No abstract

We introduce and study a novel semi‐random multigraph process, described as follows. The process starts with an empty graph on n vertices. In every round of the process, one vertex v of the graph is picked uniformly at random and independently of all previous rounds. We then choose an additional vertex (according to a strategy of our choice) and connect it by an edge to v. For various natural monotone increasing graph properties 𝒫, we prove tight upper and lower bounds on the minimum (extended over the set of all possible strategies) number of rounds required by the process to obtain, with high probability, a graph that satisfies 𝒫. Along the way, we show that the process is general enough to approximate (using suitable strategies) several well‐studied random graph models.

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 Σ, any hereditary property P of matrices over Σ, and any ǫ > 0, there exists f P (ǫ) such that for any matrix M over Σ that is ǫ-far from satisfying P, most of the f P (ǫ) × f P (ǫ) submatrices of M do not satisfy P. Here being ǫ-far from P means that one needs to modify at least an ǫ-fraction of the entries of M to make it satisfy P.However, in the above general removal lemma, f P (ǫ) grows very fast as a function of ǫ −1 , even when P 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 s × t binary matrix A and any ǫ > 0 there exists δ > 0 polynomial in ǫ, such that for any binary matrix M in which less than a δ-fraction of the s × t submatrices are equal to A, there exists a set of less than an ǫ-fraction of the entries of M that intersects every A-copy in M .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.

We investigate the adversarial robustness of streaming algorithms. In this context, an algorithm is considered robust if its performance guarantees hold even if the stream is chosen adaptively by an adversary that observes the outputs of the algorithm along the stream and can react in an online manner. While deterministic streaming algorithms are inherently robust, many central problems in the streaming literature do not admit sublinear-space deterministic algorithms; on the other hand, classical space-efficient randomized algorithms for these problems are generally not adversarially robust. This raises the natural question of whether there exist efficient adversarially robust (randomized) streaming algorithms for these problems.

No abstract