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We initiate a systematic study of mixing in non-quasirandom groups. Let A and B be two independent, high-entropy distributions over a group G. We show that the product distribution AB is statistically close to the distribution F(AB) for several choices of G and F, including: 1) G is the affine group of 2x2 matrices, and F sets the top-right matrix entry to a uniform value, 2) G is the lamplighter group, that is the wreath product of ������ and ���_{n}, and F is multiplication by a certain subgroup, 3) G is H��� where H is non-abelian, and F selects a uniform coordinate and takes a uniform conjugate of it. The obtained bounds for (1) and (2) are tight. This work is motivated by and applied to problems in communication complexity. We consider the 3-party communication problem of deciding if the product of three group elements multiplies to the identity. We prove lower bounds for the groups above, which are tight for the affine and the lamplighter groups., LIPIcs, Vol. 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), pages 80:1-80:9

Front Matter, Table of Contents, Preface, Conference Organization, LIPIcs, Vol. 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022), pages 0:i-0:xvi

LIPIcs, Volume 219, STACS 2022, Complete Volume, LIPIcs, Vol. 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022), pages 1-1044

One-bit compressed sensing (1bCS) is an extreme-quantized signal acquisition method that has been intermittently studied in the past decade. In 1bCS, linear samples of a high dimensional signal are quantized to only one bit per sample (sign of the measurement). The extreme quantization makes it an interesting case study of the more general single-index or generalized linear models. At the same time it can also be thought of as a `design' version of learning a binary linear classifier or halfspace-learning. Assuming the original signal vector to be sparse, existing results in 1bCS either aim to find the support of the vector, or approximate the signal within an $\epsilon$-ball. The focus of this paper is support recovery, which often also computationally facilitate approximate signal recovery. A \emph{universal} measurement matrix for 1bCS refers to one set of measurements that work \emph{for all} sparse signals. With universality, it is known that $\tilde{\Theta}(k^2)$ 1bCS measurements are necessary and sufficient for support recovery (where $k$ denotes the sparsity). In this work, we show that it is possible to universally recover the support with a small number of false positives with $\tilde{O}(k^{3/2})$ measurements. If the dynamic range of the signal vector is known, then with a different technique, this result can be improved to only $\tilde{O}(k)$ measurements. Other results on universal but approximate support recovery are also provided in this paper. All of our main recovery algorithms are simple and polynomial-time., Comment: 20 pages

First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem parameterized by solution size. On the other hand, FO cannot express the very simple algorithmic question whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph properties that are commonly studied in parameterized algorithmics. By adding the atomic predicates conn_k(x,y,z_1,���,z_k) that hold true in a graph if there exists a path between (the valuations of) x and y after (the valuations of) z_1,���,z_k have been deleted, we obtain separator logic FO+conn. We show that separator logic can express many interesting problems such as the feedback vertex set problem and elimination distance problems to first-order definable classes. Denote by FO+conn_k the fragment of separator logic that is restricted to connectivity predicates with at most k+2 variables (that is, at most k deletions). We show that FO+conn_{k+1} is strictly more expressive than FO+conn_k for all k ��� 0. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic FO+DP by adding the atomic predicates disjoint-paths_k[(x_1,y_1),���,(x_k,y_k)] that evaluate to true if there are internally vertex-disjoint paths between (the valuations of) x_i and y_i for all 1 ��� i ��� k. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Again we show that the fragments FO+DP_k that use predicates for at most k disjoint paths form a strict hierarchy of expressiveness. Finally, we compare the expressive power of the new logics with that of transitive-closure logics and monadic second-order logic., LIPIcs, Vol. 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022), pages 34:1-34:17

A directed acyclic graph $G=(V,E)$ is said to be $(e,d)$-depth robust if for every subset $S \subseteq V$ of $|S| \leq e$ nodes the graph $G-S$ still contains a directed path of length $d$. If the graph is $(e,d)$-depth-robust for any $e,d$ such that $e+d \leq (1-\epsilon)|V|$ then the graph is said to be $\epsilon$-extreme depth-robust. In the field of cryptography, (extremely) depth-robust graphs with low indegree have found numerous applications including the design of side-channel resistant Memory-Hard Functions, Proofs of Space and Replication, and in the design of Computationally Relaxed Locally Correctable Codes. In these applications, it is desirable to ensure the graphs are locally navigable, i.e., there is an efficient algorithm $\mathsf{GetParents}$ running in time $\mathrm{polylog} |V|$ which takes as input a node $v \in V$ and returns the set of $v$'s parents. We give the first explicit construction of locally navigable $\epsilon$-extreme depth-robust graphs with indegree $O(\log |V|)$. Previous constructions of $\epsilon$-extreme depth-robust graphs either had indegree $\tilde{\omega}(\log^2 |V|)$ or were not explicit., Comment: 12 pages, 1 figure. This is the full version of the paper published at STACS 2022. We noticed a mistake in the references for the computational intractability of the depth robustness of the graphs and fixed it

Twin-width is a new parameter informally measuring how diverse are the neighbourhoods of the graph vertices, and it extends also to other binary relational structures, e.g. to digraphs and posets. It was introduced just very recently, in 2020 by Bonnet, Kim, Thomass�� and Watrigant. One of the core results of these authors is that FO model checking on graph classes of bounded twin-width is in FPT. With that result, they also claimed that posets of bounded width have bounded twin-width, thus capturing prior result on FO model checking of posets of bounded width in FPT. However, their translation from poset width to twin-width was indirect and giving only a very loose double-exponential bound. We prove that posets of width d have twin-width at most 8d with a direct and elementary argument, and show that this bound is tight up to a constant factor. Specifically, for posets of width 2 we prove that in the worst case their twin-width is also equal 2. These two theoretical results are complemented with straightforward algorithms to construct the respective contraction sequence for a given poset., LIPIcs, Vol. 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), pages 6:1-6:13

LIPIcs, Volume 129, SoCG'19, Complete Volume, LIPIcs, Vol. 129, 35th International Symposium on Computational Geometry (SoCG 2019), pages 0-0

LIPIcs, Volume 126, STACS'19, Complete Volume, LIPIcs, Vol. 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019), pages 0-0