Improved approximations for buy-at-bulk and shallow-light $k$-Steiner trees and $(k,2)$-subgraph

Abstract

In this paper we give improved approximation algorithms for some network design problems. In the bounded-diameter or shallow-light $k$-Steiner tree problem (SL$k$ST), we are given an undirected graph $G=(V,E)$ with terminals $T\subseteq V$ containing a root $r\in T$, a cost function $c:E\rightarrow \mathbb {R}^+$, a length function $\ell :E\rightarrow \mathbb {R}^+$, a bound $L>0$ and an integer $k\ge 1$. The goal is to find a minimum $c$-cost $r$-rooted Steiner tree containing at least $k$ terminals whose diameter under $\ell$ metric is at most $L$. The input to the buy-at-bulk $k$-Steiner tree problem (BB$k$ST) is similar: graph $G=(V,E)$, terminals $T\subseteq V$ containing a root $r\in T$, cost and length functions $c,\ell :E\rightarrow \mathbb {R}^+$, and an integer $k\ge 1$. The goal is to find a minimum total cost $r$-rooted Steiner tree $H$ containing at least $k$ terminals, where the cost of each edge $e$ is $c(e)+\ell (e)\cdot f(e)$ where $f(e)$ denotes the number of terminals whose path to root in $H$ contains edge $e$. We present a bicriteria $(O(\log ^2 n),O(\log n))$-approximation for SL$k$ST: the algorithm finds a $k$-Steiner tree with cost at most $O(\log ^2 n\cdot \text{ opt }^*)$ where $\text{ opt }^*$ is the cost of an LP relaxation of the problem and diameter at most $O(L\cdot \log n)$. This improves on the algorithm of Hajiaghayi et al. (2009) (APPROX’06/Algorithmica’09) which had ratio $(O(\log ^4 n), O(\log ^2 n))$. Using this, we obtain an $O(\log ^3 n)$-approximation for BB$k$ST, which improves upon the $O(\log ^4 n)$-approximation of Hajiaghayi et al. (2009). We also consider the problem of finding a minimum cost $2$-edge-connected subgraph with at least $k$ vertices, which is introduced as the $(k,2)$-subgraph problem in Lau et al. (2009) (STOC’07/SICOMP09). This generalizes some well-studied classical problems such as the $k$-MST and the minimum cost $2$-edge-connected subgraph problems. We give an $O(\log n)$-approximation algorithm for this problem which improves upon the $O(\log ^2 n)$-approximation algorithm of Lau et al. (2009).

Other Proofs

Proof of Lemma 2

The structure of the proof is as follows. We show that the optimal value of the dual of LP-SLST in $G$ is not less than the optimal value of a dual LP for min-cost perfect matching defined in graph $F$. Therefore, by LP duality, the value of LP-SLST ($\text{ opt }^*$) is equal to the value of its dual and is greater than the value of min-cost perfect matching LP. Then we argue that from a basic feasible solution of the matching LP we can build an integral matching whose cost is not greater than the value of min-cost perfect matching LP and has at least $|T|/3$ edges. Taking into consideration the fact that graph $F$ is built with edges that are $(1 + \epsilon )$ approximation of the actual values, we conclude that $M$ costs at most $(1 + \epsilon ) \text{ opt }^*$.

Consider the following LP for the min-cost perfect matching problem (MMP) in graph $F$, along with its dual (MMD) in which $b^*(u,v)$ represents the optimal minimum $c$-cost $(u,v)$-path of length at most $2L$:

We show that the optimal solution of dual LP for SLST (D-SLST) has value at least as big as the optimal value of MMD which implies the optimal value of MMP is not greater than $\text{ opt }^*$ using LP duality. The LP D-SLST is the following:

D-SLST:

$$\begin{aligned} \max ~~ \sum \nolimits _{t\in T} \alpha _t&\nonumber \\ \sum \nolimits _{t\in T} \beta ^t_e&\le c(e) \quad e \in E\end{aligned}$$

(14)

$$\begin{aligned} \alpha _t - \sum \nolimits _{e\in p} \beta ^t_e&\le 0 \quad \quad \ t\in T, p\in \mathcal{P}_t \end{aligned}$$

