arXiv subjects

Will Rosenbaum

Publications and source records attributed to Will Rosenbaum.

At least 19 recordsLinked to original sources

arXiv2026

Bridging the Gap Between Stable Marriage and Stable Roommates: A Parameterized Algorithm for Optimal Stable Matchings

In the Stable Roommates Problem (SR), a set of $2n$ agents rank one another in a linear order. The goal is to find a matching that is stable: one that has no pair of agents who mutually prefer each other over their assigned partners. We consider the problem of finding an optimal stable matching. Agents associate weights with each of their potential partners, and the goal is to find a stable matching that minimizes the sum of the associated weights. Efficient algorithms exist for finding an optimal stable matching in the Stable Marriage Problem (SM), but the problem is NP-hard for general SR instances. In this paper, we define a notion of structural distance between SR instances and SM instances, which we call the minimum crossing distance. When an SR instance has minimum crossing distance $0$, the instance is structurally equivalent to an SM instance, and this structure can be exploited to find an optimal stable matching efficiently. More generally, we show that when an SR instance has minimum crossing distance $k$, an optimal stable matching can be computed in time $2^{O(k)} n^{O(1)}$. Thus, the optimal stable matching problem is fixed-parameter tractable (FPT) with respect to minimum crossing distance.

cs.DS↗

arXiv2026

Distributed Triangle and Simplex Enumeration in Hypergraphs

In the last decade, subgraph detection and enumeration have emerged as central problems in distributed graph algorithms. This is largely due to the problems' theoretical challenges and practical applications. In this paper, we initiate the systematic study of distributed sub-hypergraph enumeration in hypergraphs. To this end, we (1) introduce several computational models for hypergraphs that generalize the CONGEST model for graphs and evaluate their relative computational power, (2) devise algorithms for distributed triangle and simplex enumeration in our computational models and prove their optimality in two such models by showing matching lower bounds, (3) introduce classes of sparse and "everywhere sparse" hypergraphs and describe efficient distributed algorithms for triangle and simplex enumeration in these classes, and (4) describe general techniques that we believe to be useful for designing efficient algorithms in our hypergraph models.

cs.DC↗

arXiv2025

A Quadratic Lower Bound for Stable Roommates Solvability

In their seminal work on the Stable Marriage Problem (SM), Gale and Shapley introduced a generalization of SM referred to as the Stable Roommates Problem (SR). An instance of SR consists of a set of $2n$ agents, and each agent has preferences in the form of a ranked list of all other agents. The goal is to find a one-to-one matching between the agents that is stable in the sense that no pair of agents have a mutual incentive to deviate from the matching. Unlike the (bipartite) stable marriage problem, in SR, stable matchings need not exist. Irving devised an algorithm that finds a stable matching or reports that none exists in $O(n^2)$ time. In their influential 1989 text, Gusfield and Irving posed the question of whether $Ω(n^2)$ time is required for SR solvability -- the task of deciding if an SR instance admits a stable matching. In this paper we provide an affirmative answer to Gusfield and Irving's question. We show that any (randomized) algorithm that decides SR solvability requires $Ω(n^2)$ adaptive Boolean queries to the agents' preferences (in expectation). Our argument follows from a reduction from the communication complexity of the set disjointness function. The query lower bound implies quadratic time lower bounds for Turing machines, and memory access lower bounds for random access machines. Thus, we establish that Irving's algorithm is optimal (up to a logarithmic factor) in a very strong sense.

