bundle as basis for the decomposition and subsequent rounding.

• The additional $`\set{0,\tfrac{1}{2}}`$-Chvátal-Gomory cuts in the compared to the bundle allow us to round instances with only up-links and cross-links without increasing cost. Since in-links can be split into two up-links, this yields a rounding that increases the cost incurred by in-links by a factor of 2 and leaves the cost of cross-links unchanged. This replaces a rounding based on a reduction to edge cover in  that increases the cost of in-links by a factor of $`2\lambda`$ and the cost of cross-links by a factor $`\tfrac{4}{3}\tfrac{\lambda}{\lambda-1}`$ for some $`\lambda > 1`$.

For the decomposition, we will assume that we are given a shadow-complete instance $`(G,L,c)`$ and a fractional solution $`x`$ to the for $`\gamma = \ceil{\tfrac{28M}{\eps^2}}`$. An instance $`(G,L,c)`$ is shadow-complete, if for every link $`\ell \in L`$ all its shadows – links $`\ell'`$ with $`P_{\ell'} \subseteq P_\ell`$ – are also in $`L`$. ¹ We assume that the cost of a shadow $`\ell'`$ of $`\ell`$ fulfills $`c_{\ell'} \leq c_\ell`$ – otherwise, we can set $`c_{\ell'} := c_\ell`$. The decomposition begins by computing a solution $`L^h`$ that covers all heavily-covered edges

E^h := \setc{e \in E}{x(\cov(e)) \geq \tfrac{2}{\eps}}

then contracts $`L^h`$ and obtains a new tree $`\overline{G}`$ that only contains edges $`e`$ with $`x(\cov(e)) < \tfrac{2}{\eps}`$. Then, we decompose $`\overline{G}`$ and $`x`$ by splitting repeatedly along $`\alpha`$-thin edges, which are defined as follows.

For an edge $`e = \set{u,v}`$, let $`G^u`$ and $`G^v`$ be the sub-trees of $`\overline{G}`$ that are obtained by deleting $`e`$, with $`G^u`$ containing $`u`$ and $`G^v`$ containing $`v`$, respectively. An edge $`e = \set{u,v}`$ is $`\alpha`$-thin with respect to $`x`$ for an $`\alpha \in \R_{\geq 0}`$ if the costs of links on both sides of the edge are at least $`\alpha`$ (we ignore links covering $`e`$ here):

\begin{equation}
\label{eq:alphathin}
 \sum_{\ell \in L, \ell \subseteq V[G^u]} c_\ell x_\ell \geq \alpha, \quad \sum_{\ell \in L, \ell \subseteq V[G^v]} c_\ell x_\ell \geq \alpha.
\end{equation}

If $`\overline{G}`$ has an $`\alpha`$-thin edge $`e = \set{u,v}`$ with respect to $`x`$, we split $`\overline{G}`$ and $`x`$ along $`e`$:

We obtain $`x^u`$ and $`x^v`$ by splitting all links covering $`e`$ in $`x`$ into two parts, the part in $`G^u`$ and the part in $`G^v`$:

x^u_{\ell=pq}\ := \begin{cases} 
                     x_\ell                                      & \text{if } p,q \in V[G^u] \setminus\set{u},\\ 
                                         0                                           & \text{if } p \in V[G^v]\text{ or }q \in V[G^v], \\ 
                                         x_\ell + \displaystyle\sum\limits_{\substack{\ell' \in \cov(e)\\p\in\ell'}} \frac{c_{\ell'}}{c_\ell} x_{\ell'} & \text{if } p \in V[G^u], q = u,
                                        \end{cases} \quad \text{for all } \ell \in L.

