Improved approximation algorithms for $k$-connected $m$-dominating set problems

TL;DR

This paper presents improved approximation algorithms for the $k$-connected $m$-dominating set problem, achieving better ratios for unit disc-graphs and the first non-trivial ratio for general graphs, enhancing efficiency in network design.

Contribution

The authors develop new approximation algorithms with improved ratios for the $(k,m)$-connected dominating set problem in different graph classes, advancing prior work.

Findings

Achieved $O(k \\ln k)$ approximation for unit disc-graphs.

Established $O(k^2 \\ln n)$ approximation for general graphs.

Improved upon previous $O(k^2 \\ln k)$ ratio for unit disc-graphs.

Abstract

A graph is $k$-connected if it has $k$ internally-disjoint paths between every pair of 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 then $S$ is a $k$-connected $m$-dominating set, or $(k,m)$-cds for short. 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. We consider the case $m \geq k$ and obtain the following approximation ratios. For unit disc-graphs we obtain ratio $O(k\ln k)$, improving the previous ratio $O(k^2 \ln k)$. For general graphs we obtain the first non-trivial approximation ratio $O(k^2 \ln n)$.

Equations

$$|{\Gamma }_{H_r\setminus R}(A)|+|A\cap R|\ge k.$$

$$\delta w(S)\le c(F)\le \Delta w(S)$$

$$c(F)=\sum _{uv\in E}(w_u+w_v)=\sum _{v\in V}{d}_F(v)w_v.$$

Full text

1 Introduction

A graph is $k$-connected if it has $k$ internally disjoint paths between every pair of its 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 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 $S\subseteq V$.

The case $k=0$ is the $m$-Dominating Set problem. Let $\alpha_{m}$ denote the best known ratio for $m$-Dominating Set; currently $\alpha_{m}=O(1)$ in unit-disc graphs [5] and $\alpha_{m}=\ln(\Delta+m)+1<\ln\Delta+1.7$ in general graphs [4], where $\Delta$ is the maximum degree of the input graph. The $(k,m)$-CDS problem with $m\geq k$ was studied extensively. In recent papers Zhang, Zhou, Mo, and Du [15] and Fukunaga [5] obtained ratio $O(k^{2}\ln k)$ for the problem in unit-disc graphs. For unit-disc graphs and $k=2$ Zhang et al. [15] also obtained an improved ratio $\alpha_{m}+5$. In a related paper Zhang et al. [16] obtained ratio $O(k\ln\Delta)$ in general graphs with unit weights, mentionning that no non-trivial approximation algorithm for arbitrary weights is known.

Let us say that a graph with a designated set $T$ of terminals and a root node $r$ is $k$-$(T,r)$-connected if it contains $k$ internally-disjoint $rt$-paths for every $t\in T$. Our ratios for $(k,m)$-CDS are expressed in terms of $\alpha_{m}$ and the best ratio for the following known problem:

Rooted Subset $k$-Connectivity

Input: A graph $G=(V,E)$ with edge-costs/node-weights, a set $T\subseteq V$ of terminals, a root node $r\in V\setminus T$, and an integer $k$.

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

Let $\beta_{k}$ and $\beta^{\prime}_{k}$ denote the best known ratios for the Rooted Subset $k$-Connectivity problem with edge-costs and node-weights, respectively. Currently, $\beta_{m}=O(1)$ in unit-disc graphs [5], while in general graphs $\beta_{2}=2$ [3], $\beta_{3}=6\frac{2}{3}$ [13], and $\beta_{k}=O(k\ln k)$ for $k\geq 4$ [11]. We also have $\beta^{\prime}_{k}=O(k^{2}\ln n)$ by [11] and the correction of Vakilian [14] to the algorithm and the analysis of [11]; see also [6].

Our main results are summarized in the following theorem.

Theorem 1.1

Suppose that the $m$-Dominating Set problem admits ratio $\alpha_{m}$ and that the Rooted Subset $k$-Connectivity problem admits ratios $\beta_{k}$ for edge-costs and $\beta^{\prime}_{k}$ for node-weights. Then $(k,m)$-CDS with $m\geq k$ admits ratios $\alpha_{m}+\beta^{\prime}_{k}+2(k-1)=O(k^{2}\ln n)$ for general graphs and $\alpha_{m}+5\beta_{k}+2(k-1)=O(k\ln k)$ for unit-disc graphs. Furthermore, $(3,m)$-CDS on unit-disc graphs admits ratio $\alpha_{m}+5\beta_{3}=\alpha_{m}+33\frac{1}{3}$.

