Get our free extension to see links to code for papers anywhere online!Free extension: code links for papers anywhere!Free add-on: See code for papers anywhere!

Mitchell Black, Amir Nayyeri, Zhengchao Wan, Yusu Wang

Message passing graph neural networks are popular learning architectures for graph-structured data. However, it can be challenging for them to capture long range interactions in graphs. One of the potential reasons is the so-called oversquashing problem, first termed in [Alon and Yahav, 2020], that has recently received significant attention. In this paper, we analyze the oversquashing problem through the lens of effective resistance between nodes in the input graphs. The concept of effective resistance intuitively captures the "strength" of connection between two nodes by paths in the graph, and has a rich literature connecting spectral graph theory and circuit networks theory. We propose the use the concept of total effective resistance as a measure to quantify the total amount of oversquashing in a graph, and provide theoretical justification of its use. We further develop algorithms to identify edges to be added to an input graph so as to minimize the total effective resistance, thereby alleviating the oversquashing problem when using GNNs. We provide empirical evidence of the effectiveness of our total effective resistance based rewiring strategies.

Via

Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Loffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, Hao-Tsung Yang

We consider the following surveillance problem: Given a set $P$ of $n$ sites in a metric space and a set of $k$ robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency $L$ of a schedule is the maximum latency of any site, where the latency of a site $s$ is the supremum of the lengths of the time intervals between consecutive visits to $s$. When $k=1$ the problem is equivalent to the travelling salesman problem (TSP) and thus it is NP-hard. We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into $\ell$ groups, for some~$\ell \leq k$, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group at equal distance from each other. Our first main result is that approximating the optimal latency of the class of cyclic solutions can be reduced to approximating the optimal travelling salesman tour on some input, with only a $1+\varepsilon$ factor loss in the approximation factor and an $O\left(\left( k/\varepsilon \right)^k\right)$ factor loss in the runtime, for any $\varepsilon >0$. Our second main result shows that an optimal cyclic solution is a $2(1-1/k)$-approximation of the overall optimal solution. Note that for $k=2$ this implies that an optimal cyclic solution is optimal overall. The results have a number of consequences. For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a $(2(1-1/k)+\varepsilon)$-approximation of the optimal unrestricted solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting.

Via

Peyman Afshani, Mark De Berg, Kevin Buchin, Jie Gao, Maarten Loffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, Hao-Tsung Yang

We consider the problem of finding patrol schedules for $k$ robots to visit a given set of $n$ sites in a metric space. Each robot has the same maximum speed and the goal is to minimize the weighted maximum latency of any site, where the latency of a site is defined as the maximum time duration between consecutive visits of that site. The problem is NP-hard, as it has the traveling salesman problem as a special case (when $k=1$ and all sites have the same weight). We present a polynomial-time algorithm with an approximation factor of $O(k^2 \log \frac{w_{\max}}{w_{\min}})$ to the optimal solution, where $w_{\max}$ and $w_{\min}$ are the maximum and minimum weight of the sites respectively. Further, we consider the special case where the sites are in 1D. When all sites have the same weight, we present a polynomial-time algorithm to solve the problem exactly. If the sites may have different weights, we present a $12$-approximate solution, which runs in polynomial time when the number of robots, $k$, is a constant.

Via