A Fast 3-Approximation for the Capacitated Tree Cover Problem with Edge Loads

TL;DR

The paper studies the capacitated tree cover problem with edge loads and provides a near-optimal, LP-based approach. It formulates a linear program, and then gives a combinatorial $O(m\log n)$ algorithm to solve the LP exactly, followed by a linear-time rounding and splitting procedure that achieves a $3$-approximation overall. A careful analysis of the rounding step introduces seed and extension edge concepts and shows how to bound the number of resulting components, delivering both a data-dependent and a general $3$-approximation bound. The authors also prove that the LP has an integrality gap of $3$, showing the $3$-approximation cannot be improved via this LP framework in general, and they discuss the practical implications for fast, scalable network design and clustering tasks where edge loads and vertex loads constrain feasible solutions.

Abstract

The capacitated tree cover problem with edge loads is a variant of the tree cover problem, where we are given facility opening costs, edge costs and loads, as well as vertex loads. We try to find a tree cover of minimum cost such that the total edge and vertex load of each tree does not exceed a given bound. We present an $\mathcal{O}(m\log n)$ time 3-approximation algorithm for this problem. This is achieved by starting with a certain LP formulation. We give a combinatorial algorithm that solves the LP optimally in time $\mathcal{O}(m\log n)$. Then, we show that a linear time rounding and splitting technique leads to an integral solution that costs at most 3 times as much as the LP solution. Finally, we prove that the integrality gap of the LP is $3$, which shows that we can not improve the rounding step in general.

Figures (4)

• Figure 1: An illustration of the sets $Z$, $G_k$, $S_i$, $V_I$ and $\Delta_I$ for $p=3$ and $i\in[3], I\subseteq[3]$. The set $V_{\{1\}}$ for example contains the vertices in the red, the purple, the orange and the brown area, while the set $V_{\{1,2\}}$ contains only the vertices in the purple and the brown area.
• Figure 2: The cases in which extension edges can occur. Dashed edges have been rounded down, while solid ones have been rounded up. Thick edges belong to a large tree. For each edge $e$ the arrowhead points towards $v_e$.
• Figure 3: The cases in which seed edges can occur. Dashed edges have been rounded down, while solid ones have been rounded up. Thick edges belong to a large tree. For each extension edge $e$ the arrowhead points towards $v_e$. Seed edges have arrowheads on both ends.
• Figure 4: A picture showing the instance described in the proof of Theorem \ref{['sec:integrality_gap']}. The solid edges belong to the $k$-star. Edge loads are marked in blue and node loads are marked in green. The dashed edge is an example for the edges added to complete the graph.

Theorems & Definitions (22)

• Theorem 1
• Definition 2
• Lemma 3
• proof
• Corollary 4
• proof
• Theorem 5
• proof
• Claim : 1
• Claim : 2