arXiv subjects
Publications and source records attributed to Andrew McGregor.
We present a slightly simplified analysis of the asymmetric palette sparsification result by Assadi and Yazdanyar [TheoretiCS, 2026]. The motivation is mainly pedagogical; our approach avoids hypergeometric concentration bounds and extra constant factors in the palette size.
We show that a simple extension of the randomized greedy maximal independent set algorithm yields a constant approximation for the maximum matching problem. The algorithm is a simplification of an algorithm used by Assadi et al. [JACM 2026] in the context of processing data streams in the dynamic setting where edges may be inserted and deleted. In contrast to the previous work, our analysis avoids consideration of fractional matchings and yields a significantly shorter and more direct proof of the approximation factor for the basic algorithm.
We investigate the problem of compiling the generation of graph states to arbitrarily many distributed homogeneous quantum processing units (QPUs), providing a scalable partitioning algorithm and graph state generation protocol to minimize the number of Bell pairs required. Current approaches focus on the naive metric of cut edges to estimate the quantum communication cost. We show that the problem of balanced k graph partitioning, with the objective of minimizing the sizes of the maximum matchings between the partitions, leads to lower entanglement requirements across partitions. Our heuristic algorithm, BURY, partitions graph states to require fewer Bell pairs for generation than state-of-the-art k partition algorithms. Furthermore, we show that BURY reduces the cut-rank of the partitions, demonstrating that the partitioning found by our algorithm is likely to minimize the Bell pair utilization of any future improved distributed graph state generation protocol. We also discuss how our methods apply to the dynamic case where the graph state generation and measurement are performed concurrently. Our compilation approach provides a scalable foundation for reducing quantum network overhead for distributed measurement-based quantum computation (MBQC), as well as any scheme where distributed graph state generation is desired.
The ability to physically move qubits within a register allows the design of hardware-specific error-correction codes, which can achieve fault-tolerance while respecting other constraints. In particular, recent advancements have demonstrated the shuttling of electron and hole spin qubits through a quantum dot array with high fidelity. It is therefore timely to explore error correction architectures consisting merely of two parallel quantum dot arrays, an experimentally validated architecture compatible with classical wiring and control constraints. Upon such an architecture, we develop a suite of heuristic methods for compiling any Calderbank-Shor-Steane (CSS) error-correcting code's syndrome-extraction circuit to run with a reduced number of shuttling operations. We demonstrate how column-regular qLDPC codes can be compiled in a provably minimal number of shuttles that is exactly equal to the column weight of the code when Shor-style syndrome extraction is used. We provide tables stating the number of required shuttles for many contemporary codes of interest. In addition, we provide a proof of the NP hardness of minimizing the number of shuttle operations for general codes, even when using Shor syndrome extraction. We also discuss how one could get around this by placing blanks in the ancilla array to achieve minimal shuttles with Shor syndrome extraction on any CSS code, at the cost of longer ancilla arrays.
We consider the problem of reconstructing an undirected graph $G$ on $n$ vertices given multiple random noisy subgraphs or "traces". Specifically, a trace is generated by sampling each vertex with probability $p_v$, then taking the resulting induced subgraph on the sampled vertices, and then adding noise in the form of either (a) deleting each edge in the subgraph with probability $1-p_e$, or (b) deleting each edge with probability $f_e$ and transforming a non-edge into an edge with probability $f_e$. We show that, under mild assumptions on $p_v$, $p_e$ and $f_e$, if $G$ is selected uniformly at random, then $O(p_e^{-1} p_v^{-2} \log n)$ or $O((f_e-1/2)^{-2} p_v^{-2} \log n)$ traces suffice to reconstruct $G$ with high probability. In contrast, if $G$ is arbitrary, then $\exp(Ω(n))$ traces are necessary even when $p_v=1, p_e=1/2$.