$`x^v`$ is defined symmetrically. Notice that $`\tfrac{c_{\ell'}}{c_\ell} \geq 1`$: this term ensures that the cost in the sub-trees does not decrease – this is important for the bundle constraints. Figure 1 depicts this splitting. If $`G^u`$ contains an $`\alpha`$-thin edge with respect to $`x^u`$, we repeat this procedure for $`G^u`$ and $`x^u`$, and similarly for $`G^v`$ and $`x^v`$ if $`G^v`$ contains an $`\alpha`$-thin edge with respect to $`x^v`$. At the end, we obtain a set of pairs $`\setc{(G^i,x^i)}{i \in [k]}`$ such that no $`G^i`$ contains an $`\alpha`$-thin edge with respect to $`x^i`$. We refer to the contraction of $`E^h`$ and the repeated edge-splitting as $`\alpha`$-thin edge decomposition of $`G`$ and $`x`$. Notice that we have $`\supp(x^i) \cap \supp(x^j) = \emptyset`$ for $`i \neq j`$ – this allows  and us to round each $`(G^i,x^i)`$ independently from each other later on.

In the next lemma, we list important properties that the decomposition $`(G^i,x^i), i \in [k]`$ has – these properties were shown in  for the bundle , but they also apply to the . One of the key properties is that the pairs $`(G^i,x^i)`$ are $`\beta`$-simple. A pair $`(G^i,x^i)`$ is called $`\beta`$-simple for a $`\beta \in \N`$, if there exists a $`\beta`$-center $`v \in V[G^i]`$ such that removal of $`v`$ decomposes $`G^i`$ into a forest of $`t`$ trees $`K_1,\dots, K_t`$ such that for all $`j \in [t]`$:

1. the cost of links in $`K_j`$ is bounded by $`\beta`$:

   \begin{equation}
   \label{eq:beta1}
        \sum_{\ell \in L, \ell \subseteq V[K_j]} c_\ell x_\ell \leq \beta,
   \end{equation}
   

2. $`K_j`$ has at most $`\beta`$ leaves.

Notice that the trees $`K_1,\dots,K_t`$ contain neither the $`\beta`$-center $`v`$ nor edges incident to $`v`$.

The proof of this lemma can be found in Appendix [appendix:decomposition], and is based on . Properties 1 and 2 allow us to round each $`x^i`$ individually, and property 3 makes it possible to employ a rounding algorithm from . Properties 4 and 5 ensure that the decomposition costs us only a factor of $`(1+O(\eps))`$, and that the edges not contained in a $`G^i`$ can also be covered at cost $`O(\eps)\OPT`$. Thus, the approximation guarantee is dominated by how well we can round the individual $`x^i`$ solutions.

A Stronger

In this section, we introduce the which strengthens the bundle used in . We use our in the algorithmic framework described in Section [sec:decomposition] to obtain better approximation guarantees. The additional constraints that we need are Chvátal-Gomory cuts obtained from the cut .

Deriving and Separating the Odd-Cut Constraints

Derivation.

Consider an of the form $`\min \{c^\intercal x \mid Ax \geq b,\ x \in \Z^n\}`$, with $`A \in \R^{m \times n}`$ and $`b \in \R^{m}`$. A Chvátal-Gomory cut is a constraint of the form $`\lambda^\intercal A x \geq \lceil \lambda^\intercal b \rceil`$, where the vector of multipliers $`\lambda \in \R^m_{\geq 0}`$ is chosen in such a way that $`\lambda^\intercal A \in \Z^n`$. Clearly, any such constraint is valid for the integer solutions of the . It is well known that the cuts obtained for $`\lambda \in [0,1)^m`$ imply the cuts obtained for larger multipliers, hence one can assume $`\lambda \in [0,1)^m`$ without loss of generality. (A proof of this fact and more background on the Chvátal-Gomory cuts can be found, for instance, in .) In case we restrict further $`\lambda`$ to be in $`\{0,\frac{1}{2}\}^m`$, we obtain a $`\{0,\frac{1}{2}\}`$-Chvátal-Gomory cut  . These cuts are a proper specialization of the Chvátal-Gomory cuts, and are precisely the cuts that we use here.

A $`\{0,\frac{1}{2}\}`$-Chvátal-Gomory cut for the cut is any constraint of the form

\begin{equation}
\label{eq:0-1/2-cut} \sum_{e \in E[G]} \lambda_e x(\cov(e)) + \sum_{\ell \in L} \mu_\ell x_\ell \geq \Big\lceil\sum_{e \in E[G]}\lambda_e\Big\rceil
\end{equation}

where $`\lambda \in \{0,\frac{1}{2}\}^{E[G]}`$ and $`\mu \in \{0, \frac{1}{2}\}^{L}`$ are such that the coefficients in the left-hand side are all integral. Notice that for any fixed $`\lambda`$, there is a unique $`\mu`$ that achieves this.

