arXiv subjects

Junkai Song

Publications and source records attributed to Junkai Song.

9 recordsLinked to original sources

arXiv2026

Reachability in Directed Acyclic Graphs with Near-Linear Cut Queries

In the cut-query model, an algorithm is given access to a graph $G = (V, E)$ \emph{only} via cut queries. This model has seen significant attention in the undirected graph setting, with works establishing $O(n)$ cut query algorithms for computing the global minimum cut, $\widetilde{O}(n^{3/2})$ cut query algorithms for all pairs minimum cut, and many more. However, despite this vast array of progress in designing sub-quadratic query algorithms for computing properties of undirected graphs, there has been \emph{no} progress in designing such algorithms in directed graphs. Indeed, even for basic problems like whether a vertex $t$ is reachable from a vertex $s$, the cut query complexity is only known to be bounded in the interval $[\Omega(n), O(n^2 / \log n)]$. In this work, we begin a systematic study of these basic problems in directed \emph{acyclic} graphs (DAGs). In this setting, we show that reachability from a single vertex and even topological sorting are both computable in $O(n \log^3 n)$ many cut queries. As a consequence, we also obtain an algorithm which, for any \emph{arbitrary} directed graph $G$, uses only $O(n \log^3 n)$ cut queries and determines whether $G$ contains a cycle.

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arXiv2026

A Near-Optimal Parallel Algorithm for Finding Matroid Bases

We settle the classic question of the parallel complexity of computing a matroid basis, as first posed in the seminal work of Karp, Upfal, and Wigderson (FOCS 1985, JCSS 1988). Our algorithm runs in $O(n^{1/3}\log^{1/3}n)$ rounds, matching the lower bound of KUW up to a $\log^{2/3}(n)$ factor.

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arXiv2026

An $\widetilde{O} (n^{3/7})$ Round Parallel Algorithm for Matroid Bases

We study the parallel (adaptive) complexity of the classic problem of finding a basis in an $n$-element matroid, given access via an \emph{independence oracle}. In this model, the algorithm may submit polynomially many independence queries in each round, and the central question is: how many rounds are necessary and sufficient to find a basis? Karp, Upfal, and Wigderson (FOCS~1985, JCSS~1988; hereafter KUW) initiated this study, showing that $O(\sqrt{n})$ adaptive rounds suffice for any matroid, and that $\widetilde\Omega(n^{1/3})$ rounds are necessary even for partition matroids. This left a substantial gap that persisted for nearly four decades, until Khanna, Putterman, and Song (FOCS~2025; hereafter KPS) achieved $\widetilde O(n^{7/15})$ rounds, the first improvement since~KUW. In this work, we make another conceptual advance beyond KPS, giving a new algorithm that finds a matroid basis in $\widetilde O(n^{3/7})$ rounds. We develop a structural and algorithmic framework that brings a new lens to the analysis of random circuits, moving from reasoning about individual elements to understanding how dependencies span multiple elements simultaneously.

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arXiv2026

A Faster Deterministic Algorithm for Fully Dynamic Maximal Matching

In the fully dynamic maximal matching problem, the goal is to maintain a maximal matching in a graph undergoing an online sequence of edge insertions and deletions. The problem has been studied extensively in the oblivious-adversary setting, where randomized algorithms with polylogarithmic worst-case and constant amortized update time have been known for some time. A major challenge in this area has been designing an algorithm with non-trivial update time against an adaptive adversary. In a recent breakthrough, Bernstein, Bhattacharya, Kiss, and Saranurak (STOC 2025; hereafter, BBKS25) obtained the first algorithms with sublinear update time for this setting: namely, a randomized algorithm with $\tilde{O}(n^{3/4})$ amortized update time, and a deterministic algorithm with $\tilde{O}(n^{8/9})$ amortized update time. Our main result is a deterministic algorithm for fully dynamic maximal matching with amortized update time $n^{1/2+o(1)}$. A powerful tool in dynamic matching is the use of matching sparsifiers: sparse subgraphs that preserve enough information to recover matchings with desired properties. Sparsifiers, such as the EDCS data structure, have been successfully used for approximate maximum matching. For maximal matching, however, this paradigm is not as natural, since maximality must hold with respect to the entire graph. Nevertheless, BBKS25 showed that EDCS can be repurposed as a verification-and-repair mechanism for fully dynamic maximal matching against adaptive adversaries. We introduce a new deterministic framework, referred to as the subgraph system, which, in contrast to EDCS, is purpose-built for verification and maintenance of maximality. It is also designed to allow efficient recursive refinements leading to stronger and stronger parameters, that yield our deterministic algorithm with $n^{1/2+o(1)}$ amortized update time.

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arXiv2026

An $n^{2+o(1)}$ Time Algorithm for Single-Source Negative Weight Shortest Paths

We present a randomized algorithm for the single-source shortest paths (SSSP) problem on directed graphs with arbitrary real-valued edge weights that runs in $n^{2+o(1)}$ time with high probability. This result yields the first almost linear-time algorithm for the problem on dense graphs ($m = \Theta(n^2)$) and improves upon the best previously known bounds for moderately dense graphs ($m = \omega(n^{1.306})$). Our approach builds on the hop-reduction via shortcutting framework introduced by Li, Li, Rao, and Zhang (2025), which iteratively augments the graph with shortcut edges to reduce the negative hop count of shortest paths. The central computational bottleneck in prior work is the cost of explicitly constructing these shortcuts in dense regions. We overcome this by introducing a new compression technique using auxiliary Steiner vertices. Specifically, we construct these vertices to represent large neighborhoods compactly in a structured manner, allowing us to efficiently generate and propagate shortcuts while strictly controlling the growth of vertex degrees and graph size.

