The bidirected cut relaxation is the characteristic representative of the bidirected relaxations ($\mathrm{\mathcal{BCR}}$) which are a well-known class of equivalent LP-relaxations for the NP-hard Steiner Tree Problem in Graphs (STP). Although no general approximation algorithm based on $\mathrm{\mathcal{BCR}}$ with an approximation ratio better than $2$ for STP is known, it is mostly preferred in integer programming as an implementation of STP, since there exists a formulation of compact size, which turns out to be very effective in practice.
It is known that the integrality gap of $\mathrm{\mathcal{BCR}}$ is at most $2$, and a long standing open question is whether the integrality gap is less than $2$ or not. The best lower bound so far is $\frac{36}{31} \approx 1.161$ proven by Byrka et al. [BGRS13]. Based on the work of Chakrabarty et al. [CDV11] about embedding STP instances into simplices by considering appropriate dual formulations, we improve on this result by constructing a new class of instances and showing that their integrality gaps tend at least to $\frac{6}{5} = 1.2$.
More precisely, we consider the class of equivalent LP-relaxations $\mathrm{\mathcal{BCR}}^{+}$, that can be obtained by strengthening $\mathrm{\mathcal{BCR}}$ by already known straightforward Steiner vertex degree constraints, and show that the worst case ratio regarding the optimum value between $\mathrm{\mathcal{BCR}}$ and $\mathrm{\mathcal{BCR}}^{+}$ is at least $\frac{6}{5}$. Since $\mathrm{\mathcal{BCR}}^{+}$ is a lower bound for the hypergraphic relaxations ($\mathrm{\mathcal{HYP}}$), another well-known class of equivalent LP-relaxations on which the current best $(\ln(4) + \varepsilon)$-approximation algorithm for STP by Byrka et al. [BGRS13] is based, this worst case ratio also holds for $\mathrm{\mathcal{BCR}}$ and $\mathrm{\mathcal{HYP}}$.
The classical algorithm of Agrawal, Klein and Ravi [SIAM J. Comput., 24 (1995), pp. 440-456], stated in the setting of the primal-dual schema by Goemans and Williamson [SIAM J. Comput., 24 (1995), pp. 296-317] uses the undirected cut relaxation for the Steiner forest problem. Its approximation ratio is $2-\frac{1}{k}$, where $k$ is the number of terminal pairs. A variant of this algorithm more recently proposed by Könemann et al. [SIAM J. Comput., 37 (2008), pp. 1319-1341] is based on the lifted cut relaxation. In this paper, we continue this line of work and consider the bidirected cut relaxa…
The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph $G=(V,E)$ and a set $\mathcal{D}\subseteq V\times V$ of $k$ demand pairs. The aim is to compute the cheapest network $N\subseteq G$ for which there is an $s\to t$ path for each $(s,t)\in\mathcal{D}$. It is known that this problem is notoriously hard as there is no $k^{1/4-o(1)}$-approximation algorithm under Gap-ETH, even when parametrizing the runtime by $k$ [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximabi…
On the Integrality Gap of the Prize-Collecting Steiner Forest LP
在奖金收集斯坦纳森林（PCSF）问题中，我们给出了一个无向图$G=(V,E)$，边成本$\{c_e\geq 0\}_{e\in E}$，终端对$\{(s_i,t_i)\}_{i=1}^k$，以及每个终端对的惩罚$\{π_i\}_{i=1}^k$;目标是找到一个森林$F$，以最小化$c(F)+\sum_{i: (s_i,t_i)\text{ not connected in }F}π_i$。Steiner森林问题可以看作是$π_i=\infty$对所有$i$的特殊情况.人们普遍认为，自然（和充分研究）的线性规划（LP）松弛PCSF的完整性差距是最多2。我们通过证明这个LP的完整性缺口至少是$9/4$来消除这种信念。这甚至适用于平面图。我们还表明，使用此LP，不能设计一个拉格朗日乘子保持（LMP）算法的近似保证比$4$。因此，我们的结果显示了奖励收集和非奖励收集的LP弛豫的完整性间隙之间的分离（即，标准）Steiner森林，以及通过PCSF的LMP和非LMP近似算法相对于最优LP解可实现的近似比。对于奖金收集Steiner树（PCST）的特殊情形，证明了自然LP松弛存在基本可行解，且所有边变量都为正，所有值坐标至多为$1/3$.因此，我们排除了使用直接迭代舍入方法近似PCST的可能性，保证比$3$更好。
In the Directed Steiner Tree (DST) problem, we are given a directed graph $G=(V,E)$ on $n$ vertices with edge-costs $c \in \mathbb{R}_{\geq 0}^E$, a root vertex $r \in V$, and a set $K \subseteq V \setminus \{r\}$ of $k$ terminals. The goal is to find a minimum-cost subgraph of $G$ that contains a path from $r$ to every terminal $t \in K$. DST has been a notorious problem for decades as there is a large gap between the best-known polynomial-time approximation ratio of $O(k^ε)$ for any constant $ε> 0$, and the best quasi-polynomial-time approximation ratio of $O\left(\frac{\log^2 k}{\log \log k…
There has been significant recent progress on algorithms for approximating graph spanners, i.e., algorithms which approximate the best spanner for a given input graph. Essentially all of these algorithms use the same basic LP relaxation, so a variety of papers have studied the limitations of this approach and proved integrality gaps for this LP in a variety of settings. We extend these results by showing that even the strongest lift-and-project methods cannot help significantly, by proving polynomial integrality gaps even for $n^{Ω(ε)}$ levels of the Lasserre hierarchy, for both the directed a…
In the Directed Steiner Tree (DST) problem the input is a directed edge-weighted graph $G=(V,E)$, a root vertex $r$ and a set $S \subseteq V$ of $k$ terminals. The goal is to find a min-cost subgraph that connects $r$ to each of the terminals. DST admits an $O(\log^2 k/\log \log k)$-approximation in quasi-polynomial time, and an $O(k^ε)$-approximation for any fixed $ε> 0$ in polynomial-time. Resolving the existence of a polynomial-time poly-logarithmic approximation is a major open problem in approximation algorithms. In a recent work, Friggstad and Mousavi [ICALP 2023] obtained a simple and e…
We give the first constant-factor approximation algorithm for quasi-bipartite instances of Directed Steiner Tree on graphs that exclude fixed minors. In particular, for $K_r$-minor-free graphs our approximation guarantee is $O(r\cdot\sqrt{\log r})$ and, further, for planar graphs our approximation guarantee is 20. Our algorithm uses the primal-dual scheme. We employ a more involved method of determining when to buy an edge while raising dual variables since, as we show, the natural primal-dual scheme fails to raise enough dual value to pay for the purchased solution. As a consequence, we also …
We present a technique that allows for improving on some relative greedy procedures by well-chosen (non-oblivious) local search algorithms. Relative greedy procedures are a particular type of greedy algorithm that start with a simple, though weak, solution, and iteratively replace parts of this starting solution by stronger components. Some well-known applications of relative greedy algorithms include approximation algorithms for Steiner Tree and, more recently, for connectivity augmentation problems. The main application of our technique leads to a $(1.5+ε)$-approximation for Weighted Tree Au…