Let $`K := \supp(\lambda) = \{e \in E[G] \mid \lambda = \frac{1}{2}\}`$. Since $`G`$ is a tree, there exists a (not necessarily connected) set $`S \subseteq V[G]`$ such that $`K = \delta_G(S)`$. Notice that the right-hand side of [eq:0-1/2-cut] is $`\lceil |\delta_G(S)|/2 \rceil`$, and that the cut is redundant whenever $`|\delta_G(S)|`$ is even.

Let $`\pi(S)`$ denote the multiset of links $`\ell`$ such that $`P_\ell`$ intersects $`\delta_G(S)`$, in which the multiplicity of $`\ell`$ is defined as $`\lceil \frac{1}{2} |P_\ell \cap \delta_G(S)| \rceil`$ (see Figure 2). Now, assuming that $`|\delta_G(S)|`$ is odd, we can rewrite the constraint [eq:0-1/2-cut] as

\begin{equation}
\label{eq:0-1/2-cut_pi}
x(\pi(S))\geqslant \frac{|\delta_G(S)|+1}2\,.
\end{equation}

Let $`\mathcal{S}`$ denote the collection of all sets $`S`$ such that $`|\delta_G(S)|`$ is odd, and let the be the resulting from the cut after adding the odd-cut constraint [eq:0-1/2-cut_pi] for each set $`S \in \mathcal{S}`$:

\begin{lp}{min}{\sum_{\ell \in L} c_\ell x_\ell}
  \lpconstraint{x(\pi(S))}{\geq}{\frac{|\delta_G(S)|+1}{2}}{S \in \mathcal{S}}
  \finallpconstraint{x_\ell}{\geq}{0}{\ell\in L}
 \end{lp}

We remark that the first set of constraints includes in particular the covering constraint $`x(\cov(e)) \geq 1`$ for each tree edge $`e \in E[G]`$. Indeed, if $`S`$ denotes any of the two connected components arising when $`e`$ is deleted from $`G`$, we have $`\delta_G(S) = \{e\}`$ and thus $`S \in \mathcal{S}`$. The corresponding odd-cut constraint is the covering constraint.

Separation.

A nice fact that is key to our approach is that separation of the $`\{0,\frac{1}{2}\}`$-Chvátal-Gomory cuts arising from an $`\min \{c^\intercal x \mid Ax \geq b,\ x \in \Z^n\}`$ (here we take $`A \in \Z^{m \times n}`$ and $`b \in \Z^m`$) can be done in polynomial time whenever the matroid represented over $`GF(2)`$ by $`(I\,\bar{A})`$ is graphic (or co-graphic), where $`\bar{A} := A \pmod{2}`$ is the parity matrix of $`A`$. This was proved by Caprara and Fischetti , and implies directly that the constraints of the can be separated in polynomial time.

A more direct way to prove this is to rewrite inequality [eq:0-1/2-cut_pi] as the $`T`$-cut constraint ² $`x(\delta_L(S)) + y(\delta_G(S)) \geq 1`$ after introducing the slack variables $`y_e := x(\cov(e)) - 1 \geq 0`$ for $`e \in E[G]`$, and then solve the separation problem with the algorithm of Padberg and Rao .

To see this, let $`S \in \mathcal{S}`$ and let $`H`$ be the graph obtained from adding all links in $`L`$ to $`G`$. Assuming that $`y_e = x(\cov(e)) - 1`$ for all $`e \in E[G]`$, we can rewrite the odd-cut constraint [eq:0-1/2-cut_pi] as:

\begin{eqnarray*}
  x(\pi(S)) & \geq & \frac{|\delta_G(S)| + 1}2\\
  \iff \frac12\sum_{e \in \delta_G(S)}x(\cov(e)) + \frac12
  x(\delta_L(S)) & \geq &  \frac{|\delta_G(S)| + 1}2\\
  \iff \sum_{e \in \delta_G(S)} \underbrace{(x(\cov(e)) - 1)}_{=y_e} + 
  x(\delta_L(S)) & \geq & 1\\
  \iff (x,y)(\delta_H(S)) & \geq & 1\,.
\end{eqnarray*}

Above, the second form is the original expression of the odd-cut constraint as a $`\{0,\frac{1}{2}\}`$-Chvátal Gomory cut, see [eq:0-1/2-cut].

