Approximating $k$-connected $m$-dominating sets

TL;DR

This paper develops improved approximation algorithms for the $k$-connected $m$-dominating set problem in graphs, achieving better ratios especially for unit disc graphs and uniform weights, advancing the theoretical understanding of this problem.

Contribution

It introduces new approximation ratios for the $(k,m)$-CDS problem, notably reducing the ratio for general graphs to $O(k ln n)$ and providing the first sublinear and polylogarithmic ratios for unit disc graphs.

Findings

Improved approximation ratio $O(k ln n)$ for general graphs.

First sublinear ratio for the problem in unit disc graphs.

Polylogarithmic ratio $O( ln^2 k)/ ln$ for $m extgreater= (1+ ln)k$.

Abstract

A subset $S$ of nodes in a graph $G$ is a $k$-connected $m$-dominating set ($(k,m)$-cds) if the subgraph $G[S]$ induced by $S$ is $k$-connected and every $v \in V \setminus S$ has at least $m$ neighbors in $S$. In the $k$-Connected $m$-Dominating Set ($(k,m)$-CDS) problem the goal is to find a minimum weight $(k,m)$-cds in a node-weighted graph. For $m \geq k$ we obtain the following approximation ratios. For general graphs our ratio $O(k \ln n)$ improves the previous best ratio $O(k^2 \ln n)$ and matches the best known ratio for unit weights. For unit disc graphs we improve the ratio $O(k \ln k)$ to $\min\left\{\frac{m}{m-k},k^{2/3}\right\} \cdot O(\ln^2 k)$ -- this is the first sublinear ratio for the problem, and the first polylogarithmic ratio $O(\ln^2 k)/\epsilon$ when $m \geq (1+\epsilon)k$; furthermore, we obtain ratio $\min\left\{\frac{m}{m-k},\sqrt{k}\right\} \cdot O(\ln^2 k)$ for uniform weights. These results are obtained by showing the same ratios for the Subset $k$-Connectivity problem when the set $T$ of terminals is an $m$-dominating set with $m \geq k$.

Equations

$$|A\cap S_i|\ge k-(|\partial \mathbb{A}\cap T_{i-1}|+d_{G[T_i]}(\mathbb{A}))\ge k-(|\partial \mathbb{A}|+d_{G[T]}(\mathbb{A}))=k-\ell .$$

$$\frac{w(S)}{\mathsf{opt}}=\frac{O(\ln k)}{k-\ell }\left(p+\frac{(m+p{)}^2}{(m+p-\ell {)}^2}\right).$$

$$\frac{w(S)}{\mathsf{opt}}=\sum _{\ell =0}^{k-1}\frac{O(\ln \ell )}{k-\ell }\min \left\{\frac{m}{m-\ell },{\ell }^{2/3}\right\}=O({\ln }^{2}k)\cdot \min \left\{\frac{m}{m-k+1},k^{2/3}\right\}$$

$$\begin{array}[]{lllll}&\tau({\cal D}_{T})=&\min&\sum_{v\in V\setminus T}w_{v}x_{v}&\\ &&\ \mbox{s.t.}&\sum_{v\in\Gamma(A)\setminus T}x_{v}\geq 1&\forall A\in{\cal D}_{T}\\ &&&x_{v}\geq 0&\forall v\in V\setminus T\end{array}$$

$$\displaystyle\begin{array}[]{lllll}&\tau({\cal C})=&\min&\sum_{v\in V\setminus T}w_{v}x_{v}&\\ &&\ \mbox{s.t.}&\sum_{v\in\Gamma(A)\setminus T}x_{v}\geq 1&\forall A\in{\cal C}\\ &&&x_{v}\geq 0&\forall v\in V\setminus T\end{array}$$

$$\frac{w(S_r)}{\mathsf{opt}}=O\left(\frac{q+\ell }{q(k-\ell )}\right).$$

$$\frac{w(T\setminus T_0)}{\mathsf{opt}}=\frac{O(\ln k)}{k-\ell }\cdot p$$

$$\frac{w(S)}{\mathsf{opt}}=|R|\cdot O\left(\frac{q+\ell }{q(k-\ell )}\right)=\frac{O(\ln \ell )}{k-\ell }\frac{|T|}{|T|-\ell }\frac{q+\ell }{q}=\frac{O(\ln k)}{k-\ell }\frac{(m+p{)}^2}{(m+p-\ell {)}^2}$$

$$\frac{w(S)}{\mathsf{opt}}=|R|\cdot O\left(\frac{q+\ell }{q(k-\ell )}\right)=\frac{O(\ln \ell )}{k-\ell }\frac{q+\ell }{q}=\frac{O(\ln k)}{k-\ell }\frac{m+p}{m+p-\ell }$$

Full text

Keywords: $k$-connected graph; $m$-dominating set; approximation algorithm; rooted subset $k$-connectivity; subset $k$-connectivity