cs.CC↗

arXiv2024

Distributed weak independent sets in hypergraphs: Upper and lower bounds

In this paper, we consider the problem of finding weak independent sets in a distributed network represented by a hypergraph. In this setting, each edge contains a set of r vertices rather than simply a pair, as in a standard graph. A k-weak independent set in a hypergraph is a set where no edge contains more than k vertices in the independent set. We focus two variations of this problem. First, we study the problem of finding k-weak maximal independent sets, k-weak independent sets where each vertex belongs to at least one edge with k vertices in the independent set. Second we introduce a weaker variant that we call (α, β)-independent sets where the independent set is β-weak, and each vertex belongs to at least one edge with at least αvertices in the independent set. Finally, we consider the problem of finding a (2, k)-ruling set on hypergraphs, i.e. independent sets where no vertex is a distance of more than k from the nearest member of the set. Given a hypergraph H of rank r and maximum degree Δ, we provide a LLL formulation for finding an (α, β)-independent set when (β- α)^2 / (β+ α) \geq 6 \log(16 r Δ), an O(Δr / (β- α+ 1) + \log^* n) round deterministic algorithm finding an (α, β)-independent set, and a O(Δ^2(r - k) \log r + Δ\log r \log^* r + \log^* n) round algorithm for finding a k-weak maximal independent set. Additionally, we provide zero round randomized algorithms for finding (α, β) independent sets, when (β- α)^2 / (β+ α) \geq 6 c \log n + 6 for some constant c, and finding an m-weak independent set for some m \geq r / 2k where k is a given parameter. Finally, we provide lower bounds of Ω(Δ+ \log^* n) and Ω(r + \log^* n) on the problems of finding a k-weak maximal independent sets for some values of k.

cs.DC↗

arXiv2022

Packet Forwarding with a Locally Bursty Adversary

We consider packet forwarding in the adversarial queueing theory (AQT) model introduced by Borodin et al. We introduce a refinement of the AQT $(ρ, σ)$-bounded adversary, which we call a \emph{locally bursty adversary} (LBA) that parameterizes injection patterns jointly by edge utilization and packet origin. For constant ($O(1)$) parameters, the LBA model is strictly more permissive than the $(ρ, σ)$ model. For example, there are injection patterns in the LBA model with constant parameters that can only be realized as $(ρ, σ)$-bounded injection patterns with $ρ+ σ= Ω(n)$ (where $n$ is the network size). We show that the LBA model (unlike the $(ρ, σ)$ model) is closed under packet bundling and discretization operations. Thus, the LBA model allows one to reduce the study of general (uniform) capacity networks and inhomogenous packet sizes to unit capacity networks with homogeneous packets. On the algorithmic side, we focus on information gathering networks -- i.e., networks in which all packets share a common destination, and the union of packet routes forms a tree. We show that the Odd-Even Downhill (OED) forwarding protocol described independently by Dobrev et al.\ and Patt-Shamir and Rosenbaum achieves buffer space usage of $O(\log n)$ against all LBAs with constant parameters. OED is a local protocol, but we show that the upper bound is tight even when compared to centralized protocols. Our lower bound for the LBA model is in contrast to the $(ρ, σ)$-model, where centralized protocols can achieve worst-case buffer space usage $O(1)$ for $ρ, σ= O(1)$, while the $O(\log n)$ upper bound for OED is optimal only for local protocols.

cs.DC↗

arXiv2022

Training-Time Attacks against k-Nearest Neighbors

Nearest neighbor-based methods are commonly used for classification tasks and as subroutines of other data-analysis methods. An attacker with the capability of inserting their own data points into the training set can manipulate the inferred nearest neighbor structure. We distill this goal to the task of performing a training-set data insertion attack against $k$-Nearest Neighbor classification ($k$NN). We prove that computing an optimal training-time (a.k.a. poisoning) attack against $k$NN classification is NP-Hard, even when $k = 1$ and the attacker can insert only a single data point. We provide an anytime algorithm to perform such an attack, and a greedy algorithm for general $k$ and attacker budget. We provide theoretical bounds and empirically demonstrate the effectiveness and practicality of our methods on synthetic and real-world datasets. Empirically, we find that $k$NN is vulnerable in practice and that dimensionality reduction is an effective defense. We conclude with a discussion of open problems illuminated by our analysis.

