arXiv subjects

David Wajc

Publications and source records attributed to David Wajc.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotically Tight Fractional Online Matching Under Edge Arrivals

In this brief note, we close the asymptotic gap between known upper and lower bounds for fractional online matching under edge arrivals. We prove that the optimal competitive ratio for this problem is $1/2+\Theta(1/n)$. The algorithm was suggested and analyzed by OpenAI's ChatGPT Sol based on a single prompt. The presentation was streamlined over a few hours, based on a back and forth discussion with the author, who assumes responsibility for any errors.

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arXiv2025

Dimension-Free Correlated Sampling for the Hypersimplex

Sampling from multiple distributions so as to maximize overlap has been studied by statisticians since the 1950s. Since the 2000s, such correlated sampling from the probability simplex has been a powerful building block in disparate areas of theoretical computer science. We study a generalization of this problem to sampling sets from given vectors in the hypersimplex, i.e., outputting sets of size (at most) some $k$ in $[n]$, while maximizing the sampled sets' overlap. Specifically, the expected difference between two output sets should be at most $\alpha$ times their input vectors' $\ell_1$ distance. A value of $\alpha=O(\log n)$ is known to be achievable, due to Chen et al.~(ICALP'17). We improve this factor to $O(\log k)$, independent of the ambient dimension~$n$. Our algorithm satisfies other desirable properties, including (up to a $\log^* n$ factor) input-sparsity sampling time, logarithmic parallel depth and dynamic update time, as well as preservation of submodular objectives. Anticipating broader use of correlated sampling algorithms for the hypersimplex, we present applications of our algorithm to online paging, offline approximation of metric multi-labeling and swift multi-scenario submodular welfare approximating reallocation.

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arXiv2025

Chasing Submodular Objectives, and Submodular Maximization via Cutting Planes

We introduce the \emph{submodular objectives chasing problem}, which generalizes many natural and previously-studied problems: a sequence of constrained submodular maximization problems is revealed over time, with both the objective and available ground set changing at each step. The goal is to maintain solutions of high approximation and low total \emph{recourse} (number of changes), compared with exact offline algorithms for the same input sequence. For the central cardinality constraint and partition matroid constraints we provide polynomial-time algorithms achieving both optimal $(1-1/e-\epsilon)$-approximation and optimal competitive recourse for \emph{any} constant-approximation. Key to our algorithm's polynomial time, and of possible independent interest, is a new meta-algorithm for $(1-1/e-\epsilon)$-approximately maximizing the multilinear extension under general constraints, which we call {\em approximate-or-separate}. Our algorithm relies on an improvement of the round-and-separate method [Gupta-Levin SODA'20], inspired by an earlier proof by [Vondr\'ak, PhD~Thesis'07]. The algorithm, whose guarantees are similar to the influential {\em continuous greedy} algorithm [Calinescu-Chekuri-P\'al-Vondr\'ak SICOMP'11], can use any cutting plane method and separation oracle for the constraints. This allows us to introduce cutting plane methods, used for exact unconstrained submodular minimization since the '80s [Gr\"otschel/Lov\'asz/Schrijver Combinatorica'81], as a useful method for (optimal approximate) constrained submodular maximization. We show further applications of this approach to static algorithms with curvature-sensitive approximation, and to communication complexity protocols.

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arXiv2025

Online Edge Coloring: Sharp Thresholds

Vizing's theorem guarantees that every graph with maximum degree $\Delta$ admits an edge coloring using $\Delta + 1$ colors. In online settings - where edges arrive one at a time and must be colored immediately - a simple greedy algorithm uses at most $2\Delta - 1$ colors. Over thirty years ago, Bar-Noy, Motwani, and Naor [IPL'92] proved that this guarantee is optimal among deterministic algorithms when $\Delta = O(\log n)$, and among randomized algorithms when $\Delta = O(\sqrt{\log n})$. While deterministic improvements seemed out of reach, they conjectured that for graphs with $\Delta = \omega(\log n)$, randomized algorithms can achieve $(1 + o(1))\Delta$ edge coloring. This conjecture was recently resolved in the affirmative: a $(1 + o(1))\Delta$-coloring is achievable online using randomization for all graphs with $\Delta = \omega(\log n)$ [BSVW STOC'24]. Our results go further, uncovering two findings not predicted by the original conjecture. First, we give a deterministic online algorithm achieving $(1 + o(1))\Delta$-colorings for all $\Delta = \omega(\log n)$. Second, we give a randomized algorithm achieving $(1 + o(1))\Delta$-colorings already when $\Delta = \omega(\sqrt{\log n})$. Our results establish sharp thresholds for when greedy can be surpassed, and near-optimal guarantees can be achieved - matching the impossibility results of [BNMN IPL'92], both deterministically and randomly.