As Low Earth Orbit (LEO) satellite mega constellations continue to be deployed for satellite internet and recent successful experiments in satellite-based quantum entanglement distribution emerge, a natural question arises: How should we coordinate transmissions and design scalable scheduling policies for a quantum satellite internet? In this work, we consider the problem of transmission scheduling in quantum satellite networks subject to resource constraints at the satellites and ground stations. We show that the most general problem of assigning satellites to ground station pairs for entanglement distribution is NP-hard. We then propose four heuristic algorithms and evaluate their performance for Starlink mega constellation under various amount of resources and placements of the ground stations. We find that the maximum number of receivers necessary per ground station grows very slowly with the total number of deployed ground stations. Our proposed algorithms, leveraging optimal weighted b-matching and the global greedy heuristic, outperform others in entanglement distribution rate, entanglement fidelity, and handover cost metrics. While we develop these scheduling algorithms, we have also designed a software system to simulate, visualize, and evaluate satellite mega-constellations for entanglement distribution.
The maximum coverage problem is to select $k$ sets from a collection of sets such that the cardinality of the union of the selected sets is maximized. We consider $(1-1/e-ε)$-approximation algorithms for this NP-hard problem in three standard data stream models. 1. {\em Dynamic Model.} The stream consists of a sequence of sets being inserted and deleted. Our multi-pass algorithm uses $ε^{-2} k \cdot \text{polylog}(n,m)$ space. The best previous result (Assadi and Khanna, SODA 2018) used $(n +ε^{-4} k) \text{polylog}(n,m)$ space. While both algorithms use $O(ε^{-1} \log n)$ passes, our analysis shows that when $ε$ is a constant, it is possible to reduce the number of passes by a $1/\log \log n$ factor without incurring additional space. 2. {\em Random Order Model.} In this model, there are no deletions and the sets forming the instance are uniformly randomly permuted to form the input stream. We show that a single pass and $k \text{polylog}(n,m)$ space suffices for arbitrary small constant $ε$. The best previous result, by Warneke et al.~(ESA 2023), used $k^2 \text{polylog}(n,m)$ space. 3. {\em Insert-Only Model.} Lastly, our results, along with numerous previous results, use a sub-sampling technique introduced by McGregor and Vu (ICDT 2017) to sparsify the input instance. We explain how this technique and others used in the paper can be implemented such that the amortized update time of our algorithm is polylogarithmic. This also implies an improvement of the state-of-the-art insert only algorithms in terms of the update time: $\text{polylog}(m,n)$ update time suffices whereas the best previous result by Jaud et al.~(SEA 2023) required update time that was linear in $k$.
The Graphical House Allocation problem asks: how can $n$ houses (each with a fixed non-negative value) be assigned to the vertices of an undirected graph $G$, so as to minimize the "aggregate local envy", i.e., the sum of absolute differences along the edges of $G$? This problem generalizes the classical Minimum Linear Arrangement problem, as well as the well-known House Allocation Problem from Economics, the latter of which has notable practical applications in organ exchanges. Recent work has studied the computational aspects of Graphical House Allocation and observed that the problem is NP-hard and inapproximable even on particularly simple classes of graphs, such as vertex disjoint unions of paths. However, the dependence of any approximations on the structural properties of the underlying graph had not been studied. In this work, we give a complete characterization of the approximability of the Graphical House Allocation problem. We present algorithms to approximate the optimal envy on general graphs, trees, planar graphs, bounded-degree graphs, bounded-degree planar graphs, and bounded-degree trees. For each of these graph classes, we then prove matching lower bounds, showing that in each case, no significant improvement can be attained unless P = NP. We also present general approximation ratios as a function of structural parameters of the underlying graph, such as treewidth; these match the aforementioned tight upper bounds in general, and are significantly better approximations for many natural subclasses of graphs. Finally, we present constant factor approximation schemes for the special classes of complete binary trees and random graphs.
