[Paper Review] First-Order Transductions of Graphs (Invited Talk)

Ossona de Mendez, Patrice|arXiv (Cornell University)|Jan 1, 2021

Advanced Graph Theory Research23 references57 citations

TL;DR

This paper introduces twin-width, a new graph width parameter generalizing cographs and capturing major sparse and dense graph classes. It proves that first-order (FO) model checking is fixed-parameter tractable (FPT) in |φ| when a d-contraction sequence is provided, running in time f(d, |φ|) · |D|, and shows that bounded twin-width is preserved under FO interpretations and transductions, unifying and extending prior FPT results on non-monotone classes.

ABSTRACT

This paper is an extended abstract of my STACS 2021 talk "First-order transductions of graphs".

Motivation & Objective

• To define a new width parameter, twin-width, that generalizes cographs and captures diverse graph classes including minor-closed, bounded rank-width, and grid-like graphs.
• To show that FO model checking is FPT in |φ| on graphs of bounded twin-width when a d-contraction sequence is given.
• To establish closure properties of bounded twin-width under FO interpretations and transductions, unifying previous FPT results on non-monotone classes.
• To provide polynomial-time algorithms for computing d-contraction sequences on specific classes such as boolean-width-bounded graphs and d-dimensional grids.
• To explore the algorithmic and structural implications of twin-width, including connections to small classes, expanders, and group theory.

Proposed method

• Define twin-width via a sequence of vertex contractions where each pair consists of near-twins—vertices whose neighborhoods differ on at most d vertices—tracked via red edges.
• Introduce d-contractions as sequences where the red degree (number of red edges incident to any vertex) is at most d, with twin-width being the minimum such d.
• Use the Marcus-Tardos theorem on forbidden 0,1-matrices to bound the number of corners in adjacency matrices, enabling the construction of contraction sequences for structured classes.
• Design a framework based on mixed-free orderings of vertices to reduce the problem of computing contraction sequences to analyzing matrix simplicity and corner structure.
• Leverage FO interpretation and transduction closure to show that twin-width is preserved under logical operations, enabling transfer of FPT results to broader classes.
• Apply the method to specific classes: provide a (2k+1−1)-contraction sequence for boolean-width ≤k graphs and a 3d-contraction sequence for d-dimensional grids, with extensions to unit ball graphs.

Experimental results

Research questions

• RQ1 Can a graph parameter generalizing cographs be defined that captures both sparse and dense graph classes while enabling efficient FO model checking?
• RQ2 Is FO model checking FPT in |φ| on graphs of bounded twin-width when a d-contraction sequence is provided?
• RQ3 Are classes of bounded twin-width closed under FO interpretations and transductions, and can this be used to unify existing FPT algorithms?
• RQ4 Can d-contraction sequences be computed in polynomial time for natural graph classes such as grids, unit ball graphs, and boolean-width-bounded graphs?
• RQ5 What is the relationship between twin-width and other complexity measures such as expansion, small classes, and polynomial expansion?

Key findings

• Twin-width is bounded for proper minor-closed classes, bounded rank-width graphs, map graphs, Kt-free unit d-dimensional ball graphs, and posets with bounded antichains.
• A d-contraction sequence can be computed in polynomial time for boolean-width ≤k graphs, yielding a (2k+1−1)-contraction sequence.
• For d-dimensional grids of side-length n, a 3d-contraction sequence can be computed in linear time.
• The number of graphs on [n] with twin-width at most d is bounded by n!f(d)n for some function f, showing that bounded twin-width classes are small.
• Bounded twin-width is preserved under FO interpretations and transductions, enabling transfer of FPT results to broader classes.
• Maximum Independent Set on graphs of twin-width at most d admits a polynomial-time constant-approximation, as does Minimum Dominating Set under similar conditions.