arXiv subjects

Aaron Schild

Publications and source records attributed to Aaron Schild.

18 recordsLinked to original sources

arXiv2026

The Robotaxi Placement Problem: Minimizing Expected ETA for Stochastic Demand

Autonomous ride-hailing platforms must strategically position idle robotaxis to minimize the wait times of prospective riders. We formalize this as the \emph{robotaxi placement problem} ($k$-RP). Given a finite metric space and a demand distribution over its points, the goal is to position $k$ robotaxis to minimize the expected total distance in a perfect matching between the robotaxis and $k$ random riders. We present several theoretical results for this stochastic optimization problem. First, we observe that sampling robotaxi locations independently according to the demand distribution yields a randomized $2$-approximation algorithm. Second, we present an explicit inapproximability bound via a novel gap-preserving reduction from the maximum coverage problem. Furthermore, while it is not even clear whether the exact expected cost of a placement can be computed efficiently on general metrics, we design an exact polynomial-time dynamic programming algorithm for $k$-RP in tree metrics by decoupling the stochastic matching dependencies. Finally, empirical evaluations on real-world ride-hailing data reveal that a variance-reduced random placement strategy is highly effective in practice, yielding expected wait times that are very close to those obtained by computationally heavy exact algorithms for the uniform capacitated $k$-median problem.

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arXiv2025

On the Randomized Locality of Matching Problems in Regular Graphs

The main goal in distributed symmetry-breaking is to understand the locality of problems; i.e., the radius of the neighborhood that a node needs to explore in order to arrive at its part of a global solution. In this work, we study the locality of matching problems in the family of regular graphs, which is one of the main benchmarks for establishing lower bounds on the locality of symmetry-breaking problems, as well as for obtaining classification results. For approximate matching, we develop randomized algorithms to show that $(1 + \epsilon)$-approximate matching in regular graphs is truly local; i.e., the locality depends only on $\epsilon$ and is independent of all other graph parameters. Furthermore, as long as the degree $\Delta$ is not very small (namely, as long as $\Delta \geq \text{poly}(1/\epsilon)$), this dependence is only logarithmic in $1/\epsilon$. This stands in sharp contrast to maximal matching in regular graphs which requires some dependence on the number of nodes $n$ or the degree $\Delta$. We show matching lower bounds for both results. For maximal matching, our techniques further allow us to establish a strong separation between the node-averaged complexity and worst-case complexity of maximal matching in regular graphs, by showing that the former is only $O(1)$. Central to our main technical contribution is a novel martingale-based analysis for the $\approx 40$-year-old algorithm by Luby. In particular, our analysis shows that applying one round of Luby's algorithm on the line graph of a $\Delta$-regular graph results in an almost $\Delta/2$-regular graph.

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arXiv2025

Why is My Route Different Today? An Algorithm for Explaining Route Selection

Users of routing services like Apple Maps, Google Maps, and Waze frequently wonder why a given route is proposed. This question particularly arises when dynamic conditions like traffic and road closures cause unusual routes to be proposed. While many dynamic conditions may exist in a road network at any time, only a small fraction of those conditions are typically relevant to a given user's route. In this work, we introduce the concept of a simple valid explanation (SVE), which consists of a small set of traffic-laden road segments that answer the following question: Which traffic conditions cause a particular shortest traffic-aware route to differ from the shortest traffic-free route? We give an efficient algorithm for finding SVEs and show that they theoretically and experimentally lead to small and interpretable answers to the question.

