Bicriteria approximation for minimum dilation graph augmentation

TL;DR

This work studies adding $k$ edges to a given graph to minimize its dilation in a metric space, introducing a bicriteria framework that trades sparsity for dilation. It presents a main result: for every $r \ge 1$, there is an $(f,(1+\delta)g)$-bicriteria approximation with $f = 2 \sqrt[r]{2} \ k^{1/r}$ and $g = 2r$, computable in $O(n^3 (\log n + \log \tfrac{1}{\delta}))$ time; the algorithm adds $O(k^{1+1/r})$ edges to achieve dilation $2(1+\delta) r t^*$, where $t^*$ is the optimum after adding $k$ edges. The analysis leverages a greedy spanner construction augmented with a novel girth-graph auxiliary structure, and is shown to be tight under the Erdős girth conjecture. Additionally, the paper proves W[1]-hardness for achieving $(h(k),2-\varepsilon)$-bicriteria approximations via a set-cover reduction and provides a simple $(4k \log n,1)$-bicriteria scheme via a set-cover reduction, with running time $O(n^6 \log n)$. These results significantly improve dilation guarantees over prior work at a controlled cost in added edges and delineate the fundamental limits of pessimistic guarantees in this setting.

Abstract

Spanner constructions focus on the initial design of the network. However, networks tend to improve over time. In this paper, we focus on the improvement step. Given a graph and a budget $k$, which $k$ edges do we add to the graph to minimise its dilation? Gudmundsson and Wong [TALG'22] provided the first positive result for this problem, but their approximation factor is linear in $k$. Our main result is a $(2 \sqrt[r]{2} \ k^{1/r},2r)$-bicriteria approximation that runs in $O(n^3 \log n)$ time, for all $r \geq 1$. In other words, if $t^*$ is the minimum dilation after adding any $k$ edges to a graph, then our algorithm adds $O(k^{1+1/r})$ edges to the graph to obtain a dilation of $2rt^*$. Moreover, our analysis of the algorithm is tight under the Erdős girth conjecture.

Figures (6)

• Figure 1: Left: The graph $G$ (black), the greedy edge $a_i$ (blue), the path $\delta_{G^*}(a_i)$ (grey), and the edges $\delta_{G^*}(a_i) \cap S^*$ (red). Right: The girth graph $H$ and the edge $e_i$ (orange).
• Figure 2: Left: The graph $G$ (black), the optimal edges $s_1, \ldots, s_4$ (red), and the greedy edges $a_1, \ldots, a_5$ (blue). Right: The girth graph $H$ has edges $e_1, \ldots, e_5$ (orange) and a girth of $4$.
• Figure 3: Left: The graph $G$ (black), the greedy edge $a_i$ (blue), the path $\delta_{G^*}(a_i)$ (grey), and the edges $\delta_{G^*}(a_i) \cap S^*$ (red). Right: The girth graph $H$, the cycle $J$ (orange), and edge $e_i$ (dashed).
• Figure 4: The path $w_1,w_2,w_3,w_4$ is shown on the right. There is a path between $a_4(0)$ and $a_4(1)$ only using the blue edges $a_1,a_2,a_3$ and black edges in $\delta_{G^*}(a_1),\delta_{G^*}(a_2),\delta_{G^*}(a_3)$ or $\delta_{G^*}(a_4)$.
• Figure 5: The girth graph $H$ (orange), the vertices $u_{i,j} \in V(M)$ (black), the edges $M_1$ with length 1 (grey), the edges $M_2$ with length $2 \varepsilon$ (red), and the edges $M_3$ with length $\varepsilon$ (blue).

Theorems & Definitions (22)

• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 9
• Definition 10
• Lemma 11
• proof
• Lemma 11
• Lemma 12