1 Introduction

All graphs in this paper are assumed to be simple, unless stated otherwise. A (simple) graph is $k$-connected if it has $k$ pairwise internally node disjoint paths between every pair of its nodes; in this case the graph has at least $k+1$ nodes. A subset $S$ of nodes in a graph $G$ is a $k$-connected set if the subgraph $G[S]$ induced by $S$ is $k$-connected; $S$ is an $m$-dominating set if every $v\in V\setminus S$ has at least $m$ neighbors in $S$. If $S$ is both $k$-connected and $m$-dominating set then $S$ is a $k$-connected $m$-dominating set, or $(k,m)$-cds for short. A graph is a unit-disk graph if its nodes can be located in the Euclidean plane such that there is an edge between nodes $u$ and $v$ iff the Euclidean distance between $u$ and $v$ is at most $1$. We consider the following problem for $m\geq k$ both in general graphs and in unit-disc graphs.

$k$-Connected $m$-Dominating Set ($(k,m)$-CDS)

Input: A graph $G=(V,E)$ with node weights $\{w_{v}:v\in V\}$ and integers $k,m$.

Output: A minimum weight $(k,m)$-cds $S\subseteq V$.

For motivation we refer the reader to recent papers of Zhang, Zhou, Mo, and Du [10] and of Fukunaga [2], where they obtained in unit-disc graphs ratios $O(k^{3}\ln k)$ and $O(k^{2}\ln k)$, respectively. This was improved to $O(k\ln k)$ in [9], where is also given ratio $O(k^{2}\ln n)$ in general graphs. Our main results is:

Theorem 1.1

$(k,m)$-CDS* with $m\geq k$ admits the following approximation ratios: $O(k\ln n)$ in general graphs, $\min\left\{\frac{m}{m-k},k^{2/3}\right\}\cdot O(\ln^{2}k)$ in unit disc graphs, and $\min\left\{\frac{m}{m-k},\sqrt{k}\right\}\cdot O(\ln^{2}k)$ in unit disc graphs with unit weights.*

For general graphs our ratio $O(k\ln n)$ improves the previous ratio $O(k^{2}\ln n)$ of [9] and matches the best known ratio for unit weights of [11]. For unit disc graphs our ratio $\min\left\{\frac{k}{m-k},k^{2/3}\right\}\cdot O(\ln^{2}k)$ improves the previous best ratio $O(k\ln k)$ of [9]; this is the first sublinear ratio for the problem, and for any constant $\epsilon>0$ and $m=k(1+\epsilon)$ the first polylogarithmic ratio $O(\ln^{2}k)/\epsilon$.

Let us say that a graph with a set $T$ of terminals and a root $r\in T$ is $k$-$(T,r)$-connected if it has $k$ internally node disjoint $rt$-paths for every $t\in T\setminus\{r\}$. Similarly, a graph is $k$-$T$-connected if it has $k$ internally node disjoint $st$-paths for every $s,t\in T$. A reason why the case $m\geq k$ is easier than the case $m<k$ is given in the following statement (a proof can be found in [10, 2, 9]).

Lemma 1

Let $T$ be a $k$-dominating set in a graph $H=(U,F)$. If $H$ is $k$-$(T,r)$-connected then $H$ is $k$-$(U,r)$-connected; if $H$ is $k$-$T$-connected then $H$ is $k$-connected.

The above lemma implies that in the case $m\geq k$ $(k,m)$-CDS partial solutions have the property that the union of a partial solution and a feasible solution is always feasible – this enables to construct the solution iteratively. Specifically, most algorithms for the case $m\geq k$ start by computing just an $m$-dominating set $T$; the best ratios for $m$-Dominating Set are $\ln(\Delta+m)$ in general graphs [1] and $O(1)$ in unit disc graphs [2]. By invoking just these ratios, Lemma 1 enables to reduce $(k,m)$-CDS with $m\geq k$ to following (node weighted) problem:

Subset $k$-Connectivity

Input: A graph $G=(V,E)$ with node-weights $\{w_{v}:v\in V\}$, a set $T\subseteq V$ of terminals, and an integer $k$.

Output: A minimum weight $k$-$T$-connected subgraph of $G$.

The ratios for this problem are usually expressed in terms of the best known ratio $\beta$ for the following problem (in both problems we will assume w.l.o.g. that $w_{v}=0$ for all $v\in T$):

Rooted Subset $k$-Connectivity

Input: A graph $G=(V,E)$ with node-weights $\{w_{v}:v\in V\}$, a set $T\subseteq V$ of terminals, a root node $r\in T$, and an integer $k$.

Output: A minimum weight $k$-$(T,r)$-connected subgraph of $G$.