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arXiv2025

Breaking Barriers for Distributed MIS by Faster Degree Reduction

We study the problem of finding a maximal independent set (MIS) in the standard LOCAL model of distributed computing. Classical algorithms by Luby [JACM'86] and Alon, Babai, and Itai [JALG'86] find an MIS in $O(\log n)$ rounds in $n$-node graphs with high probability. Despite decades of research, the existence of any $o(\log n)$-round algorithm for general graphs remains one of the major open problems in the field. Interestingly, the hard instances for this problem must contain constant-length cycles. This is because there exists a sublogarithmic-round algorithm for graphs with super-constant girth; i.e., graphs where the length of the shortest cycle is $\omega(1)$, as shown by Ghaffari~[SODA'16]. Thus, resolving this $\approx 40$-year-old open problem requires understanding the family of graphs that contain $k$-cycles for some constant $k$. In this work, we come very close to resolving this $\approx 40$-year-old open problem by presenting a sublogarithmic-round algorithm for graphs that can contain $k$-cycles for all $k > 6$. Specifically, our algorithm finds an MIS in $O\left(\frac{\log \Delta}{\log(\log^* \Delta)} + \mathrm{poly}(\log\log n)\right)$ rounds, as long as the graph does not contain cycles of length $\leq 6$, where $\Delta$ is the maximum degree of the graph. As a result, we push the limit on the girth of graphs that admit sublogarithmic-round algorithms from $k = \omega(1)$ all the way down to a small constant $k=7$. This also implies a $o(\sqrt{\log n})$ round algorithm for MIS in trees, refuting a conjecture from the book by Barrenboim and Elkin.

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arXiv2025

Round Elimination via Self-Reduction: Closing Gaps for Distributed Maximal Matching

In this work, we present an $\Omega\left(\min\{\log \Delta, \sqrt{\log n}\}\right)$ lower bound for Maximal Matching (MM) in $\Delta$-ary trees against randomized algorithms. By a folklore reduction, the same lower bound applies to Maximal Independent Set (MIS), albeit not in trees. As a function of $n$, this is the first advancement in our understanding of the randomized complexity of the two problems in more than two decades. As a function of $\Delta$, this shows that the current upper bounds are optimal for a wide range of $\Delta \in 2^{O(\sqrt{\log n})}$, answering an open question by Balliu, Brandt, Hirvonen, Olivetti, Rabie, and Suomela [FOCS'19, JACM'21]. Moreover, our result implies a surprising and counterintuitive separation between MIS and MM in trees, as it was very recently shown that MIS in trees can be solved in $o(\sqrt{\log n})$ rounds. While MIS can be used to find an MM in general graphs, the reduction does not preserve the tree structure when applied to trees. Our separation shows that this is not an artifact of the reduction, but a fundamental difference between the two problems in trees. This also implies that MIS is strictly harder in general graphs compared to trees.

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arXiv2024

First Passage Percolation with Queried Hints

Solving optimization problems leads to elegant and practical solutions in a wide variety of real-world applications. In many of those real-world applications, some of the information required to specify the relevant optimization problem is noisy, uncertain, and expensive to obtain. In this work, we study how much of that information needs to be queried in order to obtain an approximately optimal solution to the relevant problem. In particular, we focus on the shortest path problem in graphs with dynamic edge costs. We adopt the $\textit{first passage percolation}$ model from probability theory wherein a graph $G'$ is derived from a weighted base graph $G$ by multiplying each edge weight by an independently chosen random number in $[1, \rho]$. Mathematicians have studied this model extensively when $G$ is a $d$-dimensional grid graph, but the behavior of shortest paths in this model is still poorly understood in general graphs. We make progress in this direction for a class of graphs that resemble real-world road networks. Specifically, we prove that if $G$ has a constant continuous doubling dimension, then for a given $s-t$ pair, we only need to probe the weights on $((\rho \log n )/ \epsilon)^{O(1)}$ edges in $G'$ in order to obtain a $(1 + \epsilon)$-approximation to the $s-t$ distance in $G'$. We also generalize the result to a correlated setting and demonstrate experimentally that probing improves accuracy in estimating $s-t$ distances.

