arXiv subjects

Calum MacRury

Publications and source records attributed to Calum MacRury.

At least 19 recordsLinked to original sources

arXiv2026

Approximation Algorithms for Action-Reward Query-Commit Matching

Matching problems under uncertainty arise in applications such as kidney exchange, hiring, and online marketplaces. A decision-maker must sequentially explore potential matches under local exploration constraints, while committing irrevocably to successful matches as they are revealed. The query-commit matching problem captures these challenges by modeling edges that succeed independently with known probabilities and must be accepted upon success, subject to vertex patience (time-out) constraints limiting the number of incident queries. In this work, we introduce the action-reward query-commit matching problem, a strict generalization of query-commit matching in which each query selects an action from a known action space, determining both the success probability and the reward of the queried edge. If an edge is queried using a chosen action and succeeds, it is irrevocably added to the matching, and the corresponding reward is obtained; otherwise, the edge is permanently discarded. We study the design of approximation algorithms for this problem on bipartite graphs. This model captures a broad class of stochastic matching problems, including the sequential pricing problem introduced by Pollner, Roghani, Saberi, and Wajc (EC~2022). On the positive side, Pollner et al. designed a polynomial-time approximation algorithm achieving a ratio of $0.426$ in the one-sided patience setting, which degrades to $0.395$ when both sides have bounded patience. In this work, we design computationally efficient algorithms for the action-reward query-commit in one-sided and two-sided patience settings, achieving approximation ratios of $1-1/e \approx 0.63$ and $\frac{1}{27}\!\left(19-67/e^3\right) \approx 0.58$ respectively. These results improve the state of the art for the sequential pricing problem, surpassing the previous guarantees of $0.426$ and $0.395$.

cs.DS↗

arXiv2025

Forward-backward Contention Resolution Schemes for Fair Rationing

We use contention resolution schemes (CRS) to derive algorithms for the fair rationing of a single resource when agents have stochastic demands. We aim to provide ex-ante guarantees on the level of service provided to each agent, who may measure service in different ways (Type-I, II, or III), calling for CRS under different feasibility constraints (rank-1 matroid or knapsack). We are particularly interested in two-order CRS where the agents are equally likely to arrive in a known forward order or its reverse, which is motivated by online rationing at food banks. In particular, we derive a two-order CRS for rank-1 matroids with guarantee $1/(1+e^{-1/2})\approx 0.622$, which we prove is tight. This improves upon the $1/2$ guarantee that is best-possible under a single order (Alaei, SIAM J. Comput. 2014), while achieving separation with the $1-1/e\approx 0.632$ guarantee that is possible for random-order CRS (Lee and Singla, ESA 2018). Because CRS guarantees imply prophet inequalities, this also beats the two-order prophet inequality with ratio $(\sqrt{5}-1)/2\approx 0.618$ from (Arsenis, SODA 2021), which was tight for single-threshold policies. Rank-1 matroids suffice to provide guarantees under Type-II or III service, but Type-I service requires knapsack. Accordingly, we derive a two-order CRS for knapsack with guarantee $1/3$, improving upon the $1/(3+e^{-2})\approx 0.319$ guarantee that is best-possible under a single order (Jiang et al., SODA 2022). To our knowledge, $1/3$ provides the best-known guarantee for knapsack CRS even in the offline setting. Finally, we provide an upper bound of $1/(2+e^{-1})\approx 0.422$ for two-order knapsack CRS, strictly smaller than the upper bound of $(1-e^{-2})/2\approx0.432$ for random-order knapsack CRS.