Our algorithm uses the main ideas as well as partial results from the papers of Zhang et al. [15] and Fukunaga [5]. Let us say that a graph $G$ is $k$-$T$-connected if $G$ contains $k$ internally-disjoint paths between every pair of nodes in $T$. Both papers [15, 5] consider unit-disc graphs and reduce the $(k,m)$-CDS problem with $m\geq k$ to the Subset $k$-Connectivity problem: given a graph with edge costs and a subset $T$ of terminals, find a minimum cost $k$-$T$-connected subgraph. The problem admits a trivial ratio $|T|^{2}$ for both edge-costs and node-weights, while for $|T|>k$ the best known ratios are $\frac{|T|}{|T|-k}O(k\ln k)=O(k^{2}\ln k)$ for edge-costs and $\frac{|T|}{|T|-k}O(k^{2}\ln n)=O(k^{3}\ln n)$ for node-weights [12]; see also [8]. In fact, these ratios are derived by applying $O(k)$ times the algorithm for the Rooted Subset $k$-Connectivity problem. The main reason for our improvement over the ratios of [15, 5] is a reduction to the easier Rooted Subset $k$-Connectivity problem. For small values of $k$ we present a refined reduction, but for unit disc graphs and $k=2$ the performance of our algorithm and that of [15] coincide, since for $k=2$ and edge-costs both Subset $k$-Connectivity and Rooted Subset $k$-Connectivity admit ratio $2$ [3].

2 Proof of Theorem 1.1

For an arbitrary graph $H=(U,F)$ and $u,v\in U$ let $\kappa_{H}(u,v)$ denote the maximum number of internally disjoint $uv$-paths in $H$. We say that $H$ is $k$-in-connected to $r$ if $H$ is $k$-$(U\setminus\{r\},r)$-connected, namely, if $\kappa_{H}(v,r)\geq k$ every $v\in U\setminus\{r\}$. For $A\subseteq U$ let $\Gamma_{H}(A)$ denote the set of neigbors of $A$ in $H$. The proof of the following known statement can be found in [7], see also [1, 2]; part (i) of the lemma relies on the Mader’s Undirected Critical Cycle Theorem [9].

Lemma 1

Let $H_{r}$ be $k$-in-connected to $r$ and let $R=\Gamma_{H_{r}}(r)$.

• (i)

The graph $H=H_{r}\setminus\{r\}$ can be made $k$-connected by adding a set $J$ of new edges on $R$; furthermore, if $J$ is inclusionwise-minimal then $J$ is a forest.

• (ii)

Suppose that $|R|=k$. If $k=2,3$ then $H_{r}$ is $k$-connected.

Note that an inclusionwise-minimal edge set $J$ as in Lemma 1(i) 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.

A reason why the case $m\geq k$ is easier is given in the following lemma.

Lemma 2

If a graph $H=(V,E)$ has a $k$-dominating set $T$ such that $H$ is $k$-$T$-connected then $H$ is $k$-connected.

Proof

By a known characterization of $k$-connected graphs, it is sufficient to show that $|V\setminus(A\cup B)|\geq k$ holds for any subpartition $A,B$ of $V$ such that $E$ has no edge between $A$ and $B$. If both $A\cap T,B\cap T$ are non-empty, this is so since $H$ is $k$-$T$-connected. Otherwise, if say $A\cap T=\emptyset$, then since $T$ is a $k$-dominating set we have $|\Gamma_{H}(A)|\geq k$, and the result follows. ∎

Finally, we will need the following known fact, c.f. [11].

Lemma 3

Given a pair $s,t$ of nodes in a node-weighted graph $G$, the problem of finding a minimum weight node set $P_{st}$ such that $G[P_{st}]$ has $k$ internally- disjoint $st$-paths admits a $2$-approximation algorithm.

For arbitrary $k$, we will show that the following algorithm achieves the desired approximation ratio.

We now prove that the solution computed is feasible.