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arXiv2023

Network Flow Problems with Electric Vehicles

Electric vehicle (EV) adoption in long-distance logistics faces challenges such as range anxiety and uneven distribution of charging stations. Two pivotal questions emerge: How can EVs be efficiently routed in a charging network considering range limits, charging speeds and prices? And, can the existing charging infrastructure sustain the increasing demand for EVs in long-distance logistics? This paper addresses these questions by introducing a novel theoretical and computational framework to study the EV network flow problems. We present an EV network flow model that incorporates range constraints and nonlinear charging rates, and identify conditions under which polynomial-time solutions can be obtained for optimal single EV routing, maximum flow, and minimum-cost flow problems. Our findings provide insights for optimizing EV routing in logistics, ensuring an efficient and sustainable future.

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arXiv2020

Sampling Arborescences in Parallel

We study the problem of sampling a uniformly random directed rooted spanning tree, also known as an arborescence, from a possibly weighted directed graph. Classically, this problem has long been known to be polynomial-time solvable; the exact number of arborescences can be computed by a determinant [Tut48], and sampling can be reduced to counting [JVV86, JS96]. However, the classic reduction from sampling to counting seems to be inherently sequential. This raises the question of designing efficient parallel algorithms for sampling. We show that sampling arborescences can be done in RNC. For several well-studied combinatorial structures, counting can be reduced to the computation of a determinant, which is known to be in NC [Csa75]. These include arborescences, planar graph perfect matchings, Eulerian tours in digraphs, and determinantal point processes. However, not much is known about efficient parallel sampling of these structures. Our work is a step towards resolving this mystery.

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arXiv2020

Algorithms and Hardness for Linear Algebra on Geometric Graphs

For a function $\mathsf{K} : \mathbb{R}^{d} \times \mathbb{R}^{d} \to \mathbb{R}_{\geq 0}$, and a set $P = \{ x_1, \ldots, x_n\} \subset \mathbb{R}^d$ of $n$ points, the $\mathsf{K}$ graph $G_P$ of $P$ is the complete graph on $n$ nodes where the weight between nodes $i$ and $j$ is given by $\mathsf{K}(x_i, x_j)$. In this paper, we initiate the study of when efficient spectral graph theory is possible on these graphs. We investigate whether or not it is possible to solve the following problems in $n^{1+o(1)}$ time for a $\mathsf{K}$-graph $G_P$ when $d < n^{o(1)}$: $\bullet$ Multiply a given vector by the adjacency matrix or Laplacian matrix of $G_P$ $\bullet$ Find a spectral sparsifier of $G_P$ $\bullet$ Solve a Laplacian system in $G_P$'s Laplacian matrix For each of these problems, we consider all functions of the form $\mathsf{K}(u,v) = f(\|u-v\|_2^2)$ for a function $f:\mathbb{R} \rightarrow \mathbb{R}$. We provide algorithms and comparable hardness results for many such $\mathsf{K}$, including the Gaussian kernel, Neural tangent kernels, and more. For example, in dimension $d = Ω(\log n)$, we show that there is a parameter associated with the function $f$ for which low parameter values imply $n^{1+o(1)}$ time algorithms for all three of these problems and high parameter values imply the nonexistence of subquadratic time algorithms assuming Strong Exponential Time Hypothesis ($\mathsf{SETH}$), given natural assumptions on $f$. As part of our results, we also show that the exponential dependence on the dimension $d$ in the celebrated fast multipole method of Greengard and Rokhlin cannot be improved, assuming $\mathsf{SETH}$, for a broad class of functions $f$. To the best of our knowledge, this is the first formal limitation proven about fast multipole methods.

