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Exact Learning of Weighted Graphs Using Composite Queries

Michael T. Goodrich, Songyu Liu, Ioannis Panageas

TL;DR

The paper tackles exact learning of weighted graphs with known vertices but unknown edges by leveraging composite queries that combine edge weights, distances, and connectivity information. It introduces layer-by-layer reconstruction (LBL-R), a thresholded, Voronoi-based approach that operates on weight layers G[w≥W_thr], exhaustively solving small components and applying a RECONSTRUCT routine to larger ones to recover edges iteratively. Under a Pareto-weight model with CDF F(w)=1-w^{-α}, the authors show that the outer loop terminates early, achieving a near-optimal a.a.s. query complexity of $ ilde{O}igl(igl(1+ rac{1}{ar{ ext{α}}} ext{log} Digr) D^3 n^{3/2}igr)$, where D is the maximum degree and n=|V|; this eliminates dependence on log W_max. The work extends graph reconstruction to disconnected weighted graphs and highlights the power of combining distance, edge-weight, and component queries with a layered strategy, offering a practical path to subquadratic exact learning in realistic weight models.

Abstract

In this paper, we study the exact learning problem for weighted graphs, where we are given the vertex set, $V$, of a weighted graph, $G=(V,E,w)$, but we are not given $E$. The problem, which is also known as graph reconstruction, is to determine all the edges of $E$, including their weights, by asking queries about $G$ from an oracle. As we observe, using simple shortest-path length queries is not sufficient, in general, to learn a weighted graph. So we study a number of scenarios where it is possible to learn $G$ using a subquadratic number of composite queries, which combine two or three simple queries.

Exact Learning of Weighted Graphs Using Composite Queries

TL;DR

The paper tackles exact learning of weighted graphs with known vertices but unknown edges by leveraging composite queries that combine edge weights, distances, and connectivity information. It introduces layer-by-layer reconstruction (LBL-R), a thresholded, Voronoi-based approach that operates on weight layers G[w≥W_thr], exhaustively solving small components and applying a RECONSTRUCT routine to larger ones to recover edges iteratively. Under a Pareto-weight model with CDF F(w)=1-w^{-α}, the authors show that the outer loop terminates early, achieving a near-optimal a.a.s. query complexity of , where D is the maximum degree and n=|V|; this eliminates dependence on log W_max. The work extends graph reconstruction to disconnected weighted graphs and highlights the power of combining distance, edge-weight, and component queries with a layered strategy, offering a practical path to subquadratic exact learning in realistic weight models.

Abstract

In this paper, we study the exact learning problem for weighted graphs, where we are given the vertex set, , of a weighted graph, , but we are not given . The problem, which is also known as graph reconstruction, is to determine all the edges of , including their weights, by asking queries about from an oracle. As we observe, using simple shortest-path length queries is not sufficient, in general, to learn a weighted graph. So we study a number of scenarios where it is possible to learn using a subquadratic number of composite queries, which combine two or three simple queries.

Paper Structure

This paper contains 17 sections, 14 theorems, 11 equations, 3 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Queries
  3. Related Work
  4. Our Results
  5. Preliminaries
  6. Queries with a Threshold
  7. Description of LBL-R
  8. Correctness of LBL-R
  9. Query Complexity of LBL-R
  10. Conclusion
  11. Queries without a Threshold
  12. Description of NT-R
  13. Correctness of NT-R
  14. Query Complexity of NT-R
  15. Deferred Proofs from \ref{['sec:without']}
  16. ...and 2 more sections

Key Result

Lemma 1

For connected graphs with $n$ vertices and maximum degree $D$, the randomized algorithm RECONSTRUCT has expected query complexity $\tilde{O}(D^3 n^{3/2})$.

Figures (3)

  • Figure 1: The weighted shortest path and the unweighted shortest path from $a$ to $d$ are different. In a weighted graph, the bottom path is the shortest. However, if we ignore the weights and treat it as an unweighted graph, then the top path will be the shortest with only two hops.
  • Figure 2: Using query $q_d$ alone, the two graphs will lead to the same query results for all pairs of vertices. The existence of the edge $ac$ cannot be confirmed or denied. $ac$ is called a transitive edge (\ref{['def:transitive']}) in the graph on the right.
  • Figure 3: Iteration 1 of LBL-R (\ref{['alg:layers']}). $W_{\textrm{thr}} = 2$. Edges with weight $w < W_{\textrm{thr}}$ have been discovered and are represented by dashed lines. $G[w \geq W_{\textrm{thr}}]$ has three connected components, one of which only has one vertex. Edges with weight $w \in [W_{\textrm{thr}}, 2 W_{\textrm{thr}})$ are guaranteed to be discovered and are represented by solid blue lines. Edges with weight $w \geq 2 W_{\textrm{thr}}$ will be discovered in future iterations and are represented by solid orange lines.

Theorems & Definitions (25)

  • Definition 1
  • Remark 1
  • Lemma 1: Extension of Section 2.2.2 in kannan_graph_2018
  • Lemma 2
  • proof
  • Theorem 3: Main correctness
  • proof
  • Lemma 5: Query complexity for all connected components
  • proof
  • Lemma 6: Maximum weight
  • ...and 15 more