Strong and Hiding Distributed Certification of Bipartiteness

TL;DR

It is proved that there are no strong and hiding LCPs for $2-coloring in general, unless the algorithm has access to node identifiers and uses certificates of size~$\omega(1)$.

Abstract

In this paper, we study the problem of certifying whether a graph is bipartite (i.e. $2$-colorable) with a locally checkable proof (LCP) that is able to hide a $2$-coloring from the verifier. More precisely, we say an LCP for $2$-coloring is hiding if, in a yes-instance, it is possible to assign certificates to nodes without revealing an explicit $2$-coloring. Motivated by the search for promise-free separations of extensions of the LOCAL model in the context of locally checkable labeling (LCL) problems, we also require the LCPs to satisfy what we refer to as the strong soundness property. This is a strengthening of soundness that requires that, in a no-instance (i.e., a non-$2$-colorable graph) and for every certificate assignment, the subset of accepting nodes must induce a $2$-colorable subgraph. We show that strong and hiding LCPs for $2$-coloring exist in specific graph classes and requiring only $O(\log n)$-sized certificates. Furthermore, when the input is promised to be a cycle or contains a node of degree $1$, we show the existence of strong and hiding LCPs even in an anonymous network and with constant-size certificates. Despite these upper bounds, we prove that there are no strong and hiding LCPs for $2$-coloring in general, unless the algorithm has access to node identifiers and uses certificates of size~$ω(1)$. Furthermore, in anonymous networks, the lower bound holds regardless of the certificate size. The proof relies on a Ramsey-type result as well as an argument about the realizability of subgraphs of the neighborhood graph consisting of the accepting views of an LCP. Along the way, we also give a characterization of the hiding property for the general $k$-coloring problem that appears to be a key component for future investigations in this context.

Figures (11)

• Figure 1: Example of the $r$-forgetful property for (bipartite) tori. Here we are considering $r = 3$. The shaded area indicates the intersection $N^3(u) \cap N^3(v)$ of the $3$-neighborhoods of $u$ and $v$. The orange nodes (plus $u$ itself) are those in that set that have smaller distance to $u$ than to $v$. Along the path $(v, v_1, v_2, v_3)$, the distance to every node $w$ among these orange nodes (including $u$) increases monotonically, and by the time we reach $v_3$ we have $\mathop{\mathrm{dist}}\nolimits(v_3, w) > r$.
• Figure 2: A graph $G$ and the subgraph $G_v^r$ associated with the view of a node $v$ in $G$, where $r = 2$. Notice how the edge between the nodes $x$ and $y$ is missing in $G_v^r$.
• Figure 3: Labeled instances $I_1$ and $I_2$ for the proof of \ref{['lemm:degree1hiding']}. The symbols inside the nodes represent the identifiers. The labels (certificates) are written above the nodes. Port numbers are above the edges for odd numbered nodes and bellow for even number nodes (the order is considered from left to right). For example, in $I_1,$ node $u_1$ has port $p_1$ and node $u_2$ has ports $p_1',p'_2.$ In $I_2,$ node $v_2$ has The neighborhoods of the node $u_1$ and $u_5$, which are marked in red, are identical in the two instances.
• Figure 4: Odd cycle in $\mathcal{V}(\mathcal{D},6)$ for graphs with minimum degree $1$. The center node of a view is marked in orange. The blue, green, and yellow shades correspond to the areas of the instances from \ref{['fig:instancesdeg1']}.
• Figure 5: Instances $I_1$ for the proof of \ref{['lemm:cyclehiding']}. In cyan, the center of the neighborhood we are considering for a self-loop in $\mathcal{V}(\mathcal{D},4)$

Theorems & Definitions (52)

• theorem 1
• theorem 2
• theorem 3
• theorem 4
• theorem 5
• theorem 6
• theorem 7
• lemma 1
• proof
• lemma 2