cs.DS↗

arXiv2024

Proportionally Fair Matching via Randomized Rounding

Given an edge-colored graph, the goal of the proportional fair matching problem is to find a maximum weight matching while ensuring proportional representation (with respect to the number of edges) of each color. The colors may correspond to demographic groups or other protected traits where we seek to ensure roughly equal representation from each group. It is known that, assuming ETH, it is impossible to approximate the problem with $\ell$ colors in time $2^{o(\ell)} n^{\mathcal{O}(1)}$ (i.e., subexponential in $\ell$) even on \emph{unweighted path graphs}. Further, even determining the existence of a non-empty matching satisfying proportionality is NP-Hard. To overcome this hardness, we relax the stringent proportional fairness constraints to a probabilistic notion. We introduce a notion we call $\delta$-\textsc{ProbablyAlmostFair}, where we ensure proportionality up to a factor of at most $(1 \pm \delta)$ for some small $\delta >0$ with high probability. The violation $\delta$ can be brought arbitrarily close to $0$ for some \emph{good} instances with large values of matching size. We propose and analyze simple and fast algorithms for bipartite graphs that achieve constant-factor approximation guarantees, and return a $\delta$-\textsc{ProbablyAlmostFair} matching.

cs.DS↗

arXiv2024

Online Matching and Contention Resolution for Edge Arrivals with Vanishing Probabilities

We study the performance of sequential contention resolution and matching algorithms on random graphs with vanishing edge probabilities. When the edges of the graph are processed in an adversarially-chosen order, we derive a new OCRS that is $0.382$-selectable, attaining the "independence benchmark" from the literature under the vanishing edge probabilities assumption. Complementary to this positive result, we show that no OCRS can be more than $0.390$-selectable, significantly improving upon the upper bound of $0.428$ from the literature. We also derive negative results that are specialized to bipartite graphs or subfamilies of OCRS's. Meanwhile, when the edges of the graph are processed in a uniformly random order, we show that the simple greedy contention resolution scheme which accepts all active and feasible edges is $1/2$-selectable. This result is tight due to a known upper bound. Finally, when the algorithm can choose the processing order, we show that a slight tweak to the random order -- give each vertex a random priority and process edges in lexicographic order -- results in a strictly better contention resolution scheme that is $1-\ln(2-1/e)\approx0.510$-selectable. Our positive results also apply to online matching on $1$-uniform random graphs with vanishing (non-identical) edge probabilities, extending and unifying some results from the random graphs literature.

cs.DS↗

arXiv2024

Online Contention Resolution Schemes for Network Revenue Management and Combinatorial Auctions

We study the Network Revenue Management (NRM) and Online Combinatorial Auctions (OCA) problems, in which products composed of up to L resources are allocated in an online fashion. We take a randomized rounding approach to these problems, employing the modern tool of Online Contention Resolution Schemes (OCRS) and their random-order analogues (RCRS). However in NRM and OCA, there can be negative correlation in different products being demanded, making the standard definitions of OCRS/RCRS inapplicable. This motivates a new definition of random-element OCRS/RCRS, in which batches of elements arrive, and at most one element in each batch is demanded. We first establish a simple random-element OCRS for achieving the well-known guarantee of 1/(1+L). We then prove that this guarantee is a fundamental barrier---it is tight for all prime-power values of L, using a construction based on finite affine planes. Next, we break this barrier in several settings: the standard OCRS setting, on L-partite hypergraphs, and in the RCRS setting. The first setting importantly establishes a separation between standard and random-element OCRS, showing that the negative correlation between elements being demanded in random-element OCRS can actually worsen guarantees. We also establish a separation between standard and random-element RCRS. Although the general formulations of NRM and OCA require random-element OCRS, we also identify special cases in which standard OCRS or RCRS suffice. This includes establishing new reductions from NRM with Poisson arrivals to standard OCRS, and NRM with time-homogeneous Poisson arrivals to standard RCRS.

cs.GT↗

arXiv2023

Building Hamiltonian Cycles in the Semi-Random Graph Process in Less Than $2n$ Rounds

The semi-random graph process is an adaptive random graph process in which an online algorithm is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the algorithm independently and uniformly at random. The algorithm then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a given graph property, the objective of the algorithm is to force the graph to satisfy this property asymptotically almost surely in as few rounds as possible. We focus on the property of Hamiltonicity. We present an adaptive strategy which creates a Hamiltonian cycle in $\alpha n$ rounds, where $\alpha < 1.81696$ is derived from the solution to a system of differential equations. We also show that achieving Hamiltonicity requires at least $\beta n$ rounds, where $\beta > 1.26575$.