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arXiv2024

Repairing Databases over Metric Spaces with Coincidence Constraints

Datasets often contain values that naturally reside in a metric space: numbers, strings, geographical locations, machine-learned embeddings in a Euclidean space, and so on. We study the computational complexity of repairing inconsistent databases that violate integrity constraints, where the database values belong to an underlying metric space. The goal is to update the database values to retain consistency while minimizing the total distance between the original values and the repaired ones. We consider what we refer to as \emph{coincidence constraints}, which include key constraints, inclusion, foreign keys, and generally any restriction on the relationship between the numbers of cells of different labels (attributes) coinciding in a single value, for a fixed attribute set. We begin by showing that the problem is APX-hard for general metric spaces. We then present an algorithm solving the problem optimally for tree metrics, which generalize both the line metric (i.e., where repaired values are numbers) and the discrete metric (i.e., where we simply count the number of changed values). Combining our algorithm for tree metrics and a classic result on probabilistic tree embeddings, we design a (high probability) logarithmic-ratio approximation for general metrics. We also study the variant of the problem where each individual value's allowed change is limited. In this variant, it is already NP-complete to decide the existence of any legal repair for a general metric, and we present a polynomial-time repairing algorithm for the case of a line metric.

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arXiv2024

Deterministic Online Bipartite Edge Coloring

We study online bipartite edge coloring, with nodes on one side of the graph revealed sequentially. The trivial greedy algorithm is $(2-o(1))$-competitive, which is optimal for graphs of low maximum degree, $\Delta=O(\log n)$ [BNMN IPL'92]. Numerous online edge-coloring algorithms outperforming the greedy algorithm in various settings were designed over the years (e.g., AGKM FOCS'03, BMM SODA'10, CPW FOCS'19, BGW SODA'21, KLSST STOC'22, BSVW STOC'24), all crucially relying on randomization. A commonly-held belief, first stated by [BNMN IPL'92], is that randomization is necessary to outperform greedy. Surprisingly, we refute this belief, by presenting a deterministic algorithm that beats greedy for sufficiently large $\Delta=\Omega(\log n)$, and in particular has competitive ratio $\frac{e}{e-1}+o(1)$ for all $\Delta=\omega(\log n)$. We obtain our result via a new and surprisingly simple randomized algorithm that works against adaptive adversaries (as opposed to oblivious adversaries assumed by prior work), which implies the existence of a similarly-competitive deterministic algorithm [BDBKTW STOC'90].

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arXiv2024

New Philosopher Inequalities for Online Bayesian Matching, via Pivotal Sampling

We study the polynomial-time approximability of the optimal online stochastic bipartite matching algorithm, initiated by Papadimitriou et al. (EC'21). Here, nodes on one side of the graph are given upfront, while at each time $t$, an online node and its edge weights are drawn from a time-dependent distribution. The optimal algorithm is $\textsf{PSPACE}$-hard to approximate within some universal constant. We refer to this optimal algorithm, which requires time to think (compute), as a philosopher, and refer to polynomial-time online approximations of the above as philosopher inequalities. The best known philosopher inequality for online matching yields a $0.652$-approximation. In contrast, the best possible prophet inequality, or approximation of the optimum offline solution, is $0.5$. Our main results are a $0.678$-approximate algorithm and a $0.685$-approximation for a vertex-weighted special case. Notably, both bounds exceed the $0.666$-approximation of the offline optimum obtained by Tang, Wu, and Wu (STOC'22) for the vertex-weighted problem. Building on our algorithms and the recent black-box reduction of Banihashem et al. (SODA'24), we provide polytime (pricing-based) truthful mechanisms which $0.678$-approximate the social welfare of the optimal online allocation for bipartite matching markets. Our online allocation algorithm relies on the classic pivotal sampling algorithm (Srinivasan FOCS'01, Gandhi et al. J.ACM'06), along with careful discarding to obtain negative correlations between offline nodes. Consequently, the analysis boils down to examining the distribution of a weighted sum $X$ of negatively correlated Bernoulli variables, specifically lower bounding its mass below a threshold, $\mathbb{E}[\min(1,X)]$, of possible independent interest. Interestingly, our bound relies on an imaginary invocation of pivotal sampling.

