Dariusz Dereniowski

Piotr Borowiecki, Dariusz Dereniowski, Łukasz Kuszner

In this paper, we study discrete evacuation in networks, where agents know the network topology and designated exit nodes but do not know the number and initial positions of other agents. Each agent initially occupies a distinct node and must reach any exit node. Operating in a synchronous distributed model with local communication, the agents aim to minimize the time when the last agent reaches an exit. We introduce a general algorithmic framework for constructing evacuation strategies on arbitrary graphs. As a key application, we demonstrate that the framework yields asymptotically optimal evacuation strategies -- achieving a constant competitive ratio -- for grid networks, with natural extensions to triangular and hexagonal grids.

Michał Szyfelbein, Dariusz Dereniowski

This work considers a number of optimization problems and reductive relations between them. The two main problems we are interested in are the \emph{Optimal Decision Tree} and \emph{Set Cover}. We study these two fundamental tasks under precedence constraints, that is, if a test (or set) $X$ is a predecessor of $Y$, then in any feasible decision tree $X$ needs to be an ancestor of $Y$ (or respectively, if $Y$ is added to set cover, then so must be $X$). For the Optimal Decision Tree we consider two optimization criteria: worst case identification time (height of the tree) or the average identification time. Similarly, for the Set Cover we study two cost measures: the size of the cover or the average cover time. Our approach is to develop a number of algorithmic reductions, where an approximation algorithm for one problem provides an approximation for another via a black-box usage of a procedure for the former. En route we introduce other optimization problems either to complete the `reduction landscape' or because they hold the essence of combinatorial structure of our problems. The latter is brought by a problem of finding a maximum density precedence closed subfamily, where the density is defined as the ratio of the number of items the family covers to its size. By doing so we provide $\cO^*(\sqrt{m})$-approximation algorithms for all of the aforementioned problems. The picture is complemented by a number of hardness reductions that provide $o(m^{1/12-ε})$-inapproximability results for the decision tree and covering problems. Besides giving a complete set of results for general precedence constraints, we also provide polylogarithmic approximation guarantees for two most typically studied and applicable precedence types, outforests and inforests. By providing corresponding hardness results, we show these results to be tight.

Dariusz Dereniowski, Stefan Tiegel, Przemysław Uznański et al.

We consider the problem of searching for an unknown target vertex $t$ in a (possibly edge-weighted) graph. Each \emph{vertex-query} points to a vertex $v$ and the response either admits $v$ is the target or provides any neighbor $s\not=v$ that lies on a shortest path from $v$ to $t$. This model has been introduced for trees by Onak and Parys [FOCS 2006] and for general graphs by Emamjomeh-Zadeh et al. [STOC 2016]. In the latter, the authors provide algorithms for the error-less case and for the independent noise model (where each query independently receives an erroneous answer with known probability $p<1/2$ and a correct one with probability $1-p$). We study this problem in both adversarial errors and independent noise models. First, we show an algorithm that needs $\frac{\log_2 n}{1 - H(r)}$ queries against \emph{adversarial} errors, where adversary is bounded with its rate of errors by a known constant $r<1/2$. Our algorithm is in fact a simplification of previous work, and our refinement lies in invoking amortization argument. We then show that our algorithm coupled with Chernoff bound argument leads to an algorithm for independent noise that is simpler and with a query complexity that is both simpler and asymptotically better to one of Emamjomeh-Zadeh et al. [STOC 2016]. Our approach has a wide range of applications. First, it improves and simplifies Robust Interactive Learning framework proposed by Emamjomeh-Zadeh et al. [NIPS 2017]. Secondly, performing analogous analysis for \emph{edge-queries} (where query to edge $e$ returns its endpoint that is closer to target) we actually recover (as a special case) noisy binary search algorithm that is asymptotically optimal, matching the complexity of Feige et al. [SIAM J. Comput. 1994]. Thirdly, we improve and simplify upon existing algorithm for searching of \emph{unbounded} domains due to Aslam and Dhagat [STOC 1991].