cs.LG↗

arXiv2022

Bias Reduction for Sum Estimation

In classical statistics and distribution testing, it is often assumed that elements can be sampled from some distribution $P$, and that when an element $x$ is sampled, the probability $P$ of sampling $x$ is also known. Recent work in distribution testing has shown that many algorithms are robust in the sense that they still produce correct output if the elements are drawn from any distribution $Q$ that is sufficiently close to $P$. This phenomenon raises interesting questions: under what conditions is a "noisy" distribution $Q$ sufficient, and what is the algorithmic cost of coping with this noise? We investigate these questions for the problem of estimating the sum of a multiset of $N$ real values $x_1, \ldots, x_N$. This problem is well-studied in the statistical literature in the case $P = Q$, where the Hansen-Hurwitz estimator is frequently used. We assume that for some known distribution $P$, values are sampled from a distribution $Q$ that is pointwise close to $P$. For every positive integer $k$ we define an estimator $ζ_k$ for $μ= \sum_i x_i$ whose bias is proportional to $γ^k$ (where our $ζ_1$ reduces to the classical Hansen-Hurwitz estimator). As a special case, we show that if $Q$ is pointwise $γ$-close to uniform and all $x_i \in \{0, 1\}$, for any $ε> 0$, we can estimate $μ$ to within additive error $εN$ using $m = Θ({N^{1-\frac{1}{k}} / ε^{2/k}})$ samples, where $k = \left\lceil (\log ε)/(\log γ)\right\rceil$. We show that this sample complexity is essentially optimal. Our bounds show that the sample complexity need not vary uniformly with the desired error parameter $ε$: for some values of $ε$, perturbations in its value have no asymptotic effect on the sample complexity, while for other values, any decrease in its value results in an asymptotically larger sample complexity.

cs.DS↗

arXiv2022

Finding a Winning Strategy for Wordle is NP-complete

In this paper, we give a formal definition of the popular word-guessing game Wordle. We show that, in general, determining if a given Wordle instance admits a winning strategy is NP-complete. We also show that given a Wordle instance of size $N$, a winning strategy that uses $g$ guesses in the worst case (if any) can be found in time $N^{O(g)}$.

cs.CC↗

arXiv2021

Stable Matchings with Restricted Preferences: Structure and Complexity

It is well known that every stable matching instance $I$ has a rotation poset $R(I)$ that can be computed efficiently and the downsets of $R(I)$ are in one-to-one correspondence with the stable matchings of $I$. Furthermore, for every poset $P$, an instance $I(P)$ can be constructed efficiently so that the rotation poset of $I(P)$ is isomorphic to $P$. In this case, we say that $I(P)$ realizes $P$. Many researchers exploit the rotation poset of an instance to develop fast algorithms or to establish the hardness of stable matching problems. In order to gain a parameterized understanding of the complexity of sampling stable matchings, Bhatnagar et al. [SODA 2008] introduced stable matching instances whose preference lists are restricted but nevertheless model situations that arise in practice. In this paper, we study four such parameterized restrictions; our goal is to characterize the rotation posets that arise from these models: $k$-bounded, $k$-attribute, $(k_1, k_2)$-list, $k$-range. We prove that there is a constant $k$ so that every rotation poset is realized by some instance in the first three models for some fixed constant $k$. We describe efficient algorithms for constructing such instances given the Hasse diagram of a poset. As a consequence, the fundamental problem of counting stable matchings remains $\#$BIS-complete even for these restricted instances. For $k$-range preferences, we show that a poset $P$ is realizable if and only if the Hasse diagram of $P$ has pathwidth bounded by functions of $k$. Using this characterization, we show that the following problems are fixed parameter tractable when parametrized by the range of the instance: exactly counting and uniformly sampling stable matchings, finding median, sex-equal, and balanced stable matchings.

cs.DM↗

arXiv2020

Almost Optimal Bounds for Sublinear-Time Sampling of $k$-Cliques: Sampling Cliques is Harder Than Counting