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arXiv2024

The Average-Value Allocation Problem

We initiate the study of centralized algorithms for welfare-maximizing allocation of goods to buyers subject to average-value constraints. We show that this problem is NP-hard to approximate beyond a factor of $\frac{e}{e-1}$, and provide a $\frac{4e}{e-1}$-approximate offline algorithm. For the online setting, we show that no non-trivial approximations are achievable under adversarial arrivals. Under i.i.d. arrivals, we present a polytime online algorithm that provides a constant approximation of the optimal (computationally-unbounded) online algorithm. In contrast, we show that no constant approximation of the ex-post optimum is achievable by an online algorithm.

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arXiv2024

Online Matching: A Brief Survey

Matching, capturing allocation of items to unit-demand buyers, or tasks to workers, or pairs of collaborators, is a central problem in economics. Indeed, the growing prevalence of matching-based markets, many of which online in nature, has motivated much research in economics, operations research, computer science, and their intersection. This brief survey is meant as an introduction to the area of online matching, with an emphasis on recent trends, both technical and conceptual.

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arXiv2024

Online Edge Coloring is (Nearly) as Easy as Offline

The classic theorem of Vizing (Diskret. Analiz.'64) asserts that any graph of maximum degree $\Delta$ can be edge colored (offline) using no more than $\Delta+1$ colors (with $\Delta$ being a trivial lower bound). In the online setting, Bar-Noy, Motwani and Naor (IPL'92) conjectured that a $(1+o(1))\Delta$-edge-coloring can be computed online in $n$-vertex graphs of maximum degree $\Delta=\omega(\log n)$. Numerous algorithms made progress on this question, using a higher number of colors or assuming restricted arrival models, such as random-order edge arrivals or vertex arrivals (e.g., AGKM FOCS'03, BMM SODA'10, CPW FOCS'19, BGW SODA'21, KLSST STOC'22). In this work, we resolve this longstanding conjecture in the affirmative in the most general setting of adversarial edge arrivals. We further generalize this result to obtain online counterparts of the list edge coloring result of Kahn (J. Comb. Theory. A'96) and of the recent "local" edge coloring result of Christiansen (STOC'23).

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arXiv2023

Combinatorial Stationary Prophet Inequalities

Numerous recent papers have studied the tension between thickening and clearing a market in (uncertain, online) long-time horizon Markovian settings. In particular, (Aouad and Sarita{\c{c}} EC'20, Collina et al. WINE'20, Kessel et al. EC'22) studied what the latter referred to as the Stationary Prophet Inequality Problem, due to its similarity to the classic finite-time horizon prophet inequality problem. These works all consider unit-demand buyers. Mirroring the long line of work on the classic prophet inequality problem subject to combinatorial constraints, we initiate the study of the stationary prophet inequality problem subject to combinatorially-constrained buyers. Our results can be summarized succinctly as unearthing an algorithmic connection between contention resolution schemes (CRS) and stationary prophet inequalities. While the classic prophet inequality problem has a tight connection to online CRS (Feldman et al. SODA'16, Lee and Singla ESA'18), we show that for the stationary prophet inequality problem, offline CRS play a similarly central role. We show that, up to small constant factors, the best (ex-ante) competitive ratio achievable for the combinatorial prophet inequality equals the best possible balancedness achievable by offline CRS for the same combinatorial constraints.

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arXiv2023

Simple and Asymptotically Optimal Online Bipartite Edge Coloring

We provide a simple online $\Delta(1+o(1))$-edge-coloring algorithm for bipartite graphs of maximum degree $\Delta=\omega(\log n)$ under adversarial vertex arrivals on one side of the graph. Our algorithm slightly improves the result of (Cohen, Peng and Wajc, FOCS19), which was the first, and currently only, to obtain an asymptotically optimal $\Delta(1+o(1))$ guarantee for an adversarial arrival model. More importantly, our algorithm provides a new, simpler approach for tackling online edge coloring.