math.CO↗

arXiv2023

Sharp Thresholds in Adaptive Random Graph Processes

The $\mathcal{D}$-process is a single player game in which the player is initially presented the empty graph on $n$ vertices. In each step, a subset of edges $X$ is independently sampled according to a distribution $\mathcal{D}$. The player then selects one edge $e$ from $X$, and adds $e$ to its current graph. For a fixed monotone increasing graph property $\mathcal{P}$, the objective of the player is to force the graph to satisfy $\mathcal{P}$ in as few steps as possible. Through appropriate choices of $\mathcal{D}$, the $\mathcal{D}$-process generalizes well-studied adaptive random graph processes, such as the Achlioptas process and the semi-random graph process We prove a sufficient condition for the existence of a sharp threshold for $\mathcal{P}$ in the $\mathcal{D}$-process. For the semi-random process, we use this condition to prove the existence of a sharp threshold when $\mathcal{P}$ corresponds to being Hamiltonian or to containing a perfect matching. These are the first results for the semi-random graph process which show the existence of a sharp threshold when $\mathcal{P}$ corresponds to containing a sparse spanning graph. Using a separate analytic argument, we show that each sharp threshold is of the form $C_{\mathcal{P}}n$ for some fixed constant $C_{\mathcal{P}}>0$. This answers two of the open problems proposed by Ben-Eliezer et al. (SODA 2020) in the affirmative. Unlike similar results which establish sharp thresholds for certain distributions and properties, we establish the existence of sharp thresholds without explicitly identifying asymptotically optimal strategies.

math.CO↗

arXiv2023

Random-order Contention Resolution via Continuous Induction: Tightness for Bipartite Matching under Vertex Arrivals

We introduce a new approach for designing Random-order Contention Resolution Schemes (RCRS) via exact solution in continuous time. Given a function $c(y):[0,1] \rightarrow [0,1]$, we show how to select each element which arrives at time $y \in [0,1]$ with probability exactly $c(y)$. We provide a rigorous algorithmic framework for achieving this, which discretizes the time interval and also needs to sample its past execution to ensure these exact selection probabilities. We showcase our framework in the context of online contention resolution schemes for matching with random-order vertex arrivals. For bipartite graphs with two-sided arrivals, we design a $(1+e^{-2})/2 \approx 0.567$-selectable RCRS, which we also show to be tight. Next, we show that the presence of short odd-length cycles is the only barrier to attaining a (tight) $(1+e^{-2})/2$-selectable RCRS on general graphs. By generalizing our bipartite RCRS, we design an RCRS for graphs with odd-length girth $g$ which is $(1+e^{-2})/2$-selectable as $g \rightarrow \infty$. This convergence happens very rapidly: for triangle-free graphs (i.e., $g \ge 5$), we attain a $121/240 + 7/16 e^2 \approx 0.563$-selectable RCRS. Finally, for general graphs we improve on the $8/15 \approx 0.533$-selectable RCRS of Fu et al. (ICALP, 2021) and design an RCRS which is at least $0.535$-selectable. Due to the reduction of Ezra et al. (EC, 2020), our bounds yield a $0.535$-competitive (respectively, $(1+e^{-2})/2$-competitive) algorithm for prophet secretary matching on general (respectively, bipartite) graphs under vertex arrivals.