In this work, we consider the problem of sampling a $k$-clique in a graph from an almost uniform distribution in sublinear time in the general graph query model. Specifically the algorithm should output each $k$-clique with probability $(1\pm ε)/n_k$, where $n_k$ denotes the number of $k$-cliques in the graph and $ε$ is a given approximation parameter. We prove that the query complexity of this problem is \[ Θ^*\left(\max\left\{ \left(\frac{(nα)^{k/2}}{ n_k}\right)^{\frac{1}{k-1}} ,\; \min\left\{nα,\frac{nα^{k-1}}{n_k} \right\}\right\}\right). \] where $n$ is the number of vertices in the graph, $α$ is its arboricity, and $Θ^*$ suppresses the dependence on $(\log n/ε)^{O(k)}$. Interestingly, this establishes a separation between approximate counting and approximate uniform sampling in the sublinear regime. For example, if $k=3$, $α= O(1)$, and $n_3$ (the number of triangles) is $Θ(n)$, then we get a lower bound of $Ω(n^{1/4})$ (for constant $ε$), while under these conditions, a $(1\pm ε)$-approximation of $n_3$ can be obtained by performing $\textrm{poly}(\log(n/ε))$ queries (Eden, Ron and Seshadhri, SODA20). Our lower bound follows from a construction of a family of graphs with arboricity $α$ such that in each graph there are $n_k$ cliques (of size $k$), where one of these cliques is "hidden" and hence hard to sample. Our upper bound is based on defining a special auxiliary graph $H_k$, such that sampling edges almost uniformly in $H_k$ translates to sampling $k$-cliques almost uniformly in the original graph $G$. We then build on a known edge-sampling algorithm (Eden, Ron and Rosenbaum, ICALP19) to sample edges in $H_k$, where the challenge is simulate queries to $H_k$ while being given access only to $G$.

cs.DS↗

arXiv2020

Simple Counting and Sampling Algorithms for Graphs with Bounded Pathwidth

In this paper, we consider the problem of counting and sampling structures in graphs. We define a class of "edge universal labeling problems"---which include proper $k$-colorings, independent sets, and downsets---and describe simple algorithms for counting and uniformly sampling valid labelings of graphs, assuming a path decomposition is given. Thus, we show that several well-studied counting and sampling problems are fixed parameter tractable (FPT) when parameterized by the pathwidth of the input graph. We discuss connections to counting and sampling problems for distributive lattices and, in particular, we give a new FPT algorithm for exactly counting and uniformly sampling stable matchings.

cs.DS↗

arXiv2020

PALS: Plesiochronous and Locally Synchronous Systems

Consider an arbitrary network of communicating modules on a chip, each requiring a local signal telling it when to execute a computational step. There are three common solutions to generating such a local clock signal: (i) by deriving it from a single, central clock source, (ii) by local, free-running oscillators, or (iii) by handshaking between neighboring modules. Conceptually, each of these solutions is the result of a perceived dichotomy in which (sub)systems are either clocked or fully asynchronous, suggesting that the designer's choice is limited to deciding where to draw the line between synchronous and asynchronous design. In contrast, we take the view that the better question to ask is how synchronous the system can and should be. Based on a distributed clock synchronization algorithm, we present a novel design providing modules with local clocks whose frequency bounds are almost as good as those of corresponding free-running oscillators, yet neighboring modules are guaranteed to have a phase offset substantially smaller than one clock cycle. Concretely, parameters obtained from a 15nm ASIC implementation running at 2GHz yield mathematical worst-case bounds of 30ps on phase offset for a 32x32 node grid network.