Learning causal structure is useful in many areas of artificial intelligence, including planning, robotics, and explanation. Constraint-based structure learning algorithms such as PC use conditional independence (CI) tests to infer causal structure. Traditionally, constraint-based algorithms perform CI tests with a preference for smaller-sized conditioning sets, partially because the statistical power of conventional CI tests declines rapidly as the size of the conditioning set increases. However, many modern conditional independence tests are model-based, and these tests use well-regularized models that maintain statistical power even with very large conditioning sets. This suggests an intriguing new strategy for constraint-based algorithms which may result in a reduction of the total number of CI tests performed: Test variable pairs with large conditioning sets first, as a pre-processing step that finds some conditional independencies quickly, before moving on to the more conventional strategy that favors small conditioning sets. We propose such a pre-processing step for the PC algorithm which relies on performing CI tests on a few randomly selected large conditioning sets. We perform an empirical analysis on directed acyclic graphs (DAGs) that correspond to real-world systems and both empirical and theoretical analyses for Erdős-Renyi DAGs. Our results show that Pre-Processing Plus PC (P3PC) performs far fewer CI tests than the original PC algorithm, between 0.5% to 36%, and often less than 10%, of the CI tests that the PC algorithm alone performs. The efficiency gains are particularly significant for the DAGs corresponding to real-world systems.
We study the problem of estimating the number of edges in an $n$-vertex graph, accessed via the Bipartite Independent Set query model introduced by Beame et al. (ITCS '18). In this model, each query returns a Boolean, indicating the existence of at least one edge between two specified sets of nodes. We present a non-adaptive algorithm that returns a $(1\pm ε)$ relative error approximation to the number of edges, with query complexity $\tilde O(ε^{-5}\log^{5} n )$, where $\tilde O(\cdot)$ hides $\textrm{poly}(\log \log n)$ dependencies. This is the first non-adaptive algorithm in this setting achieving $\textrm{poly}(1/ε,\log n)$ query complexity. Prior work requires $Ω(\log^2 n)$ rounds of adaptivity. We avoid this by taking a fundamentally different approach, inspired by work on single-pass streaming algorithms. Moreover, for constant $ε$, our query complexity significantly improves on the best known adaptive algorithm due to Bhattacharya et al. (STACS '22), which requires $O(ε^{-2} \log^{11} n)$ queries. Building on our edge estimation result, we give the first non-adaptive algorithm for outputting a nearly uniformly sampled edge with query complexity $\tilde O(ε^{-6} \log^{6} n)$, improving on the works of Dell et al. (SODA '20) and Bhattacharya et al. (STACS '22), which require $Ω(\log^3 n)$ rounds of adaptivity. Finally, as a consequence of our edge sampling algorithm, we obtain a $\tilde O(n\log^ 8 n)$ query algorithm for connectivity, using two rounds of adaptivity. This improves on a three-round algorithm of Assadi et al. (ESA '21) and is tight; there is no non-adaptive algorithm for connectivity making $o(n^2)$ queries.
Recent work of Acharya et al. (NeurIPS 2019) showed how to estimate the entropy of a distribution $\mathcal D$ over an alphabet of size $k$ up to $\pmε$ additive error by streaming over $(k/ε^3) \cdot \text{polylog}(1/ε)$ i.i.d. samples and using only $O(1)$ words of memory. In this work, we give a new constant memory scheme that reduces the sample complexity to $(k/ε^2)\cdot \text{polylog}(1/ε)$. We conjecture that this is optimal up to $\text{polylog}(1/ε)$ factors.
Given an $n$-point metric space $(\mathcal{X},d)$ where each point belongs to one of $m=O(1)$ different categories or groups and a set of integers $k_1, \ldots, k_m$, the fair Max-Min diversification problem is to select $k_i$ points belonging to category $i\in [m]$, such that the minimum pairwise distance between selected points is maximized. The problem was introduced by Moumoulidou et al. [ICDT 2021] and is motivated by the need to down-sample large data sets in various applications so that the derived sample achieves a balance over diversity, i.e., the minimum distance between a pair of selected points, and fairness, i.e., ensuring enough points of each category are included. We prove the following results: 1. We first consider general metric spaces. We present a randomized polynomial time algorithm that returns a factor $2$-approximation to the diversity but only satisfies the fairness constraints in expectation. Building upon this result, we present a $6$-approximation that is guaranteed to satisfy the fairness constraints up to a factor $1-ε$ for any constant $ε$. We also present a linear time algorithm returning an $m+1$ approximation with exact fairness. The best previous result was a $3m-1$ approximation. 2. We then focus on Euclidean metrics. We first show that the problem can be solved exactly in one dimension. For constant dimensions, categories and any constant $ε>0$, we present a $1+ε$ approximation algorithm that runs in $O(nk) + 2^{O(k)}$ time where $k=k_1+\ldots+k_m$. We can improve the running time to $O(nk)+ poly(k)$ at the expense of only picking $(1-ε) k_i$ points from category $i\in [m]$. Finally, we present algorithms suitable to processing massive data sets including single-pass data stream algorithms and composable coresets for the distributed processing.