Exactness of the Odd-Cut when all the In-links are Up-links

An integer matrix $`M`$ is said to be the incidence matrix of a bidirected graph if $`\sum_i |M_{ij}| \leq 2`$ for every fixed column index $`j`$. A binet matrix is any matrix of the form $`B = S^{-1} R`$ where $`M = (S\, R)`$ is the incidence matrix of a bidirected graph with full row-rank and $`R`$ is a basis of $`M`$. Binet matrices are a generalization of network matrices and were introduced by Appa and Kotnyek .

Appa   proved the following result, extending results of Edmonds and Johnson  for incidence matrices of bidirected graphs.

Pick any root $`r \in V[G]`$ in tree $`G`$. We are now ready to recall and prove Theorem [theorem:exact], which provides a key property of the that we use in our approach to approximate .

We point out that for these instances, there is a combinatorial algorithm that finds an optimum solution in strongly polynomial time. This follows from our proof of Theorem [theorem:exact] and an algorithm of Edmonds and Johnson , see also .

The Odd-Cut Bundle

As its name indicates, the contains all of the constraints of the and additionally the bundle constraints for a constant $`\gamma \in \N`$. As before, let $`\mathcal{S}`$ denote the collection of all sets $`S \subseteq V[G]`$ such that $`|\delta_G(S)|`$ is odd, let $`\mathcal{B}_\gamma`$ be the set of all $`\gamma`$-bundles, and let $`\OPT(B)`$ the cost of an integral optimal solution for the instance obtained from contracting all edges not in $`B`$ (with respect to the given costs $`c_\ell`$). Then, the is given by:

\begin{lp}{min}{\sum_{\ell \in L} c_\ell x_\ell}
  \lpconstraint{x(\pi(S))}{\geq}{\frac{|\delta_G(S)|+1}{2}}{S \in \mathcal{S}}
  \lpconstraint{\sum_{\ell \in \cov(B)} c_\ell x_\ell}{\geq}{\OPT(B)}{B \in \mathcal{B}_\gamma}
  \finallpconstraint{x_\ell}{\geq}{0}{\ell\in L}
 \end{lp}

From Lemma [lemma:constantleaves] and the discussion above, we obtain the following result.

Proof of Lemma <a href="#lemma:decomposition" data-reference-type=“ref”

data-reference=“lemma:decomposition”>[lemma:decomposition]

In this section, we prove that the properties of the decomposition in Lemma [lemma:decomposition] also hold if we use the as a basis for the decomposition instead of the bundle . The proof ideas are all due to . Notice that we use $`\gamma = \ceil{\frac{28M}{\eps^2}}`$ and $`\beta = \frac{10M}{\eps^2}`$ instead of $`\gamma = \ceil{\frac{56M}{\eps^2}}`$ and $`\beta = \frac{12M}{\eps^2}`$ in ; this is the result of slightly improved estimations. In particular, we use that:

• Links covering an edge $`e = \set{u,v}`$ can cover at most one leaf in the subtrees $`G^u`$ and $`G^v`$. This argument allows us to reduce $`\beta`$ from $`\frac{12M}{\eps^2}`$ to $`\frac{10M}{\eps^2}`$.

• The cost of links completely in a sub-tree of a $`\tfrac{10M}{\eps^2}`$-simple pair is actually bounded by $`\tfrac{4M}{\eps^2}`$, even though the pair might not be $`\tfrac{4M}{\eps^2}`$-simple due to the number of leaves of the sub-trees.

• The previous point allows us bound $`|V^h|`$ by $`\tfrac{6M}{\eps^2}`$; together with the improved $`\beta = \frac{10M}{\eps^2}`$ we obtain $`\gamma = \ceil{\frac{28M}{\eps^2}}`$.

Details can be found in the proof below.

