The authors and Fischer recently proved that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing the following (ordered) matrix removal lemma: For any finite alphabet @math , any hereditary property @math of matrices over @math , and any @math , there exists @math such that for any matrix @math over @math that is @math -far from satisfying @math , most of the @math submatrices of @math do not satisfy @math . Here being @math -far from @math means that one needs to modify at least an @math -fraction of the entries of @math to make it satisfy @math . However, in the above general removal lemma, @math grows very fast as a function of @math , even when @math is characterized by a single forbidden submatrix. In this work we establish much more efficient removal lemmas for several special cases of the above problem. In particular, we show the following: For any fixed @math binary matrix @math and any @math there exists @math polynomial in @math , such that for any binary matrix @math in which less than a @math -fraction of the @math submatrices are equal to @math , there exists a set of less than an @math -fraction of the entries of @math that intersects every @math -copy in @math . We generalize the work of Alon, Fischer and Newman [SICOMP'07] and make progress towards proving one of their conjectures. The proofs combine their efficient conditional regularity lemma for matrices with additional combinatorial and probabilistic ideas.
Very recently, the authors and Fischer @cite_26 generalized the (finite and infinite) induced graph removal lemma by obtaining an version of it, and also showed that the same type of proof can be used to obtain a removal lemma for two-dimensional (with row and column order) over a finite alphabet; this it Theorem above.
@cite_26
2611059716 2951036212 2531021157 2098322943
We consider properties of edge-colored vertex-ordered graphs, i.e., graphs with a totally ordered vertex set and a finite set of possible edge colors. We show that any hereditary property of such graphs is strongly testable, i.e., testable with a constant number of queries. We also explain how the proof can be adapted to show that any hereditary property of @math -dimensional matrices over a finite alphabet (where row and column order is not ignored) is strongly testable. The first result generalizes the result of Alon and Shapira [FOCS'05, SICOMP'08], who showed that any hereditary graph property (without vertex order) is strongly testable. The second result answers and generalizes a conjecture of Alon, Fischer and Newman [SICOMP'07] concerning testing of matrix properties. The testability is proved by establishing a removal lemma for vertex-ordered graphs. It states that for any finite or infinite family @math of forbidden vertex-ordered graphs, and any @math , there exist @math and @math so that any vertex-ordered graph which is @math -far from being @math -free contains at least @math copies of some @math (with the correct vertex order) where @math . The proof bridges the gap between techniques related to the regularity lemma, used in the long chain of papers investigating graph testing, and string testing techniques. Along the way we develop a Ramsey-type lemma for @math -partite graphs with "undesirable" edges, stating that one can find a Ramsey-type structure in such a graph, in which the density of the undesirable edges is not much higher than the density of those edges in the graph. Indyk and Sidiropoulos (2007) proved that any orientable graph of genus @math can be probabilistically embedded into a graph of genus @math with constant distortion. Viewing a graph of genus @math as embedded on the surface of a sphere with @math handles attached, Indyk and Sidiropoulos' method gives an embedding into a distribution over planar graphs with distortion @math , by iteratively removing the handles. By removing all @math handles at once, we present a probabilistic embedding with distortion @math for both orientable and non-orientable graphs. Our result is obtained by showing that the nimum-cut graph of Erickson and Har Peled (2004) has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma of Lee and Sidiropoulos (2009). In our LICS 2004 paper we introduced an approach to the study of the local structure of finite algebras and relational structures that aims at applications in the Constraint Satisfaction Problem (CSP). This approach involves a graph associated with an algebra @math or a relational structure A, whose vertices are the elements of @math (or A), the edges represent subsets of @math such that the restriction of some term operation of @math is ‘good’ on the subset, that is, act as an operation of one of the 3 types: semilattice, majority, or affine. In this paper we significantly refine and advance this approach. In particular, we prove certain connectivity and rectangularity properties of relations over algebras related to components of the graph connected by semilattice and affine edges. We also prove a result similar to 2-decomposition of relations invariant under a majority operation, only here we do not impose any restrictions on the relation. These results allow us to give a new, somewhat more intuitive proof of the bounded width theorem: the CSP over algebra @math has bounded width if and only if @math does not contain affine edges. Actually, this result shows that bounded width implies width (2,3). We also consider algebras with edges from a restricted set of types. In particular, it can be proved that type restrictions are preserved under the standard algebraic constructions. Finally, we prove that algebras without semilattice edges have few subalgebras of powers, that is, the CSP over such algebras is also polynomial time. We prove that any graph excluding Kr as a minor has can be partitioned into clusters of diameter at most Δ while removing at most O(r Δ) fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to remove O(r2 Δ) fraction of the edges. Our result is obtained by a new approach that relates the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops and robbers game on graphs excluding a fixed minor, can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter O(r) and strong-diameter partitions with padding parameter O(r2) for Kr-free graphs, O(k) for treewidth-k graphs, and O(log g) for graphs with genus g.