Stronger adversaries grow cheaper forests: online node-weighted Steiner problems

TL;DR

This work proposes a randomized online algorithm for set cover and non-metric facility location in a new adversarial model which they call semi-adaptive adversaries and obtains the first deterministic $O(\log |C| \log |F|)$-competitive algorithm for non-metric facility location.

Abstract

We propose a $O(\log k \log n)$-competitive randomized algorithm for online node-weighted Steiner forest. This is essentially optimal and significantly improves over the previous bound of $O(\log^2 k \log n)$ by Hajiaghayi et al. [2017]. In fact, our result extends to the more general prize-collecting setting, improving over previous works by a poly-logarithmic factor. Our key technical contribution is a randomized online algorithm for set cover and non-metric facility location in a new adversarial model which we call semi-adaptive adversaries. As a by-product of our techniques, we obtain the first deterministic $O(\log |C| \log |F|)$-competitive algorithm for non-metric facility location.

Figures (6)

• Figure 1: The best competitive ratios achieved by previous works for the online node-weighted Steiner tree (NWST), node-weighted Steiner forest (NWSF), and their prize collecting variants (PC). All these algorithms are polynomial time and randomized.
• Figure 2: Reducing an instance of set cover to node-weighted Steiner forest (tree): for each element $e \in X := \{e_1, \ldots, e_k\}$ in the ground set $X$ is connected to set $S \in \mathcal{S} := \{S_1, \ldots, S_n\}$ if and only if $e\in S$. Every set is connected to some distinguished vertex $r$. The demand pairs are $\{(e_1, r), (e_2, r), \ldots, (e_k, r)\}$.
• Figure 3: Augmented Greedy: Both dual balls around $s$ and $t$ of size $c$ intersect dual balls of size $c$. Connecting $s$ and $t$ by a greedy path, as well as connecting $s$ and $t$ to the respective dual balls they intersect, we reduce the number of connected components containing a dual ball of size $c$.
• Figure 4: The vertex $v^*$ is contained on the boundary of many dual balls. In other words, it would have been beneficial to select $v^*$. This corresponds to a cheap facility connected to many clients.
• Figure 5: Creating clients $(t,3)$, $(t,4)$ for the auxiliary instance of NMFL, $\ell = \log k$. Facilities are on top with corresponding (scaled) opening costs, dashed lines are possible connections to facilities with their respective cost.

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1
• proof
• Lemma 2
• proof : Proof of \ref{['lem:nwsf-competitive']}
• Lemma 3
• proof