Flexible Graph Connectivity

Abstract

Graph connectivity and network design problems are among the most fundamental problems in combinatorial optimization. The minimum spanning tree problem, the two edge-connected spanning subgraph problem ($2$ -ECSS) and the tree augmentation problem (WTAP) are all examples of fundamental well-studied network design tasks that postulate different initial states of the network and different assumptions on the reliability of network components. In this paper we motivate and study Flexible Graph Connectivity (FGC), a problem that mixes together both the modeling power and the complexities of all aforementioned problems and more. In a nutshell, FGC asks to design a connected network, while allowing to specify different reliability levels for individual edges.

In this paper we develop a general algorithmic approach for approximating FGC that yields approximation algorithms with ratios that are close to the known best bounds for many special cases, such as $2$ -ECSS and WTAP. Our algorithm and analysis combine various techniques including a weight-scaling algorithm, a charging argument that uses a variant of exchange bijections between spanning trees and a factor revealing min-max-min optimization problem.

A Proof Sketch of Theorem 1

In this section we give a sketch of an analytic upper bound of $2.523$ on the approximation ratio of Algorithm 1. For this purpose it suffices to only consider Algorithms A and C. That is, using $\alpha$-monotone exchange bijections from Sect. 2, we determine functions $f^A(\cdot )$ and $f^C(\cdot )$ for the optimization problem (2), where $f^C(\cdot )$ depends on a selection of scaling factors and some other parameters to be introduced shortly. Recall that according to Lemma 1, the selection of scaling factors in Algorithm 1 is optimal. Surprisingly, a worst-case instance for our bounds $f^A(\cdot )$ and $f^C(\cdot )$ in fact has a single threshold value which is $1/\lambda$. However, to obtain the approximation ratio of $2.523$ it is crucial to execute Algorithm 1 with all threshold values of the given instance.

Let $\mathcal {I}(N)$ be a class of instances of FGC with at most N threshold values (see Definition 1). In the following, suppose that $I \in \mathcal {I}(N)$ and recall that an optimal solution $Z^* \subseteq E(G)$ of $I$ consists of r 2-edge-connected components $C_1, C_2, \ldots , C_r$ that are joined together by safe edges $E':= \{ \overline{f}_1, \overline{f}_2, \ldots , \overline{f}_{r-1} \} \subseteq \overline{F}$ in a tree-like fashion. Moreover, for any spanning tree $T \subseteq Z^*$ we have $E'\subseteq T$.

Observe that since there is an unsafe edge for each safe edge of same weight in G, we have that each threshold value is in [0, 1]. Let $0 \le \alpha _1 \le \alpha _2 \le \ldots \le \alpha _{N} \le 1$ be the N threshold values of I in non-decreasing order. In order to prepare our analysis, we consider for $i \in \{1, 2, \ldots , N\}$ an $\alpha _i$-MST $T_i$, an $\alpha _i$-monotone exchange bijection $\varphi _i: T_i \rightarrow T$ (which exist due to Lemma 3) and a weight function $w_i := w_{\alpha _i}$. For $2 \le i \le N$ we choose $\varphi _i$ such that for each $e \in E(T_{i-1}) \cap E(T_i)$ we have $\varphi _{i-1}(e) = \varphi _i(e)$. This can be done due to Corollary 12 in [3]. In order to define the parameters of the optimization problem (2), for $1 \le i \le N$, we partition the edge set of the $\alpha _i$-MST $T_i$ into four parts $D_i$, $O_i$, $F_i$, and $S_i$ as follows.

• $D_i := \{ e \in E(T_i) \cap F \mid \varphi _i(e) \in E' \}$

• $O_i := \{ e \in E(T_i) \cap \overline{F} \mid \varphi _i(e) \in E' \}$

• $F_i := \{ e \in E(T_i) \cap F \mid \varphi _i(e) \in E(T) \setminus E' \}$

• $S_i := \{ e \in E(T_i) \cap \overline{F} \mid \varphi _i(e) \in E(T) \setminus E' \}$