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arXiv2020

Improved Distributed Approximations for Maximum Independent Set

We present improved results for approximating maximum-weight independent set ($\MaxIS$) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let $n$ and $Δ$ be the number of nodes and maximum degree, respectively, and let $\MIS(n,Δ)$ be the the running time of finding a \emph{maximal} independent set ($\MIS$) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a $Δ$-approximation for $\MaxIS$ in $O(\MIS(n,Δ)\log W)$ rounds, where $W$ is the maximum weight of a node in the graph, which can be as high as $\poly (n)$. Whether their algorithm is deterministic or randomized depends on the $\MIS$ algorithm that is used as a black-box. Our main result in this work is a randomized $(\poly(\log\log n)/ε)$-round algorithm that finds, with high probability, a $(1+ε)Δ$-approximation for $\MaxIS$ in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an \emph{exponential} speed-up in the running time over the previous best known result. Due to a lower bound of $Ω(\sqrt{\log n/\log \log n})$ that was given by Kuhn, Moscibroda and Wattenhofer [JACM, 2016] on the number of rounds for any (possibly randomized) algorithm that finds a maximal independent set (even in the LOCAL model) this result implies that finding a $(1+ε)Δ$-approximation for $\MaxIS$ is exponentially easier than $\MIS$.

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arXiv2019

Semi-Online Bipartite Matching

In this paper we introduce the \emph{semi-online} model that generalizes the classical online computational model. The semi-online model postulates that the unknown future has a predictable part and an adversarial part; these parts can be arbitrarily interleaved. An algorithm in this model operates as in the standard online model, i.e., makes an irrevocable decision at each step. We consider bipartite matching in the semi-online model, for both integral and fractional cases. Our main contributions are competitive algorithms for this problem that are close to or match a hardness bound. The competitive ratio of the algorithms nicely interpolates between the truly offline setting (no adversarial part) and the truly online setting (no predictable part).

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arXiv2019

Network design for s-t effective resistance

We consider a new problem of designing a network with small $s$-$t$ effective resistance. In this problem, we are given an undirected graph $G=(V,E)$, two designated vertices $s,t \in V$, and a budget $k$. The goal is to choose a subgraph of $G$ with at most $k$ edges to minimize the $s$-$t$ effective resistance. This problem is an interpolation between the shortest path problem and the minimum cost flow problem and has applications in electrical network design. We present several algorithmic and hardness results for this problem and its variants. On the hardness side, we show that the problem is NP-hard, and the weighted version is hard to approximate within a factor smaller than two assuming the small-set expansion conjecture. On the algorithmic side, we analyze a convex programming relaxation of the problem and design a constant factor approximation algorithm. The key of the rounding algorithm is a randomized path-rounding procedure based on the optimality conditions and a flow decomposition of the fractional solution. We also use dynamic programming to obtain a fully polynomial time approximation scheme when the input graph is a series-parallel graph, with better approximation ratio than the integrality gap of the convex program for these graphs.

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arXiv2019

A PTAS for Bounded-Capacity Vehicle Routing in Planar Graphs

The Capacitated Vehicle Routing problem is to find a minimum-cost set of tours that collectively cover clients in a graph, such that each tour starts and ends at a specified depot and is subject to a capacity bound on the number of clients it can serve. In this paper, we present a polynomial-time approximation scheme (PTAS) for instances in which the input graph is planar and the capacity is bounded. Previously, only a quasipolynomial-time approximation scheme was known for these instances. To obtain this result, we show how to embed planar graphs into bounded-treewidth graphs while preserving, in expectation, the client-to-client distances up to a small additive error proportional to client distances to the depot.

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arXiv2018

A Schur Complement Cheeger Inequality

Cheeger's inequality shows that any undirected graph $G$ with minimum nonzero normalized Laplacian eigenvalue $λ_G$ has a cut with conductance at most $O(\sqrt{λ_G})$. Qualitatively, Cheeger's inequality says that if the relaxation time of a graph is high, there is a cut that certifies this. However, there is a gap in this relationship, as cuts can have conductance as low as $Θ(λ_G)$. To better approximate the relaxation time of a graph, we consider a more general object. Instead of bounding the mixing time with cuts, we bound it with cuts in graphs obtained by Schur complementing out vertices from the graph $G$. Combinatorially, these Schur complements describe random walks in $G$ restricted to a subset of its vertices. As a result, all Schur complement cuts have conductance at least $Ω(λ_G)$. We show that unlike with cuts, this inequality is tight up to a constant factor. Specifically, there is a Schur complement cut with conductance at most $O(λ_G)$.