cs.DS↗

arXiv2023

Online Bipartite Matching in the Probe-Commit Model

We consider the classical online bipartite matching problem in the probe-commit model. In this problem, when an online vertex arrives, its edges must be probed to determine if they exist, based on known edge probabilities. A probing algorithm must respect commitment, meaning that if a probed edge exists, it must be used in the matching. Additionally, each online vertex has a patience constraint which limits the number of probes that can be made to an online vertex's adjacent edges. We introduce a new configuration linear program (LP) which we prove is a relaxation of an optimal offline probing algorithm. Using this LP, we establish the following competitive ratios which depend on the model used to generate the instance graph, and the arrival order of its online vertices: - In the worst-case instance model, an optimal $1/e$ ratio when the vertices arrive in uniformly at random (u.a.r.) order. - In the known independently distributed (i.d.) instance model, an optimal $1/2$ ratio when the vertices arrive in adversarial order, and a $1-1/e$ ratio when the vertices arrive in u.a.r. order. The latter two results improve upon the previous best competitive ratio of $0.46$ due to Brubach et al. (Algorithmica 2020), which only held in the more restricted known i.i.d. (independent and identically distributed) instance model. Our $1-1/e$-competitive algorithm matches the best known result for the prophet secretary matching problem due to Ehsani et al. (SODA 2018). Our algorithm is efficient and implies a $1-1/e$ approximation ratio for the special case when the graph is known. This is the offline stochastic matching problem, and we improve upon the $0.42$ approximation ratio for one-sided patience due to Pollner et al. (EC 2022), while also generalizing the $1-1/e$ approximation ratio for unbounded patience due to Gamlath et al. (SODA 2019).

cs.DS↗

arXiv2023

Extending Wormald's Differential Equation Method to One-sided Bounds

In this note, we formulate a "one-sided" version of Wormald's differential equation method. In the standard "two-sided" method, one is given a family of random variables which evolve over time and which satisfy some conditions including a tight estimate of the expected change in each variable over one time step. These estimates for the expected one-step changes suggest that the variables ought to be close to the solution of a certain system of differential equations, and the standard method concludes that this is indeed the case. We give a result for the case where instead of a tight estimate for each variable's expected one-step change, we have only an upper bound. Our proof is very simple, and is flexible enough that if we instead assume tight estimates on the variables, then we recover the conclusion of the standard differential equation method.

math.PR↗

arXiv2022

On (Random-order) Online Contention Resolution Schemes for the Matching Polytope of (Bipartite) Graphs

Online Contention Resolution Schemes (OCRS's) represent a modern tool for selecting a subset of elements, subject to resource constraints, when the elements are presented to the algorithm sequentially. OCRS's have led to some of the best-known competitive ratio guarantees for online resource allocation problems, with the added benefit of treating different online decisions -- accept/reject, probing, pricing -- in a unified manner. This paper analyzes OCRS's for resource constraints defined by matchings in graphs, a fundamental structure in combinatorial optimization. We consider two dimensions of variants: the elements being presented in adversarial or random order; and the graph being bipartite or general. We improve the state of the art for all combinations of variants, both in terms of algorithmic guarantees and impossibility results. Some of our algorithmic guarantees are best-known even compared to Contention Resolution Schemes that can choose the order of arrival or are offline. All in all, our results for OCRS directly improve the best-known competitive ratios for online accept/reject, probing, and pricing problems on graphs in a unified manner.

cs.DS↗

arXiv2022

A Fully Adaptive Strategy for Hamiltonian Cycles in the Semi-Random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a Hamiltonian cycle in as few rounds as possible. In particular, we present an adaptive strategy for the player which achieves it in $αn$ rounds, where $α< 2.01678$ is derived from the solution to some system of differential equations. We also show that the player cannot achieve the desired property in less than $βn$ rounds, where $β> 1.26575$. These results improve the previously best known bounds and, as a result, the gap between the upper and lower bounds is decreased from 1.39162 to 0.75102.

math.CO↗

arXiv2022

Perfect Matchings in the Semi-random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a perfect matching in as few rounds as possible. In particular, we present an adaptive strategy for the player which achieves a perfect matching in $βn$ rounds, where the value of $β< 1.206$ is derived from a solution to some system of differential equations. This improves upon the previously best known upper bound of $(1+2/e+o(1)) \, n < 1.736 \, n$ rounds. We also improve the previously best lower bound of $(\ln 2 + o(1)) \, n > 0.693 \, n$ and show that the player cannot achieve the desired property in less than $αn$ rounds, where the value of $α> 0.932$ is derived from a solution to another system of differential equations. As a result, the gap between the upper and lower bounds is decreased roughly four times.