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arXiv2023

Near-Optimal Dynamic Rounding of Fractional Matchings in Bipartite Graphs

We study dynamic $(1-\epsilon)$-approximate rounding of fractional matchings -- a key ingredient in numerous breakthroughs in the dynamic graph algorithms literature. Our first contribution is a surprisingly simple deterministic rounding algorithm in bipartite graphs with amortized update time $O(\epsilon^{-1} \log^2 (\epsilon^{-1} \cdot n))$, matching an (unconditional) recourse lower bound of $\Omega(\epsilon^{-1})$ up to logarithmic factors. Moreover, this algorithm's update time improves provided the minimum (non-zero) weight in the fractional matching is lower bounded throughout. Combining this algorithm with novel dynamic \emph{partial rounding} algorithms to increase this minimum weight, we obtain several algorithms that improve this dependence on $n$. For example, we give a high-probability randomized algorithm with $\tilde{O}(\epsilon^{-1}\cdot (\log\log n)^2)$-update time against adaptive adversaries. (We use Soft-Oh notation, $\tilde{O}$, to suppress polylogarithmic factors in the argument, i.e., $\tilde{O}(f)=O(f\cdot \mathrm{poly}(\log f))$.) Using our rounding algorithms, we also round known $(1-\epsilon)$-decremental fractional bipartite matching algorithms with no asymptotic overhead, thus improving on state-of-the-art algorithms for the decremental bipartite matching problem. Further, we provide extensions of our results to general graphs and to maintaining almost-maximal matchings.

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arXiv2023

Online Dependent Rounding Schemes for Bipartite Matchings, with Applications

We introduce the abstract problem of rounding an unknown fractional bipartite $b$-matching $\bf{x}$ revealed online (e.g., output by an online fractional algorithm), exposed node-by-node on~one~side. The objective is to maximize the \emph{rounding ratio} of the output matching $M$, which is the minimum over all fractional $b$-matchings $\bf{x}$, and edges $e$, of the ratio $\Pr[e\in M]/x_e$. In analogy with the highly influential offline dependent rounding schemes of Gandhi et al.~(FOCS'02, JACM'06), we refer to such algorithms as \emph{online dependent rounding schemes} (ODRSes). This problem, with additional restrictions on the possible inputs $\bf{x}$, has played a key role in recent developments in online computing. We provide the first generic $b$-matching ODRSes that impose no restrictions on $\bf{x}$. Specifically, we provide ODRSes with rounding ratios of $0.646$ and $0.652$ for $b$-matchings and simple matchings, respectively. This breaks the natural barrier of $1-1/e$, prevalent for online matching problems, and numerous online problems more broadly. Using our ODRSes, we provide a number of algorithms with similar better-than-$(1-1/e)$ ratios for several problems in online edge coloring, stochastic optimization, and more. Our techniques, which have already found applications in several follow-up works (Patel and Wajc SODA'24, Blikstad et al.~SODA'25, Braverman et al.~SODA'25, and Aouad et al.~2024), include periodic use of \emph{offline} contention resolution schemes (in online algorithm design), grouping nodes, and a new scaling method which we call \emph{group discount and individual markup}.

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arXiv2022

Dynamic Matching with Better-than-2 Approximation in Polylogarithmic Update Time

We present dynamic algorithms with polylogarithmic update time for estimating the size of the maximum matching of a graph undergoing edge insertions and deletions with approximation ratio strictly better than $2$. Specifically, we obtain a $1+\frac{1}{\sqrt{2}}+\epsilon\approx 1.707+\epsilon$ approximation in bipartite graphs and a $1.973+\epsilon$ approximation in general graphs. We thus answer in the affirmative the major open question first posed in the influential work of Onak and Rubinfeld (STOC'10) and repeatedly asked in the dynamic graph algorithms literature. Our randomized algorithms also work against an adaptive adversary and guarantee worst-case polylog update time, both w.h.p. Our algorithms are based on simulating new two-pass streaming matching algorithms in the dynamic setting. Our key new idea is to invoke the recent sublinear-time matching algorithm of Behnezhad (FOCS'21) in a white-box manner to efficiently simulate the second pass of our streaming algorithms, while bypassing the well-known vertex-update barrier.