Currently, $\beta=O(k^{2}\ln|T|)$ [7]. From previous work it can be deduced that Subset $k$-Connectivity with $|T|\geq k$ admits ratio $\beta+k^{2}$. Add a new root node $r$ connected to a set $R\subseteq T$ of $k$ nodes by edges of cost zero. Then compute a $\beta$-approximate solution to the obtained Rooted Subset $k$-Connectivity instance. Finally, augment this solution by computing for every $u,v\in R$ a min-weight set of $k$ internally disjoint $uv$-paths. For the $(k,m)$-CDS problem with $m\geq k$ this already gives ratio $\beta+k^{2}=O(k^{2}\ln|T|)$ in general graphs. For the special case when $T$ is a $k$-dominating set the ratio $\beta+k^{2}$ was improved in [9] to $\beta+k-1$, since then in the final step it is sufficient to compute a min-weight set of $k$ internally disjoint $uv$-paths for pairs that form a forest on $R$.

We now consider unit disc graphs. Zhang et. al. [10] showed that any $k$-connected unit disc graph has a $k$-connected spanning subgraph of maximum degree $\leq 5k$. This implies that the node weighted case is reduced with a loss of factor $O(k)$ to the case of node induced edge costs – when $c_{uv}=w_{u}+w_{v}$ for every edge $e=uv\in E$. The edge costs version of Subset $k$-Connectivity admits ratio $O(k^{2}\ln k)$, which gives ratio $O(k^{3}\ln k)$ for $(k,m)$-CDS with $m\geq k$ in unit disc graphs. Fukunaga [2] obtained ratio $O(k^{2}\ln k)$ using a different approach – he considered the Rooted Subset Connectivity Augmentation problem, when $G[T]$ is $\ell$-$(T,r)$-connected and we seek a minimum weight $S\subseteq V\setminus T$ such that $G[T\cup S]$ is $(\ell+1)$-$(T,r)$-connected. In [7] it is shown that the augmentation problem decomposes into $O(k)$ “uncrossable” subproblems, and Fukunaga [2] designed a primal-dual $O(1)$-approximation algorithm for such an uncrossable subproblem in unit disc graphs. This gives ratio $O(\ell)$ for Rooted Subset Connectivity Augmentation in unit disc graphs. Furthermore, using the so called “backward augmentation analysis” Fukunaga showed that since his approximation is w.r.t. an LP, then sequentially increasing the $T$-connectivity by $1$ invokes only a factor of $O(\ln k)$, thus obtaining ratio $O(k\ln k)$ for Rooted Subset Connectivity Augmentation. He then combined this result with a decomposition of the Subset $k$-Connectivity problem into $k$ Rooted Subset $k$-Connectivity problems, and obtained ratio $O(k^{2}\ln k)$. As was mentioned, in [9] it is proved that ratio $\beta$ for Rooted Subset $k$-Connectivity implies ratio $\beta+k-1$ for $(k,m)$-CDS with $m\geq k$, which improves the ratio to $O(k\ln k)$.

However, it seems that previous reductions and methods alone do not enable to obtain ratio better than $O(k^{2}\ln|T|)$ in general graphs, or a sublinear ratio in unit disc graphs. These algorithm rely on the ratios and decompositions for the Rooted/Subset $k$-Connectivity problems from [7, 8], but these do not consider the specific feature relevant to $(k,m)$-CDS – that the set $T$ of terminals is an $m$-dominating set; note that then Subset $k$-Connectivity is equivalent to the problem of finding the lightest $k$-connected subgraph containing $T$, by Lemma 1. Here we change this situation by asking the following question:

If the set $T$ of terminals is an $m$-dominating set with $m\geq k$, what approximation ratios can we achieve for (node weighted) Subset $k$-Connectivity?

Our answer to this question is given in the following theorem, which is of independent interest, and note that it implies Theorem 1.1.

Theorem 1.2

The (node weighted) Subset $k$-Connectivity problem such that $T$ is an $m$-dominating set with $m\geq k$ admits the following approximation ratios: $O(k\ln n)$ in general graphs, $O(\ln^{2}k)\cdot\min\left\{\frac{m}{m-k+1},k^{2/3}\right\}$ in unit disc graphs, and $O(\ln^{2}k)\cdot\min\left\{\frac{m}{m-k+1},\sqrt{k}\right\}$ in unit disc graphs with unit weights.

In the proof of Theorem 1.2 we use several results and ideas from previous works [7, 8, 10, 2, 9]. As was mentioned, the best ratios for the Subset $k$-Connectivity are derived via reductions of [8, 9] from the ratios for the Rooted Subset $k$-Connectivity problem, so we will consider the latter problem; the currently best known ratio for this problem is $O(k^{2}\ln|T|)$ [7]. The algorithm of [7] works in $\ell$ iterations, where at iteration $\ell=0,\ldots,k-1$ it considers the augmentation problem of increasing the connectivity from $\ell$ to $\ell+1$. This is equivalent to covering a certain family ${\cal F}$ of “tight sets” (a.k.a. “deficient sets”), and the algorithm of [7] decomposes this problem into $O(\ell)$ uncrossable family covering problems; the ratio for covering each uncrossable family is $O(\ln n)$ in general graphs [7] and $O(1)$ in unit disc graphs [2].

