Graph Reconstruction via MIS Queries

Abstract

In the Graph Reconstruction (GR) problem, a player initially only knows the vertex set $V$ of an input graph $G=(V, E)$ and is required to learn its set of edges $E$. To this end, the player submits queries to an oracle and must deduce $E$ from the oracle's answers. In this paper, we initiate the study of GR via Maximal Independent Set (MIS) queries, a more powerful variant of Independent Set (IS) queries. Given a query $U \subseteq V$, the oracle responds with any, potentially adversarially chosen, maximal independent set $I \subseteq U$ in the induced subgraph $G[U]$. We show that, for GR, MIS queries are strictly more powerful than IS queries when parametrized by the maximum degree $\Delta$ of the input graph. We give tight (up to poly-logarithmic factors) upper and lower bounds for this problem: 1. We observe that the simple strategy of taking uniform independent random samples of $V$ and submitting those to the oracle yields a non-adaptive randomized algorithm that executes $O(\Delta^2 \cdot \log n)$ queries and succeeds with high probability. Furthermore, combining the strategy of taking uniform random samples of $V$ with the probabilistic method, we show the existence of a deterministic non-adaptive algorithm that executes $O(\Delta^3 \cdot \log(\frac{n}{\Delta}))$ queries. 2. Regarding lower bounds, we prove that the additional $\Delta$ factor when going from randomized non-adaptive algorithms to deterministic non-adaptive algorithms is necessary. We show that every non-adaptive deterministic algorithm requires $\Omega(\Delta^3 / \log^2 \Delta)$ queries. For arbitrary randomized adaptive algorithms, we show that $\Omega(\Delta^2)$ queries are necessary in graphs of maximum degree $\Delta$, and that $\Omega(\log n)$ queries are necessary, even when the input graph is an $n$-vertex cycle.