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arXiv2022

Improved Online Contention Resolution for Matchings and Applications to the Gig Economy

Motivated by applications in the gig economy, we study approximation algorithms for a \emph{sequential pricing problem}. The input is a bipartite graph $G=(I,J,E)$ between individuals $I$ and jobs $J$. The platform has a value of $v_j$ for matching job $j$ to an individual worker. In order to find a matching, the platform can consider the edges $(i j) \in E$ in any order and make a one-time take-it-or-leave-it offer of a price $\pi_{ij} = w$ of its choosing to $i$ for completing $j$. The worker accepts the offer with a known probability $ p_{ijw} $; in this case the job and the worker are irrevocably matched. What is the best way to make offers to maximize revenue and/or social welfare? The optimal algorithm is known to be NP-hard to compute (even if there is only a single job). With this in mind, we design efficient approximations to the optimal policy via a new Random-Order Online Contention Resolution Scheme (RO-OCRS) for matching. Our main result is a 0.456-balanced RO-OCRS in bipartite graphs and a 0.45-balanced RO-OCRS in general graphs. These algorithms improve on the recent bound of $\frac{1}{2}(1-e^{-2})\approx 0.432$ of [BGMS21], and improve on the best known lower bounds for the correlation gap of matching, despite applying to a significantly more restrictive setting. As a consequence of our OCRS results, we obtain a $0.456$-approximate algorithm for the sequential pricing problem. We further extend our results to settings where workers can only be contacted a limited number of times, and show how to achieve improved results for this problem, via improved algorithms for the well-studied stochastic probing problem.

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arXiv2021

The Stationary Prophet Inequality Problem

We study a continuous and infinite time horizon counterpart to the classic prophet inequality, which we term the stationary prophet inequality problem. Here, copies of a good arrive and perish according to Poisson point processes. Buyers arrive similarly and make take-it-or-leave-it offers for unsold items. The objective is to maximize the (infinite) time average revenue of the seller. Our main results are pricing-based policies which (i) achieve a $1/2$-approximation of the optimal offline policy, which is best possible, and (ii) achieve a better than $(1-1/e)$-approximation of the optimal online policy. Result (i) improves upon bounds implied by recent work of Collina et al. (WINE'20), and is the first optimal prophet inequality for a stationary problem. Result (ii) improves upon a $1-1/e$ bound implied by recent work of Aouad and Sarita\c{c} (EC'20), and shows that this prevalent bound in online algorithms is not optimal for this problem.

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arXiv2021

Beating the Folklore Algorithm for Dynamic Matching

The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence $2$-approximate) matching in $O(n)$ worst-case update time in $n$-node graphs. We present the first deterministic algorithm which outperforms the folklore algorithm in terms of {\em both} approximation ratio and worst-case update time. Specifically, we give a $(2-\Omega(1))$-approximate algorithm with $O(m^{3/8})=O(n^{3/4})$ worst-case update time in $n$-node, $m$-edge graphs. For sufficiently small constant $\epsilon>0$, no deterministic $(2+\epsilon)$-approximate algorithm with worst-case update time $O(n^{0.99})$ was known. Our second result is the first deterministic $(2+\epsilon)$-approximate weighted matching algorithm with $O_\epsilon(1)\cdot O(\sqrt[4]{m}) = O_\epsilon(1)\cdot O(\sqrt{n})$ worst-case update time. Our main technical contributions are threefold: first, we characterize the tight cases for \emph{kernels}, which are the well-studied matching sparsifiers underlying much of the $(2+\epsilon)$-approximate dynamic matching literature. This characterization, together with multiple ideas -- old and new -- underlies our result for breaking the approximation barrier of $2$. Our second technical contribution is the first example of a dynamic matching algorithm whose running time is improved due to improving the \emph{recourse} of other dynamic matching algorithms. Finally, we show how to use dynamic bipartite matching algorithms as black-box subroutines for dynamic matching in general graphs without incurring the natural $\frac{3}{2}$ factor in the approximation ratio which such approaches naturally incur.

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