However, a more careful analysis of the [7] algorithm reveals that in fact the number of uncrossable families is $O(\ell/q)+1$, where $q$ is the minimum number of terminals in a tight set. Specifically, the algorithm has an “inflation phase” that works towards reaching $q\geq\ell+1$ – in which case the entire family of tight sets becomes uncrossable, by repeatedly covering $O(\ell/q)$ uncrossable families to double $q$. Hence if $q_{0}$ is the initial value of $q$, the total number of uncrossable families that the algorithm covers is $1$ plus order of $\frac{\ell}{q_{0}}\left(1+\frac{1}{2}+\frac{1}{4}+\cdots\right)=O(\ell/q_{0})$. Note that a large part of the uncrossable families are covered at the beginning – when $q$ is small. One of our contributions is designing a different “lighter” inflation algorithms for increasing the parameter $q$. These algorithms just aim to cover the inclusion minimal tight sets by adding a light set $S$ of nodes, and then add $S$ to the set $T$ of terminals; if $T$ is a $k$-dominating set then adding any set $S$ to $T$ does not make the problem harder, by Lemma 1.

Our algorithms for covering inclusion minimal tight sets reduce the problem to a set covering type problem. In the case of general graphs the reduction is to a special case considered in [5] of the Submodular Covering problem; the ratio invoked by this procedure is only $O(\ln n)$ and if we apply it $p=\max\{2k-m-1,1\}$ times then we get $q\geq m-\ell+p(k-\ell)\geq k$ for all $\ell=0,\ldots,k-1$. In fact, we apply this procedure before considering the augmentation problems, but it guarantees that $q\geq k$ through all augmentation iterations. The same procedure applies in the case of unit disc graphs, but to avoid the dependence on $n$ in the ratio we use a different procedure. Specifically, we use the result of Zhang et. al. [10] that minimally $k$-connected unit disc graph has maximum degree $\leq 5k$, to reduce the problem of covering the family of tight sets to the Set-Cover problem with soft capacities. This approach gives ratio $\min\left\{\frac{m}{m-k},k^{2/3}\right\}\cdot O(\ln^{2}k)$.

In the rest of the paper we prove Theorem 1.2; Section 2 considers general graphs and Section 3 considers unit disc graphs.

2 General graphs

While edge-cuts of a graph correspond to node subsets, a natural way to represent a node-cut of a graph is by a pair of sets, as follows.

Definition 1

An ordered pair $\mathbb{A}=(A,A^{+})$ of subsets of $V$ with $A\subseteq A^{+}$ is called a biset; the set $\partial\mathbb{A}=A^{+}\setminus A$ is called the cut of $\mathbb{A}$. We say that $\mathbb{A}$ is a $(T,r)$-biset if $A\cap T\neq\emptyset$ and $r\in V\setminus A^{+}$. An edge covers a biset $\mathbb{A}$ if it has one end in $A$ and the other in $V\setminus A^{+}$. For an edge set/graph $J$ let $d_{J}(\mathbb{A})$ denote the number of edges in $J$ that cover $\mathbb{A}$.

By Menger’s Theorem $G$ is $k$-$(T,r)$-connected iff $|\partial\mathbb{A}|+d_{G[T]}(\mathbb{A})\geq k$ holds for every $(T,r)$-biset $\mathbb{A}$. Given a Rooted Subset $k$-Connectivity instance, we say that a $(T,r)$-biset $\mathbb{A}$ is a deficient biset if $|\partial\mathbb{A}|+d_{G[T]}(\mathbb{A})\leq k-1$.

We use the algorithm from [7] for Rooted Subset $k$-Connectivity. Two deficient bisets $\mathbb{A},\mathbb{B}$ are $T$-dependent if $A\cap T\subseteq\partial\mathbb{A}$ or $B\cap T\subseteq\partial\mathbb{B}$. A Rooted Subset $k$-Connectivity instance is $T$-independence-free if no pair of bisets are $T$-independent. We have the following from previous work [7].

Theorem 2.1 ([7])

$T$-independence-free Rooted Subset $k$-Connectivity instances admit ratio $O(k\ln|T|)$.

Clearly, a sufficient condition for an instance to be $T$-independence-free is:

Lemma 2

If for a Rooted Subset $k$-Connectivity instance $|A\cap T|\geq k$ holds for every deficient biset $\mathbb{A}$, then the instance is $T$-independence-free.

In the next two lemmas we show how to find an $O(k\ln n)$-approximate set $S\subseteq V\setminus T$ such that adding $S$ to $T$ result in a $T$-independence-free instance.

Lemma 3 (Inflation Lemma for general graphs)

There exists a polynomial time algorithm that given an instance of Rooted Subset $k$-Connectivity finds $S\subseteq V\setminus T$ such that $|A\cap S|\geq k-(|\partial\mathbb{A}|+d_{G[T\cup S]}(\mathbb{A}))$ holds for any $(T,r)$-biset $\mathbb{A}$, and $w(S)=O(\ln\Delta)\cdot{\sf opt}$.