We consider directed graph algorithms in a streaming setting, focusing on problems concerning orderings of the vertices. This includes such fundamental problems as topological sorting and acyclicity testing. We also study the related problems of finding a minimum feedback arc set (edges whose removal yields an acyclic graph), and finding a sink vertex. We are interested in both adversarially-ordered and randomly-ordered streams. For arbitrary input graphs with edges ordered adversarially, we show that most of these problems have high space complexity, precluding sublinear-space solutions. Some lower bounds also apply when the stream is randomly ordered: e.g., in our most technical result we show that testing acyclicity in the $p$-pass random-order model requires roughly $n^{1+1/p}$ space. For other problems, random ordering can make a dramatic difference: e.g., it is possible to find a sink in an acyclic tournament in the one-pass random-order model using polylog$(n)$ space whereas under adversarial ordering roughly $n^{1/p}$ space is necessary and sufficient given $Θ(p)$ passes. We also design sublinear algorithms for the feedback arc set problem in tournament graphs; for random graphs; and for randomly ordered streams. In some cases, we give lower bounds establishing that our algorithms are essentially space-optimal. Together, our results complement the much maturer body of work on algorithms for undirected graph streams.
In the beautifully simple-to-state problem of trace reconstruction, the goal is to reconstruct an unknown binary string $x$ given random "traces" of $x$ where each trace is generated by deleting each coordinate of $x$ independently with probability $p<1$. The problem is well studied both when the unknown string is arbitrary and when it is chosen uniformly at random. For both settings, there is still an exponential gap between upper and lower sample complexity bounds and our understanding of the problem is still surprisingly limited. In this paper, we consider natural parameterizations and generalizations of this problem in an effort to attain a deeper and more comprehensive understanding. We prove that $\exp(O(n^{1/4} \sqrt{\log n}))$ traces suffice for reconstructing arbitrary matrices. In the matrix version of the problem, each row and column of an unknown $\sqrt{n}\times \sqrt{n}$ matrix is deleted independently with probability $p$. Our results contrasts with the best known results for sequence reconstruction where the best known upper bound is $\exp(O(n^{1/3}))$. An optimal result for random matrix reconstruction: we show that $Θ(\log n)$ traces are necessary and sufficient. This is in contrast to the problem for random sequences where there is a super-logarithmic lower bound and the best known upper bound is $\exp({O}(\log^{1/3} n))$. We show that $\exp(O(k^{1/3}\log^{2/3} n))$ traces suffice to reconstruct $k$-sparse strings, providing an improvement over the best known sequence reconstruction results when $k = o(n/\log^2 n)$. We show that $\textrm{poly}(n)$ traces suffice if $x$ is $k$-sparse and we additionally have a "separation" promise, specifically that the indices of 1's in $x$ all differ by $Ω(k \log n)$.
