No distributed quantum advantage for approximate graph coloring

TL;DR

An almost complete characterization of the hardness of c-coloring χ-chromatic graphs with distributed algorithms, for a wide range of models of distributed computing, and shows that these problems do not admit any distributed quantum advantage.

Abstract

We give an almost complete characterization of the hardness of $c$-coloring $χ$-chromatic graphs with distributed algorithms, for a wide range of models of distributed computing. In particular, we show that these problems do not admit any distributed quantum advantage. To do that: 1) We give a new distributed algorithm that finds a $c$-coloring in $χ$-chromatic graphs in $\tilde{\mathcal{O}}(n^{\frac{1}α})$ rounds, with $α= \bigl\lfloor\frac{c-1}{χ- 1}\bigr\rfloor$. 2) We prove that any distributed algorithm for this problem requires $Ω(n^{\frac{1}α})$ rounds. Our upper bound holds in the classical, deterministic LOCAL model, while the near-matching lower bound holds in the non-signaling model. This model, introduced by Arfaoui and Fraigniaud in 2014, captures all models of distributed graph algorithms that obey physical causality; this includes not only classical deterministic LOCAL and randomized LOCAL but also quantum-LOCAL, even with a pre-shared quantum state. We also show that similar arguments can be used to prove that, e.g., 3-coloring 2-dimensional grids or $c$-coloring trees remain hard problems even for the non-signaling model, and in particular do not admit any quantum advantage. Our lower-bound arguments are purely graph-theoretic at heart; no background on quantum information theory is needed to establish the proofs.

Figures (8)

• Figure 1: Construction for the $\operatorname{\mathsf{rand-LOCAL}}$ model. For any $T(n)$-round algorithm $\mathcal{A}$ solving the problem, there is an $i^{\star} \in [9]$ (in the figure, $i^\star = 4$) such that $\Pr[\mathcal{A} \text{ fails in } G^{(i^\star)}] \ge 1/9$. Then, $\Pr[\mathcal{A} \text{ fails on } H_N ] \ge 1 - (1-1/9)^N$ by independence, where $H_N$ is an admissible instance. As long as $\lvert V(H_N)\rvert \le n$, this gives the lower bound.
• Figure 2: Construction for the $\operatorname{\mathsf{NS-LOCAL}}$ model. We start with $N$ copies $G_1, \dots, G_N$ of $G$ and consider their disjoint union. We prove that, in this specific graph, there is already a combination of indices $\mathbf{x} = (x_1, \dots, x_N) \in [9]^N$ (in the figure, $\mathbf{x} = (4,3,\dots,9)$) for which $\Pr[\mathcal{A} \text{ fails on } \bigcup_{j \in [N]} G_j^{(x_j)} ] \ge 1 - (1-1/9)^N$. Then, property (3) of \ref{['def:intro:cheating-graph']} ensures that we can construct an admissible instance $H_\mathbf{x}$ as shown in the figure, with $\lvert V(H_\mathbf{x})\rvert = n$. By the properties of the $\operatorname{\mathsf{NS-LOCAL}}$ model, since $H_\mathbf{x}$ and $\bigsqcup_{i \in [N]} G_i$ share the same local view around $\bigcup_{j \in [N]} G_j^{(x_j)}$, $\mathcal{A}$ fails on $H_\mathbf{x}$ too with at least the same probability.
• Figure 4: Tensor product $K_2 \times K_3$.
• Figure 5: The $2$-join of $K_2^{(1)}$ and $K_3^{(2)}$, with all the connections. Full lines represent edges within the tensor product graphs plus the starting and ending graph. Dotted lines represent edges among these graphs. $K_2^{(1)} \times K_2^{(2)} \times \{1\}$ and $K_2 ^{(1)}\times K_2^{(2)} \times \{2\}$ are two copies of the tensor product $K_2^{(1)} \times K_2^{(2)}$.
• Figure 6: Representation of the $2$-join of two copies $K_3^{(1)}$ and $K_3^{(2)}$ of $K_3$. Here, for the sake of visibility, only some edges are represented. The yellow node in $K_3^{(1)}$ is connected to all nodes in the yellow area of $K_3^{(1)} \times K_3^{(2)} \times \{1\}$; similarly, the green node in $K_3^{(2)}$ is connected to all nodes in the green area of $K_3^{(1)} \times K_3^{(1)} \times \{2\}$. The blue nodes of $K_3^{(1)} \times K_3^{(2)} \times \{1,2\}$ are connected to all nodes in the blue areas of $K_3^{(1)} \times K_3^{(2)} \times \{1,2\}$. Other connections can be deduced by symmetry.

Theorems & Definitions (77)

• Theorem 1.1
• Definition 1.2: Non-signaling distribution, informal version
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1: Cheating graph, informal version
• Theorem 4.1
• Definition 4.1: Network decomposition
• Definition 4.2: $(\lambda, d)$-clustering
• Theorem 4.3