Proof

The Centered Rooted Subset $k$-Connectivity problem is a particular case of the Rooted Subset $k$-Connectivity problem when all nodes of positive weight are neighbors of the root. This problem admits ratio $O(\ln\Delta)$ [5], where here $\Delta$ is the maximum degree of a neighbor of the root. We use this in our algorithm as follows:

Let $S^{*}$ and $S^{*}_{c}$ be optimal solutions to Rooted Subset $k$-Connectivity and the constructed Centered Rooted Subset $k$-Connectivity instances, respectively. For every $t\in T$ fix some set of $k$ internally disjoint $rt$-paths in the graph $G[T\cup S^{*}]$, and obtain a set $P_{t}$ by picking for each path the node in $S^{*}$ that is closest to $t$ on this path, if such a node exists. Let $P=\cup_{t\in T}P_{t}$. Then $P$ is a feasible solution to the constructed Centered Rooted Subset $k$-Connectivity instance, since for each $t\in T$, $G^{\prime}$ has $|P_{t}|$ internally disjoint $rt$-paths of length $2$ each that go through $P_{t}$, and $k-|P_{t}|$ paths that have all nodes in $T$. Furthermore, since $P\subseteq S^{*}$, $w(P)\leq w(S^{*})$. Thus $w(S^{*}_{c})\leq w(P)\leq w(S^{*})$, implying that $w(S)=O(\ln\Delta)\cdot w(S^{*})$.

Now let $\mathbb{A}$ be a $(T,r)$-biset on $T\cup S$. Then:

• •

$d_{G^{\prime}[T\cup S]}(\mathbb{A})=|A\cap S|+d_{G[T\cup S]}(\mathbb{A})$ by the construction.

• •

$|\partial\mathbb{A}|+d_{G^{\prime}[T\cup S]}(\mathbb{A})\geq k$ since $G^{\prime}[T\cup S]$ is $k$-$(T,r)$-connected.

Combining we get that $|\partial\mathbb{A}|+d_{G[T\cup S]}(\mathbb{A})+|A\cap S|\geq k$, as claimed. ∎

Our algorithms use the following simple procedure – Algorithm 2, that sequentially adds $p$ sets $S_{1},\ldots,S_{p}$ to an $m$-dominating set $T=T_{0}$ with $m\geq k$; in the case of general graphs considered in this section, each $S_{i}$ is as in Lemma 3.

Lemma 4

Suppose that we are given a Rooted Subset $k$-Connectivity instance such that $T$ is an $m$-dominating set in $G$ with $m\geq k$. If at each iteration $i$ at step 2 of Algorithm 2 we add to $T$ a set $S=S_{i}$ is as in Lemma 3, then at the end of the algorithm $w(T\setminus T_{0})=O(p\ln\Delta)\cdot{\sf opt}$, and $|A\cap T|\geq m-\ell+p(k-\ell)$ holds for any biset $\mathbb{A}$ on $T$ with $|\partial\mathbb{A}|+d_{G[T]}(\mathbb{A})=\ell\leq k-1$. In particular, if $p\geq\max\{2k-m-1,1\}$ then the resulting instance is $T$-independence-free.

Proof

The bound $w(T\setminus T_{0})=O(p\ln\Delta)\cdot{\sf opt}$ follows from Lemma 1 and the bound $w(S)=O(\ln\Delta)\cdot{\sf opt}$ in Lemma 3.

Let $\mathbb{A}$ be a biset as in the lemma. Let $T_{i}=T_{0}\cup S_{1}\cup\cdots\cup S_{i}$ be the set stored in $T$ at the end of iteration $i$, where $T_{0}$ is the initial set. Applying Lemma 3 on $T_{i-1}$ and $S_{i}$ we get

[TABLE]

In particular $A\cap S_{1}\neq\emptyset$. Any $v\in A\cap S_{1}$ has in $G[T]$ at least $m$ neighbors in $T_{0}$, and at most $\ell$ of them are not in $A$; thus $v$ has at least $m-\ell$ neighbors in $A\cap T_{0}$, so $|A\cap T_{0}|\geq m-\ell$. Since $T_{0},S_{1},\ldots,S_{p}$ are pairwise disjoint we get $|A\cap T|\geq|A\cap T_{0}|+\sum_{i=1}^{p}|A\cap S_{i}|\geq m-\ell+p(k-\ell)$. ∎

The proof of the following known statement can be found in [4], and the second part follows from Mader’s Undirected Critical Cycle Theorem [6].

Lemma 5

Let $H_{r}=(U,F)$ be a $k$-$(U,r)$-connected graph and $R$ the set of neighbors of $r$ in $H_{r}$. The graph $H=H_{r}\setminus\{r\}$ can be made $k$-connected by adding a set $J$ of new edges on $R$, and if $J$ is inclusion minimal then $J$ is a forest.

Note that an inclusion minimal edge set $J$ as in Lemma 5 can be computed in polynomial time, by starting with $J$ being a clique on $R$ and repeatedly removing from $J$ an edge $e$ if $H\cup(J\setminus e)$ remains $k$-connected.