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arXiv2018

Spectral Subspace Sparsification

We introduce a new approach to spectral sparsification that approximates the quadratic form of the pseudoinverse of a graph Laplacian restricted to a subspace. We show that sparsifiers with a near-linear number of edges in the dimension of the subspace exist. Our setting generalizes that of Schur complement sparsifiers. Our approach produces sparsifiers by sampling a uniformly random spanning tree of the input graph and using that tree to guide an edge elimination procedure that contracts, deletes, and reweights edges. In the context of Schur complement sparsifiers, our approach has two benefits over prior work. First, it produces a sparsifier in almost-linear time with no runtime dependence on the desired error. We directly exploit this to compute approximate effective resistances for a small set of vertex pairs in faster time than prior work (Durfee-Kyng-Peebles-Rao-Sachdeva '17). Secondly, it yields sparsifiers that are reweighted minors of the input graph. As a result, we give a near-optimal answer to a variant of the Steiner point removal problem. A key ingredient of our algorithm is a subroutine of independent interest: a near-linear time algorithm that, given a chosen set of vertices, builds a data structure from which we can query a multiplicative approximation to the decrease in the effective resistance between two vertices after identifying all vertices in the chosen set to a single vertex with inverse polynomial additional additive error in near-constant time.

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arXiv2017

An almost-linear time algorithm for uniform random spanning tree generation

We give an $m^{1+o(1)}β^{o(1)}$-time algorithm for generating a uniformly random spanning tree in an undirected, weighted graph with max-to-min weight ratio $β$. We also give an $m^{1+o(1)}ε^{-o(1)}$-time algorithm for generating a random spanning tree with total variation distance $ε$ from the true uniform distribution. Our second algorithm's runtime does not depend on the edge weights. Our $m^{1+o(1)}β^{o(1)}$-time algorithm is the first almost-linear time algorithm for the problem --- even on unweighted graphs --- and is the first subquadratic time algorithm for sparse weighted graphs. Our algorithms improve on the random walk-based approach given in Kelner-Mądry and Mądry-Straszak-Tarnawski. We introduce a new way of using Laplacian solvers to shortcut a random walk. In order to fully exploit this shortcutting technique, we prove a number of new facts about electrical flows in graphs. These facts seek to better understand sets of vertices that are well-separated in the effective resistance metric in connection with Schur complements, concentration phenomena for electrical flows after conditioning on partial samples of a random spanning tree, and more.

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arXiv2017

Localization of Electrical Flows

We show that in any graph, the average length of a flow path in an electrical flow between the endpoints of a random edge is $O(\log^2 n)$. This is a consequence of a more general result which shows that the spectral norm of the entrywise absolute value of the transfer impedance matrix of a graph is $O(\log^2 n)$. This result implies a simple oblivious routing scheme based on electrical flows in the case of transitive graphs.

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arXiv2013

Interdiction Problems on Planar Graphs

Interdiction problems are leader-follower games in which the leader is allowed to delete a certain number of edges from the graph in order to maximally impede the follower, who is trying to solve an optimization problem on the impeded graph. We introduce approximation algorithms and strong NP-completeness results for interdiction problems on planar graphs. We give a multiplicative $(1 + ε)$-approximation for the maximum matching interdiction problem on weighted planar graphs. The algorithm runs in pseudo-polynomial time for each fixed $ε> 0$. We also show that weighted maximum matching interdiction, budget-constrained flow improvement, directed shortest path interdiction, and minimum perfect matching interdiction are strongly NP-complete on planar graphs. To our knowledge, our budget-constrained flow improvement result is the first planar NP-completeness proof that uses a one-vertex crossing gadget.

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