Introduction

The tree augmentation problem (weighted or unweighted) is a fundamental and intensively studied problem in the area of network design, see for example the surveys by Khuller  and Kortsarz and Nutov . While already in the unweighted case the problem is known to be -hard, the best algorithms for and achieve approximation factors of $`2`$ and $`\tfrac{3}{2}`$ respectively. One of the main open questions about these problems is to improve the quality of approximation algorithms.

Adjiashvili  recently managed to push the approximation guarantee for below $`2`$ in case the link costs are bounded by a constant $`M`$, which was the first improvement in over 35 years that did not restrict the structure of the tree or the links. His algorithm is based on an that strengthens the standard for the problem. Letting $`\cov(e)`$ denote the set of links connecting distinct connected components of $`G - e`$, for each tree edge $`e \in E`$, the standard for is

\begin{alignat}
{4}
 & \text{min}    &       & \sum_{\ell \in L}           c_\ell &x_\ell &       & \label{eq:cut1}\\
 & \text{s.\,t.} & \quad & \sum_{\mathclap{\ell \in \cov(e)}} &x_\ell &\geq 1 & \qquad &\text{for all } e\in E,\label{eq:cut2}\\
 &               &       &                                    &x_\ell &\geq 0 & \qquad &\text{for all } \ell \in L.\label{eq:cut3}
\end{alignat}

This is known as the cut .

Our results.

We add to the cut all its , thus obtaining a new for that we call the . The is key to our approach. It has three extremely useful properties:

• One can solve the efficiently, even though separating is -hard in general  .

• The is compatible with the decomposition approach of  to split the given instance and solution into well-structured independent instances together with their own local solutions.

• The is exact if for a certain choice of root $`r \in V[G]`$, every link $`\ell`$ connects either two different connected components of $`G - r`$ (in which case $`\ell`$ is called a cross-link) or some node of $`G`$ to one of its ancestors (in which case $`\ell`$ is called an up-link).

We prove the last property by establishing that the constraint matrix of the cut is an integral binet matrix. These matrices were introduced by Appa and Kotnyek  as a generalization of network matrices. Relying on earlier work by Edmonds and Johnson , Appa   proved that the integer hull of polyhedra of the form $`\setc{x}{Ax \geq b,\ x \geq 0}`$ can be described by whenever $`A`$ is an integral binet matrix and $`b`$ is an integer vector. This results in the following theorem.

Although the alone might be sufficient to obtain a $`\tfrac{3}{2}`$-approximation for (or maybe even better approximation), we combine the odd-cut with the bundle constraints from , resulting in the . This last is the one that we use in our algorithm.

We follow the decomposition approach of . After splitting the given instance and its optimum solution into independent rooted instances and corresponding solutions (respectively), at an extra cost of $`\varepsilon \OPT`$, Adjiashvili applies to each one of the local instances two distinct procedures producing feasible solutions, whose cost is bounded in terms of the local solution.

One of the two procedures of  produces an integer solution of cost at most $`c^\intercal x^\mathrm{in} + 2 c^\intercal x^\mathrm{cr} + \varepsilon \OPT`$, where $`c^\intercal x^\mathrm{in}`$ is the local cost on in-links (defined as all the links that are not cross-links) and $`c^\intercal x^\mathrm{cr}`$ is the local cost on cross-links. This is the part of the analysis where bundle constraints are used. We keep this procedure as is in our algorithm.

Using Theorem [theorem:exact], we improve the other procedure of  to obtain an integer solution of cost at most $`2 c^\intercal x^\mathrm{in} + c^\intercal x^\mathrm{cr}`$. This gives a significant improvement in the approximation factor since combining both procedures, we can construct an integer solution in each of the local instances of cost at most

\begin{align*}
&\ \min\left\{c^\intercal x^\mathrm{in} + 
2 c^\intercal x^\mathrm{cr} + \delta,
2 c^\intercal x^\mathrm{in} + 
c^\intercal x^\mathrm{cr}\right\}\\
\leq &\
\frac12 \left(c^\intercal x^\mathrm{in} + 
2 c^\intercal x^\mathrm{cr} + \delta\right)
+ \frac12 \left(2 c^\intercal x^\mathrm{in} + 
c^\intercal x^\mathrm{cr}\right)\\
\leq&\ \frac32 \left(c^\intercal x^\mathrm{in} + 
c^\intercal x^\mathrm{cr}\right) + \delta,
\end{align*}

where $`\delta`$ is a small quantity whose sum across the local instances is at most $`\varepsilon \OPT`$. This yields our main result.