Our algorithm for general graphs is as follows.

Except step 2, the algorithm is identical to the algorithm of [9] – the only difference is that step 2 improves the factor invoked by step 3. In [9] it is also proved that at the end of the algorithm $T\cup S\cup P$ is a $k$-connected set. The dominating terms in the ratio are invoked by steps 2 and 3, and they are both $O(k\ln n)$, while step 5 invokes just ratio $k-1$; thus the overall ratio is $O(k\ln n)$.

This concludes the proof of Theorem 1.2 for general graphs.

3 Unit disc graphs

Our goal in this section is to prove the following:

Lemma 6

Consider a Subset $k$-Connectivity instance on unit disc graph $G=(V,E)$ where $T$ is an $m$-dominating set with $m\geq k$ and $G[T]$ is $\ell$-connected, $\ell\leq k-1$. Then for any integer $1\leq p\leq\ell+1$ there exists a polynomial time algorithm that computes $S\subseteq V\setminus T$ such that $G[T\cup S]$ is $(\ell+1)$-connected and

[TABLE]

Furthermore, in the case of unit weights $\frac{w(S)}{\sf opt}=\frac{O(\ln k)}{k-\ell}\left(p+\frac{m+p}{m+p-\ell}\right)$.

Let us show that Lemma 6 implies the unit disc part of Theorem 1.2. We can apply the Lemma 6 algorithm sequentially, starting with an $O(1)$-approximate $m$-dominating set $T=T_{0}$, and at iteration $\ell=0,\ldots,k-1$ add to $T$ a set $S=S_{\ell}$ as in the lemma. In the case of arbitrary weights choosing $p=1$ if $m-\ell\geq\ell^{2/3}$ and $p=\ell^{2/3}$ otherwise gives $\frac{w(S_{\ell})}{\sf opt}=\frac{O(\ln\ell)}{k-\ell}\min\left\{\frac{m}{m-\ell},\ell^{2/3}\right\}$. Then denoting $S=S_{0}\cup\cdots\cup S_{k-1}$ we get:

[TABLE]

In the case of unit weights, choosing $p=1$ if $m-\ell\geq\sqrt{\ell}$ and $p=\sqrt{\ell}$ otherwise gives $\frac{w(S_{\ell})}{\sf opt}=\frac{O(\ln\ell)}{k-\ell}\min\left\{\frac{m}{m-\ell},\sqrt{\ell}\right\}$, and then by a similar analysis we get $\frac{w(S)}{\sf opt}=O(\ln^{2}k)\cdot\min\left\{\frac{m}{m-k+1},\sqrt{k}\right\}$.

In the rest of this section we prove Lemma 6, so let $G$, $T$, and $\ell$ be as in the lemma. We need some definitions and known facts concerning biset families.

Definition 2

A biset $\mathbb{A}$ on $V$ is deficient (w.r.t. $T$) if $A\cap T\neq\emptyset\neq T\setminus A^{+}$, $d_{G[T]}(\mathbb{A})=0$, and $|\partial\mathbb{A}|=\ell$. Let ${\cal D}_{T}$ denote the set of deficient bisets. We say that $v\in V\setminus T$ covers $\mathbb{A}\in{\cal D}_{T}$ if $v\in\Gamma(A)\setminus T$, where $\Gamma(A)$ denotes the set of neighbors of $A$ in $G$; $S\subseteq V\setminus T$ covers ${\cal F}\subseteq{\cal D}_{T}$ if every $\mathbb{A}\in{\cal F}$ is covered by some $v\in S$.

Note that $\mathbb{A}\in{\cal D}_{T}$ if and only if $\partial\mathbb{A}\subseteq T$ is a minimum node cut of $G[T]$, and $A\cap T$ is a union of some, but not all, connected components of $G[T]\setminus\partial\mathbb{A}$. The following lemma is known, c.f. [7, 2].

Lemma 7

$G[T\cup S]$* is $(\ell+1)$-connected if and only if $S$ covers ${\cal D}_{T}$.*

Thus we have the following LP-relaxation for the problem of finding a min-weight cover of ${\cal D}_{T}$:

[TABLE]

Note that if $\mathbb{A}\in{\cal D}_{T}$ then $\Gamma(A)\setminus T=\Gamma(A)\setminus\partial\mathbb{A}$, and thus the constraint $\sum_{v\in\Gamma(A)\setminus T}x_{v}\geq 1$ is equivalent to $\sum_{v\in\Gamma(A)\setminus\partial\mathbb{A}}x_{v}\geq 1$.

Definition 3

We say that $\mathbb{A}$** contains $\mathbb{B}$** and write $\mathbb{A}\subseteq\mathbb{B}$ if $A\subseteq B$ and $A^{+}\subseteq B^{+}$. Inclusion minimal members of a biset family ${\cal F}$ are called ${\cal F}$-cores.

Theorem 3.1 (Zhang, Zhou, Mo, & Du [10])

Any $k$-connected unit disc graph has a $k$-connected spanning subgraph of maximum degree $\leq 5k$.