math.CO↗

arXiv2021

Prophet Matching Meets Probing with Commitment

We consider the online stochastic matching problem for bipartite graphs where edges adjacent to an online node must be probed to determine if they exist, based on known edge probabilities. Our algorithms respect commitment, in that if a probed edge exists, it must be used in the matching. We study this matching problem subject to a downward-closed constraint on each online node's allowable edge probes. Our setting generalizes the commonly studied patience (or time-out) constraint which limits the number of probes that can be made to an online node's adjacent edges. We introduce a new LP that we prove is a relaxation of an optimal offline probing algorithm (the adaptive benchmark) and which overcomes the limitations of previous LP relaxations. (1) A tight $\frac{1}{2}$ ratio when the stochastic graph is generated from a known stochastic type graph where the $t^{th}$ online node is drawn independently from a known distribution $\scr{D}_{π(t)}$ and $π$ is chosen adversarially. We refer to this setting as the known i.d. stochastic matching problem with adversarial arrivals. (2) A $1-1/e$ ratio when the stochastic graph is generated from a known stochastic type graph where the $t^{th}$ online node is drawn independently from a known distribution $\scr{D}_{π(t)}$ and $π$ is a random permutation. We refer to this setting as the known i.d. stochastic matching problem with random order arrivals. Our results improve upon the previous best competitive ratio of $0.46$ in the known i.i.d. setting against the standard adaptive benchmark. Moreover, we are the first to study the prophet secretary matching problem in the context of probing, where we match the best known classical result.

cs.DM↗

arXiv2021

Improved Guarantees for Offline Stochastic Matching via New Ordered Contention Resolution Schemes

Matching is one of the most fundamental and broadly applicable problems across many domains. In these diverse real-world applications, there is often a degree of uncertainty in the input which has led to the study of stochastic matching models. Here, each edge in the graph has a known, independent probability of existing derived from some prediction. Algorithms must probe edges to determine existence and match them irrevocably if they exist. Further, each vertex may have a patience constraint denoting how many of its neighboring edges can be probed. We present new ordered contention resolution schemes yielding improved approximation guarantees for some of the foundational problems studied in this area. For stochastic matching with patience constraints in general graphs, we provide a 0.382-approximate algorithm assuming each vertex has patience at least $2$. Under this assumption, we improve upon the previous best 0.31-approximation of Baveja et al. (2018). When the vertices do not have patience constraints, we describe a 0.432-approximate random order probing algorithm with several corollaries such as an improved guarantee for the Prophet Secretary problem under Edge Arrivals. Finally, for the special case of bipartite graphs with unit patience constraints on one of the partitions, we show a 0.632-approximate algorithm that improves on the recent $1/3$-guarantee of Hikima et al. (2021).

cs.DS↗

arXiv2021

Greedy Approaches to Online Stochastic Matching

Within the context of stochastic probing with commitment, we consider the online stochastic matching problem; that is, the one-sided online bipartite matching problem where edges adjacent to an online node must be probed to determine if they exist based on edge probabilities that become known when an online vertex arrives. If a probed edge exists, it must be used in the matching (if possible). We consider the competitiveness of online algorithms in both the adversarial order model (AOM) and the random order model (ROM). More specifically, we consider a bipartite stochastic graph $G = (U,V,E)$ where $U$ is the set of offline vertices, $V$ is the set of online vertices and $G$ has edge probabilities $(p_{e})_{e \in E}$ and edge weights $(w_{e})_{e \in E}$. Additionally, $G$ has probing constraints $(\scr{C}_{v})_{v \in V}$, where $\scr{C}_v$ indicates which sequences of edges adjacent to an online vertex $v$ can be probed. We assume that $U$ is known in advance, and that $\scr{C}_v$, together with the edge probabilities and weights adjacent to an online vertex are only revealed when the online vertex arrives. This model generalizes the various settings of the classical bipartite matching problem, and so our main contribution is in making progress towards understanding which classical results extend to the stochastic probing model.