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arXiv2025

Optimal Parallel Basis Finding in Graphic and Related Matroids

We study the parallel complexity of finding a basis of a graphic matroid under independence-oracle access. Karp, Upfal, and Wigderson (FOCS 1985, JCSS 1988) initiated the study of this problem and established two algorithms for finding a spanning forest: one running in $O(\log m)$ rounds with $m^{\Theta(\log m)}$ queries, and another, for any $d \in \mathbb{Z}^+$, running in $O(m^{2/d})$ rounds with $\Theta(m^d)$ queries. A key open question they posed was whether one could simultaneously achieve polylogarithmic rounds and polynomially many queries. We give a deterministic algorithm that uses $O(\log m)$ adaptive rounds and $\mathrm{poly}(m)$ non-adaptive queries per round to return a spanning forest on $m$ edges, and complement this result with a matching $\Omega(\log m)$ lower bound for any (even randomized) algorithm with $\mathrm{poly}(m)$ queries per round. Thus, the adaptive round complexity for graphic matroids is characterized exactly, settling this long-standing problem. Beyond graphs, we show that our framework also yields an $O(\log m)$-round, $\mathrm{poly}(m)$-query algorithm for any binary matroid satisfying a smooth circuit counting property, implying, among others, an optimal $O(\log m)$-round parallel algorithms for finding bases of cographic matroids.

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arXiv2025

On the Parallel Complexity of Finding a Matroid Basis

A fundamental question in parallel computation, posed by Karp, Upfal, and Wigderson (FOCS 1985, JCSS 1988), asks: \emph{given only independence-oracle access to a matroid on $n$ elements, how many rounds are required to find a basis using only polynomially many queries?} This question generalizes, among others, the complexity of finding bases of linear spaces, partition matroids, and spanning forests in graphs. In their work, they established an upper bound of $O(\sqrt{n})$ rounds and a lower bound of $\widetilde{\Omega}(n^{1/3})$ rounds for this problem, and these bounds have remained unimproved since then. In this work, we make the first progress in narrowing this gap by designing a parallel algorithm that finds a basis of an arbitrary matroid in $\tilde{O}(n^{7/15})$ rounds (using polynomially many independence queries per round) with high probability, surpassing the long-standing $O(\sqrt{n})$ barrier. Our approach introduces a novel matroid decomposition technique and other structural insights that not only yield this general result but also lead to a much improved new algorithm for the class of \emph{partition matroids} (which underlies the $\widetilde\Omega(n^{1/3})$ lower bound of Karp, Upfal, and Wigderson). Specifically, we develop an $\tilde{O}(n^{1/3})$-round algorithm, thereby settling the round complexity of finding a basis in partition matroids.

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arXiv2024

Colorful Priority $k$-Supplier

In the Priority $k$-Supplier problem the input consists of a metric space $(F \cup C, d)$ over set of facilities $F$ and a set of clients $C$, an integer $k > 0$, and a non-negative radius $r_v$ for each client $v \in C$. The goal is to select $k$ facilities $S \subseteq F$ to minimize $\max_{v \in C} \frac{d(v,S)}{r_v}$ where $d(v,S)$ is the distance of $v$ to the closes facility in $S$. This problem generalizes the well-studied $k$-Center and $k$-Supplier problems, and admits a $3$-approximation [Plesn\'ik, 1987, Bajpai et al., 2022. In this paper we consider two outlier versions. The Priority $k$-Supplier with Outliers problem [Bajpai et al., 2022] allows a specified number of outliers to be uncovered, and the Priority Colorful $k$-Supplier problem is a further generalization where clients are partitioned into $c$ colors and each color class allows a specified number of outliers. These problems are partly motivated by recent interest in fairness in clustering and other optimization problems involving algorithmic decision making. We build upon the work of [Bajpai et al., 2022] and improve their $9$-approximation Priority $k$-Supplier with Outliers problem to a $1+3\sqrt{3}\approx 6.196$-approximation. For the Priority Colorful $k$-Supplier problem, we present the first set of approximation algorithms. For the general case with $c$ colors, we achieve a $17$-pseudo-approximation using $k+2c-1$ centers. For the setting of $c=2$, we obtain a $7$-approximation in random polynomial time, and a $2+\sqrt{5}\approx 4.236$-pseudo-approximation using $k+1$ centers.

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arXiv2023

Online Matching with Stochastic Rewards: Advanced Analyses Using Configuration Linear Programs

Mehta and Panigrahi (2012) proposed Online Matching with Stochastic Rewards, which generalizes the Online Bipartite Matching problem of Karp, Vazirani, and Vazirani (1990) by associating the edges with success probabilities. This new feature captures the pay-per-click model in online advertising. Recently, Huang and Zhang (2020) studied this problem under the online primal dual framework using the Configuration Linear Program (LP), and got the best known competitive ratios of the Stochastic Balance algorithm. Their work suggests that the more expressive Configuration LP is more suitable for this problem than the Matching LP. This paper advances the theory of Configuration LP in two directions. Our technical contribution includes a characterization of the joint matching outcome of an offline vertex and \emph{all its neighbors}. This characterization may be of independent interest, and is aligned with the spirit of Configuration LP. By contrast, previous analyses of Ranking generally focus on only one neighbor. Second, we designed a Stochastic Configuration LP that captures a stochastic benchmark proposed by Goyal and Udwani (2020), who used a Path-based LP. The Stochastic Configuration LP is smaller and simpler than the Path-based LP. Moreover, using the new LP we improved the competitive ratio of Stochastic Balance from $0.596$ to $0.611$ when the success probabilities are infinitesimal, and to $0.613$ when the success probabilities are further equal.

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