Lemma 4

The computed solution is feasible, namely, at the end of the algorithm $T\cup S\cup P$ is a $(k,m)$-cds.

Proof

Since $T$ is an $m$-dominating set, so is any superset of $T$. Thus the node set $T\cup S\cup P$ returned by the algorithm is an $m$-dominating set.

It remains to prove that $T\cup S\cup P$ is a $k$-connected set. We first prove that the graph $H_{r}$ computed at step 3 is $k$-in-connected to $r$. By Menger’s Theorem, $\kappa_{H_{r}}(v,r)\geq k$ iff for all $A\subseteq T\cup S$ with $v\in A$

[TABLE]

Let $\emptyset\neq A\subseteq T\cup S$. If $A\cap T\neq\emptyset$ then (1) holds since $H_{r}$ is $k$-$(T,r)$-connected. If $A\cap S\neq\emptyset$ then $|\Gamma_{H_{r}\setminus R}(A)|\geq m\geq k$, since $T$ is an $m$-dominating set and thus every node in $A\cap S$ has at least $m$ neighbors in $T$. In both cases, (1) holds, hence $H_{r}$ is $k$-in-connected to $r$.

The graph $H\cup J$ is $k$-connected, which implies that the graph $G[T\cup S\cup P]$ is $(T\cup S)$-$k$-connected and thus $T$-$k$-connected. Furthermore, $T$ is a $k$-dominating set, since $m\geq k$. Applying Lemma 2 on the graph $G[T\cup S\cup P]$ we get that this graphs is $k$-connected, as required. ∎

Lemma 5

Algorithm 1 has ratio $\alpha_{m}+\beta^{\prime}_{k}+2(k-1)$.

Proof

Let $S^{*}$ be an optimal solution to $(k,m)$-CDS. Clearly, $w(T)\leq\alpha_{m}w(S^{*})\leq\beta^{\prime}_{k}w(S^{*})$. We claim that $w(S)\leq\beta^{\prime}_{k}w(S^{*}\setminus T)$. For this note that $S^{*}\setminus T$ is a feasible solution to the problem considered at step 3 of the algorithm, while $S$ is a $\beta^{\prime}_{k}$-approximate solution. For the same reason, for each $uv\in J$ the set $S^{*}\setminus(T\cup S)$ is a feasible solution to the problem considered at step 5, while the set $P_{uv}$ computed is a $2$-approximate solution; thus $w(P_{uv})\leq 2w(S^{*}\setminus(T\cup S))$. Finally, note that $|J|\leq k-1$, and thus $w(P)\leq 2(k-1)w(S^{*})$. The lemma follows. ∎

This concludes the proof of the case of general $k$ and general graphs. Let us now consider unit disc graphs. Then we use the following result of [15].

Theorem 2.1 (Zhang, Zhou, Mo, and Du [15])

Any $k$-connected unit-disc graph has a $k$-connected spanning subgraph of maximum degree at most $5$ if $k=2$, and at most $5k$ if $k\geq 3$.

Note that any $k$-connected graph has minimum degree $k$. Thus Theorem 2.1 implies that when searching for a $k$-connected subgraph in a unit disc graph, one can convert node-weights to edge-costs while invoking in the ratio only a factor of $5/2$ in the case $k=2$ and $5$ in the case $k\geq 3$. Specifically, given node weights $\{w_{v}:v\in V\}$ define edge-costs $c_{uv}=w_{u}+w_{v}$. Then for any subgraph $(S,F)$ of $G$ with maximum degree $\Delta$ and minimum degree $\delta$ we have:

[TABLE]

since $w_{v}\geq 0$ for all $v\in V$ and since

[TABLE]

We may use this conversion in some steps of our algorithm, and specifically in step 3, which concludes the proof of the case of general $k$ and unit-disc graphs.

In the case $k=3$ we use a result of Mader [10] that any edge-minimal $k$-connected graph has at least $\frac{(k-1)n+2}{2k-1}$ nodes of degree $k$. At step 3 of the algorithm we “guess” such a node $r$ and the $3$ edges incident to $r$ in some edge-minimal optimal solution, remove from $G$ all other edges incident to $r$, and run step 3 while omitting steps 4 and 5. By Lemma 1(ii) the graph $G[S\cup T]$ will be already $3$-connected.