(15)

$$\begin{aligned} \alpha _t, \beta ^t_e&\ge 0 \quad \quad \ e \in E, t\in T \end{aligned}$$

(16)

Let $y^*_t$ be an optimal solution for MMD and $d_L(u, v)$ be the shortest path between $u$ and $v$ with regard to cost function $c$ in $G$ and of length at most $L$ (note that the bound on length here is $L$ and not $2L$ as in computing $b^*(u,v)$ for MMD). We make a ball $B_t$ of radius $y^*_t$ (using $d_L(u,v)$ function) around each $t \in T$ in $G$. More formally, $B_t$ contains all the nodes $v$ with $d_L(v,t) \le y^*_t$. For the edges $e = (u,v)$ which at least one of $d_L(u, t) < y^*_t$ or $d_L(v, t) < y^*_t$ is true, $g^t(e)$ is the fraction of $e$ contained in ball $B_t$ and is defined as: $g^t(e) = min\{\frac{y^*_t - min\{d_L(u,t), d_L(v,t)\}}{c(e)}, 1\}$. Basically, $g^t(e)$ is the maximum fraction of edge $e$ such that there is a path from $t$ to that point of length at most $L$ whose cost is no more than $y^*_t$.

Define $\hat{\beta }^t_e = g^t(e) \cdot c(e)$ and $\hat{\alpha }_t = y^*_t$. In the following we prove that $\hat{\beta }$ and $\hat{\alpha }$ form a feasible solution to D-SLST. It is clear that $\hat{\beta }$ and $\hat{\alpha }$ do not violate Constraints 16. The main observation here is that balls $\{B_t\}_{t\in T}$ are not overlapping: by way of contradiction suppose that there is an edge $e$ (or a positive fraction of $e$) that belongs to two balls $B_t$ and $B_{t'}$. This implies there is a point along that common section of $e$ such that there is a path from that point to each of $t$ and $t'$ whose length is smaller than $L$ and whose cost is smaller than $y^*_t$ and $y^*_{t'}$, respecitvely. Therefore, there is a path from $t$ to $t'$ of length at most $2L$ whose cost is smaller than $y^*_t+y^*_{t'}$, which violates Constraint $y_t + y_{t'} \le b^*(t,t')$ in MMD. This observation directly shows that Constraints 14 are not violated. Note that $r$ is also in $V(F)$ so the ball $B_r$ is also disjoint from the other balls. As a result, each path $p\in \mathcal{P}_t$ consists of at least one part in $B_t$ and one part in $B_r$, therefore $p$ is longer than the radius of $B_t$ which makes Constraints 15 be tight. Thus, $\hat{\alpha }$ and $\hat{\beta }$ are feasible solution to D-SLST with value at least $\sum _{u \in V(F)} y^*_u$ and hence D-SLST has value at least as big as that of MMD.

Now we show how to find an integral matching containing at least $|T|/3$ nodes. Notice that there is no odd-set constraints in MMP which makes it integral (the integral LP with odd set constraints is known as Edmond’s matching polytope). It is well known that in a basic feasible solution to MMP all $x_{u,v}$ are in the set $\{0, \frac{1}{2}, 1 \}$ and the edges with value $\frac{1}{2}$ make odd cycles Schrijver (2003). This can be proved from the fact that any basic feasible solution cannot be written as convex combination of two other feasible solutions.

Let $x^*$ be a basic feasible solution to MMP. We add all the edges $e$ with $x^*_e = 1$ to $M$. Moreover, from each odd cycle $O$, it is easy to see that we can add at least $\frac{|O|}{3}$ of its edges to $M$ such that the total cost of added edges is less than $\sum _{e \in O} x^*_e \cdot c(e)$ taking into account that $x^*_e = \frac{1}{2}$ for all $e \in O$. Therefore, $M$ has at least $\frac{V(F)}{3}$ edges whose cost is not more than the MMP’s value. As we showed that the value of MMP is not greater than $\text{ opt }^*$ and as we are able to find a $(1+\epsilon )$-approximation to $b^*(u,v)$ for each edge of $F$, the proof of lemma follows.$\square$