Graph Reconstruction via MIS Queries

TL;DR

Graph Reconstruction via MIS queries addresses reconstructing a graph G=(V,E) using a maximal independent set (MIS) oracle on vertex subsets. The authors develop MIS-based strategies to learn non-edges by observing MIS responses on randomly sampled induced subgraphs, enabling edge determination by complement; they provide both randomized non-adaptive and deterministic non-adaptive algorithms with provable bounds, and show how to make them adaptive with only O(log Δ) rounds. They establish tight (up to poly-log factors) upper bounds O(Δ^2 log n) for randomized and O(Δ^3 log(n/Δ)) for deterministic non-adaptive MIS GR, and accompanying lower bounds Ω(Δ^2) and Ω(log n) for adaptive randomized and Ω(Δ^3 / log^2 Δ) for non-adaptive deterministic cases. Recent improvements by others sharpen several lower bounds, reinforcing a near-complete separation between adaptive/non-adaptive and randomized/deterministic MIS GR in the Δ-parameterized setting, with MIS queries proving strictly more powerful than IS queries in this regime.

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 $Δ$ 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 $O(Δ^2 \cdot \log n)$ 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 $O(Δ^3 \cdot \log(\frac{n}Δ))$ queries. 2. Regarding lower bounds, we prove that the additional $Δ$ factor when going from randomized non-adaptive algorithms to deterministic non-adaptive algorithms is necessary. We show that every non-adaptive deterministic algorithm requires $Ω(Δ^3 / \log^2 Δ)$ queries. For arbitrary randomized adaptive algorithms, we show that $Ω(Δ^2)$ queries are necessary in graphs of maximum degree $Δ$, and that $Ω(\log n)$ queries are necessary, even when the input graph is an $n$-vertex cycle.

Theorems & Definitions (26)

• Theorem 1
• proof
• Definition 1: Witness
• Lemma 1
• proof
• Definition 2: $\Delta$-Query-Scheme
• Lemma 2
• proof
• Lemma 3
• proof