Lemma 8 (Inflation Lemma for unit disc graphs)

There exists a polynomial time algorithm that computes $S\subseteq V\setminus T$ that covers the family ${\cal C}$ of cores of ${\cal D}_{T}$ and $w(S)=O(\ln k)\cdot\frac{{\sf opt}}{k-\ell}$.

Proof

The problem of covering ${\cal C}$ is essentially a (weighted) Set-Cover problem where for each $v\in V\setminus T$ the corresponding set has weight $w_{v}$ and consists of the cores covered by $v$. Then the greedy algorithm for Set-Cover computes a solution of weight $O(\ln|{\cal C}|)$ times the value of the standard Set-Cover LP

[TABLE]

For any $S^{\prime}\subseteq V\setminus T$ such that $G[T\cup S^{\prime}]$ is $k$-connected, any tight set $A$ has at least $k-\ell$ neighbors in $G[T\cup S^{\prime}]$, hence if $x^{\prime}$ is a characteristic vector of $S^{\prime}$ then $\frac{x^{\prime}}{k-\ell}$ is a feasible solution to the LP. Consequently, $\tau({\cal C})\leq\frac{\sf opt}{k-\ell}$.

In [[3], Lemma 3.5, Case II] it is proved that if $C\cap C^{\prime}$ for two distinct cores $\mathbb{C},\mathbb{C}^{\prime}$ then there is $P\subseteq V$ with $|P|\leq\ell+1$ such that $P\cap C$ for every $\mathbb{C}\in{\cal C}$. In this case $G$ has at most $\ell(\ell+1)$ distinct cores, since for every core $\mathbb{C}$ there is $s\in P\cap C$ and $t\in V\setminus C^{+}$, and for each $(s,t)\in P\times P$ there is at most one such core. Hence in the case $|{\cal C}|\leq\ell(\ell+1)$ we get a solution of weight $O(\ln\ell)\cdot\tau({\cal C})=O(\ln\ell)\cdot\frac{\sf opt}{k-\ell}$.

In the case $|{\cal C}_{\cal F}|>\ell(\ell+1)$ we have $C\cap C^{\prime}=\emptyset$ for any $\mathbb{C},\mathbb{C}^{\prime}\in{\cal C}$, and relying on Theorem 3.1 we modify this reduction such that every $v\in V\setminus T$ can cover at most $5k$ cores; this is essentially the Set-Cover with (soft) capacities problem. Specifically, for each pair $(v,J)$ where $v\in V\setminus S$ and $J$ is a set of at most $5k$ edges incident to $v$, we add a new node $v_{J}$ of weight $w_{v}$ with corresponding copies of the edges in $J$. In the obtained Set-Cover instance the maximum size of a set is at most $5k$, since the ${\cal F}$-cores are pairwise disjoint. Note that we do not need to construct this Set-Cover instance explicitly to run the greedy algorithm – we just need to determine for each $v\in V$ the maximum number of at most $5k$ not yet covered cores that can be covered by $v$. Since the ${\cal F}$-cores are pairwise disjoint, this can be done in polynomial time. Note that during the greedy algorithm we may pick pairs $(v,J)$ and $(v,J^{\prime})$ with distinct $J,J^{\prime}$ but with the same node $v$, but this only makes the solution lighter. Since in the Set-Cover instance the maximum set size is $5k$, the computed solution has weight $O(\ln k)\cdot\tau$, where here $\tau$ is an optimal LP-value of the modified instance. Now we argue in the same way as before that $\tau\leq\frac{{\sf opt}}{k-\ell}$. Consider a feasible solution $S^{\prime}\subseteq V\setminus T$ and an edge $J^{\prime}$ such that $G[T]\cup S^{\prime}\cup J^{\prime}$ is a spanning $k$-connected subgraph of $G[T\cup S^{\prime}]$ and $\deg_{J^{\prime}}(v)\leq 5k$ for all $v\in S^{\prime}$; such $J^{\prime}$ exists by Theorem 3.1. Let $x^{\prime}$ be the characteristic vector of the pairs $(v,J^{\prime}_{v})$ where $v\in S^{\prime}$ and $J^{\prime}_{v}$ is the set of edges in $J^{\prime}$ incident to $v$. Any tight set $A$ has at least $k-\ell$ neighbors in $G[T]\cup S^{\prime}\cup J^{\prime}$, hence $\frac{x^{\prime}}{k-\ell}$ is a feasible solution to the LP. Consequently, $\tau\leq\frac{{\sf opt}}{k-\ell}$. ∎

Corollary 1

If at step 2 of Algorithm 2 we add $S=S_{i}$ is as in Lemma 8, then at the end of the algorithm $w(T\setminus T_{0})=O(p\ln k)\cdot\tau^{*}$ and $|A\cap T|\geq m-\ell+p$ holds for any $\mathbb{A}\in{\cal D}_{T}$.

