On Distributed Computation of the Minimum Triangle Edge Transversal

TL;DR

The paper studies distributed computation of MTET, the minimum set of edges whose removal makes a graph triangle-free. It proves fundamental limits: MTET is a global problem with a $\Omega(D)$ lower bound in LOCAL and a near-quadratic lower bound in CONGEST, while enabling fast approximations through reductions to MHVC. It provides a $$(1+\epsilon)$$-approximation in $\mathrm{poly}(\log n)$ rounds in LOCAL, and efficient $3$- and $(3+\epsilon)$-approximations in $O(\log n)$ and $O(\log n/\log\log n)$ rounds, with a $\Delta$-overhead for CONGEST via MHVC reductions. The work introduces novel gadgets (ring-of-triangles, bit-gadgets) and leverages 2-party communication complexity to establish tight lower bounds, advancing both the theory and practice of distributed triangle-related computations.

Abstract

The distance of a graph from being triangle-free is a fundamental graph parameter, counting the number of edges that need to be removed from a graph in order for it to become triangle-free. Its corresponding computational problem is the classic minimum triangle edge transversal problem, and its normalized value is the baseline for triangle-freeness testing algorithms. While triangle-freeness testing has been successfully studied in the distributed setting, computing the distance itself in a distributed setting is unknown, to the best of our knowledge, despite being well-studied in the centralized setting. This work addresses the computation of the minimum triangle edge transversal in distributed networks. We show with a simple warm-up construction that this is a global task, requiring $Ω(D)$ rounds even in the $\mathsf{LOCAL}$ model with unbounded messages, where $D$ is the diameter of the network. However, we show that approximating this value can be done much faster. A $(1+ε)$-approximation can be obtained in $\text{poly}\log{n}$ rounds, where $n$ is the size of the network graph. Moreover, faster approximations can be obtained, at the cost of increasing the approximation factor to roughly 3, by a reduction to the minimum hypergraph vertex cover problem. With a time overhead of the maximum degree $Δ$, this can also be applied to the $\mathsf{CONGEST}$ model, in which messages are bounded. Our key technical contribution is proving that computing an exact solution is ``as hard as it gets'' in $\mathsf{CONGEST}$, requiring a near-quadratic number of rounds. Because this problem is an edge selection problem, as opposed to previous lower bounds that were for node selection problems, major challenges arise in constructing the lower bound, requiring us to develop novel ingredients.

Figures (7)

• Figure 1: The family of lower bound graphs for minimum vertex cover AbboudCKP21.
• Figure 2: An illustration of a line of triangles. The bold red edges are the only optimal solution of triangle edge transversal.
• Figure 3: Two illustrations of $t$-rings of triangles. The numbers inside the triangles indicates their order. The bold red edges show an optimal solution to triangle edge transversal, and the dashed blue edges show the other optimal solution.
• Figure 4: The basic structure of the family of lower bound graphs for deciding the size of the minimum triangle edge transversal, with many edges and nodes omitted for clarity. See the additional figures for more detailed illustrations.
• Figure 5: The bit-gadget layer of connections, each bit node is connected to the corresponding nodes in the bit gadget with respect to the binary representation of that node.

Theorems & Definitions (43)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 1: A $t$-line of triangles
• Claim 1
• proof
• Definition 2: A $t$-ring of triangles
• Claim 2
• proof