cs.DS↗

arXiv2021

The Phase Transition of Discrepancy in Random Hypergraphs

Motivated by the Beck-Fiala conjecture, we study the discrepancy problem in two related models of random hypergraphs on $n$ vertices and $m$ edges. In the first (edge-independent) model, a random hypergraph $H_1$ is constructed by fixing a parameter $p$ and allowing each of the $n$ vertices to join each of the $m$ edges independently with probability $p$. In the parameter range in which $pn \rightarrow \infty$ and $pm \rightarrow \infty$, we show that with high probability (w.h.p.) $H_1$ has discrepancy at least $\Omega(2^{-n/m} \sqrt{pn})$ when $m = O(n)$, and at least $\Omega(\sqrt{pn \log\gamma })$ when $m \gg n$, where $\gamma = \min\{ m/n, pn\}$. In the second (edge-dependent) model, $d$ is fixed and each vertex of $H_2$ independently joins exactly $d$ edges uniformly at random. We obtain analogous results for this model by generalizing the techniques used for the edge-independent model with $p=d/m$. Namely, for $d \rightarrow \infty$ and $dn/m \rightarrow \infty$, we prove that w.h.p. $H_{2}$ has discrepancy at least $\Omega(2^{-n/m} \sqrt{dn/m})$ when $m = O(n)$, and at least $\Omega(\sqrt{(dn/m) \log\gamma})$ when $m \gg n$, where $\gamma =\min\{m/n, dn/m\}$. Furthermore, we obtain nearly matching asymptotic upper bounds on the discrepancy in both models (when $p=d/m$), in the dense regime of $m \gg n$. Specifically, we apply the partial colouring lemma of Lovett and Meka to show that w.h.p. $H_{1}$ and $H_{2}$ each have discrepancy $O( \sqrt{dn/m} \log(m/n))$, provided $d \rightarrow \infty$, $d n/m \rightarrow \infty$ and $m \gg n$. This result is algorithmic, and together with the work of Bansal and Meka characterizes how the discrepancy of each random hypergraph model transitions from $\Theta(\sqrt{d})$ to $o(\sqrt{d})$ as $m$ varies from $m=\Theta(n)$ to $m \gg n$.

math.CO↗

arXiv2021

Bipartite Stochastic Matching: Online, Random Order, and I.I.D. Models

Within the context of stochastic probing with commitment, we consider the online stochastic matching problem; that is, the one sided online bipartite matching problem where edges adjacent to an online node must be probed to determine if they exist, based on known edge probabilities. If a probed edge exists, it must be used in the matching (if possible). We study this problem in the generality of a patience (or budget) constraint which limits the number of probes that can be made to edges adjacent to an online node. The patience constraint results in modelling and computational efficiency issues that are not encountered in the special cases of unit patience and full (i.e., unlimited) patience. The stochastic matching problem leads to a variety of settings. Our main contribution is to provide a new LP relaxation and a unified approach for establishing new and improved competitive bounds in three different input model settings (namely, adversarial, random order, and known i.i.d.). In all these settings, the algorithm does not have any control on the ordering of the online nodes. We establish competitive bounds in these settings, all of which generalize the standard non-stochastic setting when edges do not need to be probed (i.e., exist with certainty). All of our competitive ratios hold for arbitrary edge probabilities and patience constraints: (1) A $1-1/e$ ratio when the stochastic graph is known, offline vertices are weighted and online arrivals are adversarial. (2) A $1-1/e$ ratio when the stochastic graph is known, edges are weighted, and online arrivals are given in random order (i.e., in ROM, the random order model). (3) A $1-1/e$ ratio when online arrivals are drawn i.i.d. from a known stochastic type graph and edges are weighted. (4) A (tight) $1/e$ ratio when the stochastic graph is unknown, edges are weighted and online arrivals are given in random order.

cs.DS↗