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Communication Dans Un Congrès Année : 2023

Twin-width V: linear minors, modular counting, and matrix multiplication

Résumé

We continue developing the theory around the twin-width of totally ordered binary structures (or equivalently, matrices over a finite alphabet), initiated in the previous paper of the series. We first introduce the notion of parity and linear minors of a matrix, which consists of iteratively replacing consecutive rows or consecutive columns with a linear combination of them. We show that a matrix class (i.e., a set of matrices closed under taking submatrices) has bounded twin-width if and only if its linear-minor closure does not contain all matrices. We observe that the fixed-parameter tractable (FPT) algorithm for first-order model checking on structures given with an O(1)-sequence (certificate of bounded twin-width) and the fact that first-order transductions of bounded twin-width classes have bounded twin-width, both established in Twin-width I, extend to first-order logic with modular counting quantifiers. We make explicit a win-win argument obtained as a by-product of Twin-width IV, and somewhat similar to bidimensionality, that we call rank-bidimensionality. This generalizes the seminal work of Guillemot and Marx [SODA '14], which builds on the Marcus-Tardos theorem [JCTA '04]. It works on general matrices (not only on classes of bounded twin-width) and, for example, yields FPT algorithms deciding if a small matrix is a parity or a linear minor of another matrix given in input, or exactly computing the grid or mixed number of a given matrix (i.e., the maximum integer k such that the row set and the column set of the matrix can be partitioned into k intervals, with each of the k 2 defined cells containing a non-zero entry, or two distinct rows and two distinct columns, respectively). Armed with the above-mentioned extension to modular counting, we show that the twin-width of the product of two conformal matrices A, B (i.e., whose dimensions are such that AB is defined) over a finite field is bounded by a function of the twin-width of A, of B, and of the size of the field. Furthermore, if A and B are n × n matrices of twin-width d over Fq, we show that AB can be computed in time O d,q (n 2 log n). We finally present an ad hoc algorithm to efficiently multiply two matrices of bounded twinwidth, with a single-exponential dependence in the twin-width bound. More precisely, pipelined to observations and results of Pilipczuk et al. [STACS '22], we obtain the following. If the inputs are given in a compact tree-like form (witnessing twin-width at most d), called twin-decomposition of width d, then two n × n matrices A, B over F2 can be multiplied in time 4 d+o(d) n, in the sense that a twin-decomposition of their product AB, with width 2 d+o(d) , is output within that time, and each entry of AB can be queried in time O d (log log n). Furthermore, for every ε > 0, the query time can be brought to constant time O(1/ε) if the running time is increased to near-linear 4 d+o(d) n 1+ε. Notably, the running time is sublinear (essentially square root) in the number of (non-zero) entries.
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Dates et versions

hal-04292966 , version 1 (17-11-2023)

Identifiants

Citer

Edouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, Stéphan Thomassé. Twin-width V: linear minors, modular counting, and matrix multiplication. STACS 2023 ( Symposium on Theoretical Aspects of Computer Science ), Mar 2023, Hambourg, Germany. ⟨10.4230/LIPIcs.STACS.2023.15⟩. ⟨hal-04292966⟩
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