Proof

We have $|A\cap S_{i}|\geq 1$ for all $i$. In particular $A\cap S_{1}\neq\emptyset$. Any $v\in A\cap S_{1}$ has in $G[T]$ at least $m$ neighbors in $T_{0}$, and at most $\ell$ of them are not in $A$; thus $v$ has at least $m-\ell$ neighbors in $A\cap T_{0}$, so $|A\cap T_{0}|\geq m-\ell$. Since $T_{0},S_{1},\ldots,S_{p}$ are pairwise disjoint we get $|A\cap T|\geq|A\cap T_{0}|+\sum_{i=1}^{p}|A\cap S_{i}|\geq m-\ell+p$. ∎

Now we decompose the problem of covering ${\cal D}_{T}$ into several subproblems. For $r\in T$ let ${\cal D}_{(T,r)}=\{\mathbb{A}\in{\cal D}_{T}:r\in T\setminus A^{+}\}$.

Theorem 3.2 ([8])

Given an $\ell$-$T$-connected graph with $|T|\geq\ell+1$, one can find in polynomial time $R\subseteq T$ of size $|R|=O\left(\frac{|T|}{|T|-\ell}\ln\ell\right)$ such that ${\cal D}=\cup_{r\in R}{\cal D}_{(T,r)}$.

We now describe how to cover the family ${\cal D}_{(T,r)}$ for given $r\in T$.

Definition 4

The intersection and the union of two bisets $\mathbb{A},\mathbb{B}$ are defined by $\mathbb{A}\cap\mathbb{B}=(A\cap B,A^{+}\cap B^{+})$ and $\mathbb{A}\cup\mathbb{B}=(A\cup B,A^{+}\cup B^{+})$. The biset $\mathbb{A}\setminus\mathbb{B}$ is defined by $\mathbb{A}\setminus\mathbb{B}=(A\setminus B^{+},A^{+}\setminus B)$. A biset family ${\cal F}$ is called:

• •

uncrossable if $\mathbb{A}\cap\mathbb{B},\mathbb{A}\cup\mathbb{B}\in{\cal F}$ or if $\mathbb{A}\setminus\mathbb{B},\mathbb{B}\setminus\mathbb{A}\in{\cal F}$ for all $\mathbb{A},\mathbb{B}\in{\cal F}$.

• •

$T$-intersecting if $\mathbb{A}\cap\mathbb{B},\mathbb{A}\cup\mathbb{B}\in{\cal F}$ for any $\mathbb{A},\mathbb{B}\in{\cal F}$ with $A\cap B\cap T\neq\emptyset$.

• •

$T$-co-crossing if $\mathbb{A}\setminus\mathbb{B},\mathbb{B}\setminus\mathbb{A}\in{\cal F}$ for any $\mathbb{A},\mathbb{B}\in{\cal F}$ with $A\cap B^{*}\cap T\neq\emptyset$ and $B\cap A^{*}\cap T\neq\emptyset$.

Lemma 9 ([7])

${\cal D}_{(T,r)}$* is $T$-intersecting and $T$-co-crossing for any $r\in T$.*

Theorem 3.3 ([7])

There exists a polynomial time algorithm that given a $T$-intersecting $T$-co-crossing biset family ${\cal F}$ sequentially finds $O\left(\frac{q+\ell}{q}\right)$ $T$-intersecting uncrossable subfamilies of ${\cal F}$, such that the union of covers of these subfamilies covers ${\cal F}$, where $q=\min\{|A\cap T|:\mathbb{A}\in{\cal F}\}$ and $\displaystyle\ell=\max_{\mathbb{A}\in{\cal F}}|\partial\mathbb{A}\cap T|$.

Theorem 3.4 (Fukunaga [2])

If ${\cal F}$ is a $T$-intersecting uncrossable subfamily of ${\cal D}_{T}$ then there exists a polynomial time algorithm that computes a cover $S$ of ${\cal F}$ of weight $w(S)\leq\frac{15\sf opt}{k-\ell}$.

Combining Lemma 9 with Theorems 3.3 and 3.4 we get:

Corollary 2

For any $r\in T$, there exists a polynomial time algorithm that computes a cover $S_{r}$ of ${\cal D}_{(T,r)}$ such that if $q=\min\{|A\cap T|:\mathbb{A}\in{\cal D}_{(T,r)}\}$ then

[TABLE]

The algorithm for unit disc graphs is as follows:

We bound the weight of each of the sets computed. Let $T_{0}$ denote the initial set stored in $T$. By Lemma 8, at the end of step 1 we have

[TABLE]

Now we bound the weight of the set $S$ computed in steps $3$ to $6$:

[TABLE]

The overall weight of the augmenting set computed is as claimed in Lemma 6.

In the case of unit weights, we add arbitrary $\ell$ nodes to $T$; this step invokes an additive term of $O(1)$ to the ratio. Then we will have $|R|=O(\ln\ell)$ and thus

[TABLE]

The overall weight of the augmenting set computed is as claimed in Lemma 6.

This concludes the proof of Lemma 6, and thus also the proof of Theorem 1.2.