arXiv 2507.044732025-07cs.DS

通过分解技术为割相对可生存网络设计提供紧界保证

Tight Guarantees for Cut-Relative Survivable Network Design via a Decomposition Technique

Nikhil Kumar, JJ Nan, Chaitanya Swamy

AI总结：

针对割相对可生存网络设计问题（CR-SNDP），提出新颖分解技术以解决其割需求函数非弱超模的难题，给出紧2-近似算法，并展示相对SNDP问题的新硬度结果。

摘要：

在经典的可生存网络设计问题（SNDP）中，给定无向图$G=(V,E)$、非负边成本以及若干$(s_i,t_i,r_i)$元组（其中$s_i,t_i∈V$，$r_i∈\text{Z}_+$），我们寻求一个最小成本的边子集$H⊆E$，使得即使任意$r_i-1$条边失效，每个$s_i$-$t_i$对仍保持连通。已知SNDP可通过弱超模的割需求函数$f$等价建模，即寻找一个最小成本边集，使其跨越每个割$S⊆V$的边数至少为$f(S)$。\n最近，Dinitz等人提出了SNDP的一个变体，该变体针对$G$强制要求相对容错水平，目标是找到一个与$G$自身容错性至少相当的解$H$。他们基于路径和故障集对其进行形式化，由此产生路径相对SNDP。沿着这一思路，我们引入一种新的相对网络设计模型——割相对SNDP（CR-SNDP），其目标是选择最小成本边子集，尽可能满足给定的（弱超模）割需求函数，即跨越每个割$S⊆V$的边数为$\text{min}\{f(S),|δ_G(S)|\}$。\n与SNDP不同，SNDP的割相对和路径相对版本并不等价。CR-SNDP（以及路径相对SNDP）的割需求函数不是弱超模的，且自然线性规划松弛的极点解不一定对应紧割约束的层状族。因此，标准技术无法直接用于设计该问题的近似算法。我们开发了一种新颖的分解技术来规避这一困难，并利用它给出CR-SNDP的紧2-近似算法。我们还展示了这些相对SNDP问题的新硬度结果。

英文摘要：

In the classical \emph{survivable-network-design problem} (SNDP), we are given an undirected graph $G = (V, E)$, non-negative edge costs, and some $(s_i,t_i,r_i)$ tuples, where $s_i,t_i\in V$ and $r_i\in\mathbb{Z}_+$. We seek a minimum-cost subset $H \subseteq E$ such that each $s_i$-$t_i$ pair remains connected even if any $r_i-1$ edges fail. It is well-known that SNDP can be equivalently modeled using a weakly-supermodular \emph{cut-requirement function} $f$, where we seek a minimum-cost edge-set containing at least $f(S)$ edges across every cut $S \subseteq V$. Recently, Dinitz et al. proposed a variant of SNDP that enforces a \emph{relative} level of fault tolerance with respect to $G$, where the goal is to find a solution $H$ that is at least as fault-tolerant as $G$ itself. They formalize this in terms of paths and fault-sets, which gives rise to \emph{path-relative SNDP}. Along these lines, we introduce a new model of relative network design, called \emph{cut-relative SNDP} (CR-SNDP), where the goal is to select a minimum-cost subset of edges that satisfies the given (weakly-supermodular) cut-requirement function to the maximum extent possible, i.e., by picking $\min\{f(S),|δ_G(S)|\}$ edges across every cut $S\subseteq V$. Unlike SNDP, the cut-relative and path-relative versions of SNDP are not equivalent. The resulting cut-requirement function for CR-SNDP (as also path-relative SNDP) is not weakly supermodular, and extreme-point solutions to the natural LP-relaxation need not correspond to a laminar family of tight cut constraints. Consequently, standard techniques cannot be used directly to design approximation algorithms for this problem. We develop a \emph{novel decomposition technique} to circumvent this difficulty and use it to give a \emph{tight $2$-approximation algorithm for CR-SNDP}. We also show new hardness results for these relative-SNDP problems.

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摘要：

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