This result is the best known for with link costs bounded by a constant, and a significant improvement over the previously known $`\approx 1.96418+\eps`$-approximation . For , it is the best -based result and matches the results of the -based algorithm of , while only being the $`\eps`$-term worse than the best overall algorithm  . Finally, we point out that while the approximation factors are improving, the proofs are actually getting simpler, which we see as another indication of the power behind our approach.

Related work.

Frederickson and JáJá  showed that is -hard even if the tree has constant diameter and the link costs are either 1 or 2. For , Cheriyan   proved that the problem is -hard even if the links form a cycle on the leaves of the tree. Kortsarz, Krauthgamer and Lee  then showed that even is hard, meaning that these problems have no , unless $`\PT = \NP`$.

For , there are several 2-approximations known, the first given by Frederickson and Jájá  and then simplified by Khuller and Thurimella . The primal-dual approach of Goemans   and the iterative rounding algorithm of Jain  also give a 2-approximation for this problem. Until recently, the factor of 2 was best known for general trees. Adjiashvili  then managed to give a $`\delta+\eps`$-approximation for the case where all link costs are bound by a constant, for any small $`\eps > 0`$ and $`\delta = \tfrac{8(23+3\sqrt{5})}{121} \approx 1.96418`$.

For , the best known approximation is a $`\tfrac{3}{2}`$-approximation algorithm by Kortsarz and Nutov , which is purely combinatorial. There is also a $`\tfrac{3}{2}+\eps`$-approximation for any $`\eps > 0`$, given by Cheriyan and Gao , which is also combinatorial but whose analysis is based on an relaxation, showing an integrality gap of $`\tfrac{3}{2}+\eps`$ for the . As far as -based algorithms are concerned, the state of the art was a $`\tfrac{7}{4}`$-approximation due to Kortsarz and Nutov  until Adjiashvili  gave a $`\tfrac{5}{3}+\eps`$-approximation based on the bundle , for any small $`\eps > 0`$.

For special classes of trees, there are better approximation guarantees known. For example, Cohen and Nutov  described a $`1+\ln 2 \approx 1.6931`$-approximation for for trees with constant radius. For the special case of where every link ends in two leaves of the tree, Maduel and Nutov  gave a $`\tfrac{17}{12}`$-approximation. In case that the tree has radius 3 or 2, they could strengthen the bound to $`\tfrac{11}{8}`$ and $`\tfrac{4}{3}`$, respectively.

In terms of lower bounds on integrality gaps, Cheriyan   conjectured that the cut has an integrality gap of $`\tfrac{4}{3}`$ for . However, this conjecture has been refuted by Cheriyan  , who proved that the integrality gap of the cut is at least $`\tfrac{3}{2}`$, even for .

Proof of Lemma <a href="#lemma:david" data-reference-type=“ref”

data-reference=“lemma:david”>[lemma:david]

\begin{equation*}
  y_{\ell=uv} := \begin{cases} x_\ell & \ell \in \supp(\xin), r \not\in \ell\\ x_\ell + \sum_{\substack{\ell' \in \supp(\xcr)\\v \in \ell'}} x_{\ell'} & u=r,v\in V[H]\setminus\set{r} \\ 0 & \text{else} \end{cases} \quad \text{ for all } \ell \in L.
\end{equation*}

\begin{equation*}
  z_{\ell} := \begin{cases} y_\ell & \ell \subseteq V[\overline{H}]\\ 0 & \text{else} \end{cases} \quad \text{ for all } \ell \in L.
\end{equation*}

\begin{equation*}
     \OPT(E[\overline{H}]) \leq c^\tp z + |V^h \cap V[\overline{H}]|.
\end{equation*}

c^\tp z = \sum_{\ell \in \cov(\overline{H})} c_\ell z_\ell \geq \sum_{\ell \in \cov(\overline{H})}   c_\ell x_\ell \geq \OPT(E[\overline{H}])