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- W2518846372 abstract "The graph removal lemma is an important structural result in graph theory. It has many different variants and is closely related to property testing and other areas. Our main aim is to develop removal lemmas of the same spirit for two dimensional matrices. These are statements of the following type: fix a finite family $F$ of matrices over some alphabet $Gamma$. Suppose that for an $n times n$ matrix $M$ over $Gamma$, for any couple of integers $s,t > 0$ at most $o(n^{s+t})$ of the $s times t$ submatrices of $M$ are equal to matrices from $F$. Then one can modify no more than $o(n^2)$ entries in $M$ to make it $F$-free (that is, after the modification no submatrix of $M$ is equal to a matrix from $F$). As a representative example, one of our main results is the following: fix an $s times t$ binary matrix $A$. For any $epsilon > 0$ there exists $delta > 0$ such that for any $n times n$ binary matrix $M$ that contains at most $delta n^{s+t}$ copies of $A$, there exists a set of $epsilon n^2$ entries of $M$ that intersects every $A$-copy in $M$. Moreover, $delta^{-1}$ is polynomial in $epsilon^{-1}$. The major difficulty is that the rows and the columns of a matrix are ordered. These are the first removal lemma type results for two dimensional graph-like objects with a predetermined order. Our results have direct consequences in matrix property testing: they imply that for several types of families $F$ and choices of the alphabet $Gamma$, one can determine with good probability whether a given matrix $M$ is $F$-free or $epsilon$-far from $F$-freeness (i.e., one needs to change at least an $epsilon$-fraction of its entries to make it $F$-free) by sampling only a constant number of entries in $M$. In particular, we generalize an efficient induced removal lemma for bipartite graphs of Alon, Fischer and Newman, making progress towards settling one of their open problems." @default.
- W2518846372 created "2016-09-23" @default.
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- W2518846372 date "2016-09-14" @default.
- W2518846372 modified "2023-09-27" @default.
- W2518846372 title "Removal Lemmas for Matrices." @default.
- W2518846372 hasPublicationYear "2016" @default.
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