cs.DC↗

arXiv2020

Seeing Far vs. Seeing Wide: Volume Complexity of Local Graph Problems

Consider a graph problem that is locally checkable but not locally solvable: given a solution we can check that it is feasible by verifying all constant-radius neighborhoods, but to find a solution each node needs to explore the input graph at least up to distance $Ω(\log n)$ in order to produce its output. We consider the complexity of such problems from the perspective of volume: how large a subgraph does a node need to see in order to produce its output. We study locally checkable graph problems on bounded-degree graphs. We give a number of constructions that exhibit tradeoffs between deterministic distance, randomized distance, deterministic volume, and randomized volume: - If the deterministic distance is linear, it is also known that randomized distance is near-linear. In contrast, we show that there are problems with linear deterministic volume but only logarithmic randomized volume. - We prove a volume hierarchy theorem for randomized complexity: among problems with linear deterministic volume complexity, there are infinitely many distinct randomized volume complexity classes between $Ω(\log n)$ and $O(n)$. This hierarchy persists even when restricting to problems whose randomized and deterministic distance complexities are $Θ(\log n)$. - Similar hierarchies exist for polynomial distance complexities: for any $k, \ell \in N$ with $k \leq \ell$, there are problems whose randomized and deterministic distance complexities are $Θ(n^{1/\ell})$, randomized volume complexities are $Θ(n^{1/k})$, and whose deterministic volume complexities are $Θ(n)$. Additionally, we consider connections between our volume model and massively parallel computation (MPC). We give a general simulation argument that any volume-efficient algorithm can be transformed into a space-efficient MPC algorithm.

cs.DC↗

arXiv2019

Fault Tolerant Gradient Clock Synchronization

Synchronizing clocks in distributed systems is well-understood, both in terms of fault-tolerance in fully connected systems and the dependence of local and global worst-case skews (i.e., maximum clock difference between neighbors and arbitrary pairs of nodes, respectively) on the diameter of fault-free systems. However, so far nothing non-trivial is known about the local skew that can be achieved in topologies that are not fully connected even under a single Byzantine fault. Put simply, in this work we show that the most powerful known techniques for fault-tolerant and gradient clock synchronization are compatible, in the sense that the best of both worlds can be achieved simultaneously. Concretely, we combine the Lynch-Welch algorithm [Welch1988] for synchronizing a clique of $n$ nodes despite up to $f<n/3$ Byzantine faults with the gradient clock synchronization (GCS) algorithm by Lenzen et al. [Lenzen2010] in order to render the latter resilient to faults. As this is not possible on general graphs, we augment an input graph $\mathcal{G}$ by replacing each node by $3f+1$ fully connected copies, which execute an instance of the Lynch-Welch algorithm. We then interpret these clusters as supernodes executing the GCS algorithm, where for each cluster its correct nodes' Lynch-Welch clocks provide estimates of the logical clock of the supernode in the GCS algorithm. By connecting clusters corresponding to neighbors in $\mathcal{G}$ in a fully bipartite manner, supernodes can inform each other about (estimates of) their logical clock values. This way, we achieve asymptotically optimal local skew, granted that no cluster contains more than $f$ faulty nodes, at factor $O(f)$ and $O(f^2)$ overheads in terms of nodes and edges, respectively. Note that tolerating $f$ faulty neighbors trivially requires degree larger than $f$, so this is asymptotically optimal as well.

cs.DC↗

arXiv2019

With Great Speed Come Small Buffers: Space-Bandwidth Tradeoffs for Routing

We consider the Adversarial Queuing Theory (AQT) model, where packet arrivals are subject to a maximum average rate $0\leρ\le1$ and burstiness $σ\ge0$. In this model, we analyze the size of buffers required to avoid overflows in the basic case of a path. Our main results characterize the space required by the average rate and the number of distinct destinations: we show that $O(k d^{1/k})$ space suffice, where $d$ is the number of distinct destinations and $k=\lfloor 1/ρ\rfloor$; and we show that $Ω(\frac 1 k d^{1/k})$ space is necessary. For directed trees, we describe an algorithm whose buffer space requirement is at most $1 + d' + σ$ where $d'$ is the maximum number of destinations on any root-leaf path.

