Cover Time and Broadcast Time

We introduce a new technique for bounding the cover time of random walks by relating it to the runtime of randomized broadcast. In particular, we strongly confirm for dense graphs the intuition of Chandra et al. (1997) that ``the cover time of the graph is an appropriate metric for the performance of certain kinds of randomized broadcast algorithms\u27\u27. In more detail, our results are as follows: begin{itemize} item For any graph $G=(V,E)$ of size $n$ and minimum degree $delta$, we have $mathcal{R}(G)= mathcal{O}(frac{|E|}{delta} cdot log n)$, where $mathcal{R}(G)$ denotes the quotient of the cover time and broadcast time. This bound is tight for binary trees and tight up to logarithmic factors for many graphs including hypercubes, expanders and lollipop graphs. item For any $delta$-regular (or almost $delta$-regular) graph $G$ it holds that $mathcal{R}(G) = Omega(frac{delta^2}{n} cdot frac{1}{log n})$. Together with our upper bound on $mathcal{R}(G)$, this lower bound strongly confirms the intuition of Chandra et al.~for graphs with minimum degree $Theta(n)$, since then the cover time equals the broadcast time multiplied by $n$ (neglecting logarithmic factors). item Conversely, for any $delta$ we construct almost $delta$-regular graphs that satisfy $mathcal{R}(G) = mathcal{O}(max { sqrt{n},delta } cdot log^2 n)$. Since any regular expander satisfies $mathcal{R}(G) = Theta(n)$, the strong relationship given above does not hold if $delta$ is polynomially smaller than $n$. end{itemize} Our bounds also demonstrate that the relationship between cover time and broadcast time is much stronger than the known relationships between any of them and the mixing time (or the closely related spectral gap)

On Approximate Reconfigurability of Label Cover

Given a two-prover game $G$ and its two satisfying labelings $\psi_\mathsf{s}$ and $\psi_\mathsf{t}$, the Label Cover Reconfiguration problem asks whether $\psi_\mathsf{s}$ can be transformed into $\psi_\mathsf{t}$ by repeatedly changing the value of a vertex while preserving any intermediate labeling satisfying $G$. We consider an optimization variant of Label Cover Reconfiguration by relaxing the feasibility of labelings, referred to as Maxmin Label Cover Reconfiguration: we are allowed to transform by passing through any non-satisfying labelings, but required to maximize the minimum fraction of satisfied edges during transformation from $\psi_\mathsf{s}$ to $\psi_\mathsf{t}$. Since the parallel repetition theorem of Raz (SIAM J. Comput., 1998), which implies NP-hardness of Label Cover within any constant factor, produces strong inapproximability results for many NP-hard problems, one may think of using Maxmin Label Cover Reconfiguration to derive inapproximability results for reconfiguration problems. We prove the following results on Maxmin Label Cover Reconfiguration, which display different trends from those of Label Cover and the parallel repetition theorem: (1) Maxmin Label Cover Reconfiguration can be approximated within a factor of nearly $\frac{1}{4}$ for restricted graph classes, including slightly dense graphs and balanced bipartite graphs. (2) A naive parallel repetition of Maxmin Label Cover Reconfiguration does not decrease the optimal objective value. (3) Label Cover Reconfiguration on projection games can be decided in polynomial time. The above results suggest that a reconfiguration analogue of the parallel repetition theorem is unlikely.Comment: 11 page

Approximating Subdense Instances of Covering Problems

We study approximability of subdense instances of various covering problems on graphs, defined as instances in which the minimum or average degree is Omega(n/psi(n)) for some function psi(n)=omega(1) of the instance size. We design new approximation algorithms as well as new polynomial time approximation schemes (PTASs) for those problems and establish first approximation hardness results for them. Interestingly, in some cases we were able to prove optimality of the underlying approximation ratios, under usual complexity-theoretic assumptions. Our results for the Vertex Cover problem depend on an improved recursive sampling method which could be of independent interest

Deciding first-order properties of nowhere dense graphs

Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion. We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes. At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption). As a by-product, we give an algorithmic construction of sparse neighbourhood covers for nowhere dense graphs. This extends and improves previous constructions of neighbourhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those. Our proofs are based on a new game-theoretic characterisation of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a "rank-preserving" version of Gaifman's locality theorem.Comment: 30 page

Minimum Path Cover: The Power of Parameterization

Computing a minimum path cover (MPC) of a directed acyclic graph (DAG) is a fundamental problem with a myriad of applications, including reachability. Although it is known how to solve the problem by a simple reduction to minimum flow, recent theoretical advances exploit this idea to obtain algorithms parameterized by the number of paths of an MPC, known as the width. These results obtain fast [M\"akinen et al., TALG] and even linear time [C\'aceres et al., SODA 2022] algorithms in the small-width regime. In this paper, we present the first publicly available high-performance implementation of state-of-the-art MPC algorithms, including the parameterized approaches. Our experiments on random DAGs show that parameterized algorithms are orders-of-magnitude faster on dense graphs. Additionally, we present new pre-processing heuristics based on transitive edge sparsification. We show that our heuristics improve MPC-solvers by orders-of-magnitude