We present algorithms for the Max-Cover and Max-Unique-Cover problems in the data stream model. The input to both problems are $m$ subsets of a universe of size $n$ and a value $k\in [m]$. In Max-Cover, the problem is to find a collection of at most $k$ sets such that the number of elements covered by at least one set is maximized. In Max-Unique-Cover, the problem is to find a collection of at most $k$ sets such that the number of elements covered by exactly one set is maximized. Our goal is to design single-pass algorithms that use space that is sublinear in the input size. Our main algorithmic results are: If the sets have size at most $d$, there exist single-pass algorithms using $\tilde{O}(d^{d+1} k^d)$ space that solve both problems exactly. This is optimal up to polylogarithmic factors for constant $d$. If each element appears in at most $r$ sets, we present single pass algorithms using $\tilde{O}(k^2 r/ε^3)$ space that return a $1+ε$ approximation in the case of Max-Cover. We also present a single-pass algorithm using slightly more memory, i.e., $\tilde{O}(k^3 r/ε^{4})$ space, that $1+ε$ approximates Max-Unique-Cover. In contrast to the above results, when $d$ and $r$ are arbitrary, any constant pass $1+ε$ approximation algorithm for either problem requires $Ω(ε^{-2}m)$ space but a single pass $O(ε^{-2}mk)$ space algorithm exists. In fact any constant-pass algorithm with an approximation better than $e/(e-1)$ and $e^{1-1/k}$ for Max-Cover and Max-Unique-Cover respectively requires $Ω(m/k^2)$ space when $d$ and $r$ are unrestricted. En route, we also obtain an algorithm for a parameterized version of the streaming Set-Cover problem.
We study the problem of learning the causal relationships between a set of observed variables in the presence of latents, while minimizing the cost of interventions on the observed variables. We assume access to an undirected graph $G$ on the observed variables whose edges represent either all direct causal relationships or, less restrictively, a superset of causal relationships (identified, e.g., via conditional independence tests or a domain expert). Our goal is to recover the directions of all causal or ancestral relations in $G$, via a minimum cost set of interventions. It is known that constructing an exact minimum cost intervention set for an arbitrary graph $G$ is NP-hard. We further argue that, conditioned on the hardness of approximate graph coloring, no polynomial time algorithm can achieve an approximation factor better than $Θ(\log n)$, where $n$ is the number of observed variables in $G$. To overcome this limitation, we introduce a bi-criteria approximation goal that lets us recover the directions of all but $εn^2$ edges in $G$, for some specified error parameter $ε> 0$. Under this relaxed goal, we give polynomial time algorithms that achieve intervention cost within a small constant factor of the optimal. Our algorithms combine work on efficient intervention design and the design of low-cost separating set systems, with ideas from the literature on graph property testing.
Hierarchical clustering is a fundamental task often used to discover meaningful structures in data, such as phylogenetic trees, taxonomies of concepts, subtypes of cancer, and cascades of particle decays in particle physics. Typically approximate algorithms are used for inference due to the combinatorial number of possible hierarchical clusterings. In contrast to existing methods, we present novel dynamic-programming algorithms for \emph{exact} inference in hierarchical clustering based on a novel trellis data structure, and we prove that we can exactly compute the partition function, maximum likelihood hierarchy, and marginal probabilities of sub-hierarchies and clusters. Our algorithms scale in time and space proportional to the powerset of $N$ elements which is super-exponentially more efficient than explicitly considering each of the (2N-3)!! possible hierarchies. Also, for larger datasets where our exact algorithms become infeasible, we introduce an approximate algorithm based on a sparse trellis that compares well to other benchmarks. Exact methods are relevant to data analyses in particle physics and for finding correlations among gene expression in cancer genomics, and we give examples in both areas, where our algorithms outperform greedy and beam search baselines. In addition, we consider Dasgupta's cost with synthetic data.
Diversity is an important principle in data selection and summarization, facility location, and recommendation systems. Our work focuses on maximizing diversity in data selection, while offering fairness guarantees. In particular, we offer the first study that augments the Max-Min diversification objective with fairness constraints. More specifically, given a universe $U$ of $n$ elements that can be partitioned into $m$ disjoint groups, we aim to retrieve a $k$-sized subset that maximizes the pairwise minimum distance within the set (diversity) and contains a pre-specified $k_i$ number of elements from each group $i$ (fairness). We show that this problem is NP-complete even in metric spaces, and we propose three novel algorithms, linear in $n$, that provide strong theoretical approximation guarantees for different values of $m$ and $k$. Finally, we extend our algorithms and analysis to the case where groups can be overlapping.