The parameters of (2) are given as follows. For $1 \le i \le N$ we let $E^{\bar{F}}_i$ (resp., $E^F_i$) be the set of edges in $E'$ (resp., $E(T) - E'$) that have threshold value $\alpha _i$. That is, $E^{\bar{F}}_i := \{ e \in E'\mid \alpha _e = \alpha _i \}$ and $E^F_i := \{ e \in E(T) - E'\mid \alpha _e = \alpha _i \}$. For $1 \le i \le N$ we let $\beta _i = w(E^{\bar{F}}_i)/{\text {OPT}(I)}$ and $\gamma _i = w(E^F_i) /{\text {OPT}(I)}$ be the fraction of the weight of the optimal solution that is contributed by the edges in $E^{\bar{F}}_i$ (resp., $E^F_i$). Finally, let $\xi \in [0, 1]$ be the the fraction of the weight of the optimal solution that does not correspond to the tree T; i.e., $\xi := \frac{w(Z^*) - w(T)}{\text {OPT}(I)}$. The following properties of $\beta _i$, $\gamma _i$, $1 \le i \le N$, are readily verified:

• 1. $\beta _1, \beta _2, \ldots \beta _{N}, \gamma _1, \gamma _2, \ldots \gamma _{N}, \xi \in [0, 1]$,

• 2. $\sum _{j=1}^{N} \beta _j = \frac{w(E')}{\text {OPT}(I)}$,

• 3. $\sum _{j=1}^{N} \gamma _j = \frac{w(T - E')}{\text {OPT}(I)}$ and

• 4. $\xi = 1 - \sum _{j = 1}^{N} \beta _j - \sum _{j=1}^{N} \gamma _j$.

We now bound the cost of the solutions $Z_i^C$ and $Z^A$ returned by Algorithm C (resp., Algorithm A) in terms of the parameters.

Lemma 4

Suppose we run Algorithm 1 with the optimal threshold values $W = \{ \alpha _i \}_{1 \le i \le N}$. Let $Z_i^C$ be the solution to the instance $(G, w_i, \overline{F})$ of FGC computed by Algorithm C in Algorithm 1. Then

$$\begin{aligned}&w(Z_i^C) \le \left( 1 + \sum _{j = 1}^{i-1} (\lambda -1 + \lambda \alpha _{j}) \beta _j + (\lambda -1) \cdot \sum _{j=1}^{N} \gamma _j + \sum _{j=i}^{N} \frac{\gamma _j}{\alpha _j} \right) \cdot \mathrm{OPT}(I).\end{aligned}$$

Lemma 5

Suppose we run Algorithm 1 with the optimal threshold values $W = \{ \alpha _i \}_{1 \le i \le N}$. Let $Z^A$ be the solution to the instance $(G, w, \overline{F})$ of FGC computed by Algorithm A in Algorithm 1. Then

$$\begin{aligned} w(Z^A) \le \left( \lambda + \lambda \cdot \sum _{j = 1}^{N} \alpha _{j} \beta _j \right) \cdot \mathrm{OPT}(I).\end{aligned}$$

With the bounds from Lemmas 4 and 5 and by applying standard techniques we can simplify problem (2) to

$$\begin{aligned} \begin{aligned} \max \;&\lambda \cdot \left( 1 + \sum _{j=1}^{N} \alpha _j \hat{\beta _j} \right) \\ \text {subject to }&\sum _{j=1}^{N} \hat{\beta _j} \cdot (1 + \alpha _j (\lambda - 1 + \lambda \alpha _j) ) = 1,\\&0 \le \alpha _1 \le \alpha _2 \le \ldots \le \alpha _N \le 1,\\&\hat{\beta _j} \in [0, 1] \text { for all } j \in \{1, \ldots , N \}. \end{aligned} \end{aligned}$$

(3)

Theorem 3

The approximation guarantee of Algorithm 1 for instances with at most N threshold values is upper bounded by the optimal value of optimization problem (3).

We solve Problem (3) analytically and observe that the optimal value does not depend on N. Hence we obtain the claimed approximation ratio of 2.523 for $\lambda = 2$.

Theorem 4

Algorithm 1 has an approximation guarantee of $\frac{\lambda \cdot (\lambda + 2 \sqrt{\lambda })}{2\sqrt{\lambda } + \lambda -1}$.