cs.DS↗

arXiv2019

The Arboricity Captures the Complexity of Sampling Edges

In this paper, we revisit the problem of sampling edges in an unknown graph $G = (V, E)$ from a distribution that is (pointwise) almost uniform over $E$. We consider the case where there is some a priori upper bound on the arboriciy of $G$. Given query access to a graph $G$ over $n$ vertices and of average degree $d$ and arboricity at most $α$, we design an algorithm that performs $O\!\left(\fracα{d} \cdot \frac{\log^3 n}{\varepsilon}\right)$ queries in expectation and returns an edge in the graph such that every edge $e \in E$ is sampled with probability $(1 \pm \varepsilon)/m$. The algorithm performs two types of queries: degree queries and neighbor queries. We show that the upper bound is tight (up to poly-logarithmic factors and the dependence in $\varepsilon$), as $Ω\!\left(\fracα{d} \right)$ queries are necessary for the easier task of sampling edges from any distribution over $E$ that is close to uniform in total variational distance. We also prove that even if $G$ is a tree (i.e., $α= 1$ so that $\fracα{d}=Θ(1)$), $Ω\left(\frac{\log n}{\log\log n}\right)$ queries are necessary to sample an edge from any distribution that is pointwise close to uniform, thus establishing that a $\mathrm{poly}(\log n)$ factor is necessary for constant $α$. Finally we show how our algorithm can be applied to obtain a new result on approximately counting subgraphs, based on the recent work of Assadi, Kapralov, and Khanna (ITCS, 2019).

cs.CC↗

arXiv2018

A Stable Marriage Requires Communication

The Gale-Shapley algorithm for the Stable Marriage Problem is known to take $Θ(n^2)$ steps to find a stable marriage in the worst case, but only $Θ(n \log n)$ steps in the average case (with $n$ women and $n$ men). In 1976, Knuth asked whether the worst-case running time can be improved in a model of computation that does not require sequential access to the whole input. A partial negative answer was given by Ng and Hirschberg, who showed that $Θ(n^2)$ queries are required in a model that allows certain natural random-access queries to the participants' preferences. A significantly more general - albeit slightly weaker - lower bound follows from Segal's general analysis of communication complexity, namely that $Ω(n^2)$ Boolean queries are required in order to find a stable marriage, regardless of the set of allowed Boolean queries. Using a reduction to the communication complexity of the disjointness problem, we give a far simpler, yet significantly more powerful argument showing that $Ω(n^2)$ Boolean queries of any type are indeed required for finding a stable - or even an approximately stable - marriage. Notably, unlike Segal's lower bound, our lower bound generalizes also to (A) randomized algorithms, (B) allowing arbitrary separate preprocessing of the women's preferences profile and of the men's preferences profile, (C) several variants of the basic problem, such as whether a given pair is married in every/some stable marriage, and (D) determining whether a proposed marriage is stable or far from stable. In order to analyze "approximately stable" marriages, we introduce the notion of "distance to stability" and provide an efficient algorithm for its computation.

cs.GT↗

arXiv2018

Lower Bounds for Approximating Graph Parameters via Communication Complexity

In a celebrated work, Blais, Brody, and Matulef developed a technique for proving property testing lower bounds via reductions from communication complexity. Their work focused on testing properties of functions, and yielded new lower bounds as well as simplified analyses of known lower bounds. Here, we take a further step in generalizing the methodology of Blais et al. to analyze the query complexity of graph parameter estimation problems. In particular, our technique decouples the lower bound arguments from the representation of the graph, allowing it to work with any query type. We illustrate our technique by providing new simpler proofs of previously known tight lower bounds for the query complexity of several graph problems: estimating the number of edges in a graph, sampling edges from an almost-uniform distribution, estimating the number of triangles (and more generally, $r$-cliques) in a graph, and estimating the moments of the degree distribution of a graph. We also prove new lower bounds for estimating the edge connectivity of a graph and estimating the number of instances of any fixed subgraph in a graph. We show that the lower bounds for estimating the number of triangles and edge connectivity also hold in a strictly stronger computational model that allows access to uniformly random edge samples.

cs.CC↗