arXiv subjects

Joakim Blikstad

Publications and source records attributed to Joakim Blikstad.

At least 19 recordsLinked to original sources

arXiv2026

Deterministic Edge Coloring with few Colors in CONGEST

As the main contribution of this work we present deterministic edge coloring algorithms in the CONGEST model. In particular, we present an algorithm that edge colors any $n$-node graph with maximum degree $Δ$ with with $(1+\varepsilon)Δ+O(\sqrt{\log n})$ colors in $\tilde{O}(\log^{2.5} n+\log^2 Δ\log n)$ rounds. This brings the upper bound polynomially close to the lower bound of $Ω(\log n/\log\log n)$ rounds that also holds in the more powerful LOCAL model [Chang, He, Li, Pettie, Uitto; SODA'18]. As long as $Δ\geq c\sqrt{\log n}$ our algorithm uses fewer than $2Δ-1$ colors and to the best of our knowledge is the first polylogarithmic-round CONGEST algorithm achieving this for any range of $Δ$. As a corollary we also improve the complexity of edge coloring with $2Δ-1$ colors for all ranges of $Δ$ to $\tilde{O}(\log^{2.5} n+\log^2 Δ\log n)$. This improves upon the previous $O(\log^8 n)$-round algorithm from [Fischer, Ghaffari, Kuhn; FOCS'17]. Our approach builds on a refined analysis and extension of the online edge-coloring algorithm of Blikstad, Svensson, Vintan, and Wajc [FOCS'25], and more broadly on new connections between online and distributed graph algorithms. We show that their algorithm exhibits very low locality and, if it can additionally have limited local access to future edges (as distributed algorithms can), it can be derandomized for smaller degrees. Under this additional power, we are able to bypass classical online lower bounds and translate the results to efficient distributed algorithms. This leads to our CONGEST algorithm for $(1+\varepsilon)Δ+O(\sqrt{\log n})$-edge coloring. Since the modified online algorithm can be implemented more efficiently in the LOCAL model, we also obtain (marginally) improved complexity bounds in that model.

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arXiv2026

Incremental Submodular Maximization: Better Than Greedy

We consider submodular maximization under increasing cardinality constraint and ask for a good incremental solution, i.e., an ordering of the ground set such that each prefix of the ordering yields a good solution for its respective cardinality. A classical result in this setting is that the greedy algorithm achieves a competitive ratio, i.e., an approximation guarantee across all cardinalities, of $\mathrm{e}/(\mathrm{e}-1) \approx 1.582$. No better general guarantee was previously known. We present an adaptive scaling algorithm achieving a competitive ratio of $1.373$. We complement our result by a deterministic lower bound of $1.25$ on the best possible competitive ratio for incremental submodular maximization.

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arXiv2026

Revisiting Diameter in Directed Graphs

The reachability diameter ($\mathrm{ReachDiam}$) of a directed graph is the maximum distance over all pairs $u,v$ where $v$ is reachable from $u$. This notion is present in the definition of shortcut sets, and the name was recently coined in that context by Haeupler, Jiang, and Saranurak [SOSA 2026]. While this is a very natural notion of diameter in directed graphs, and especially DAGs, it is so far not computationally explored. Other definitions of diameter in directed graphs are either trivial (infinite) in graphs that are not strongly connected (e.g., the classical definition) or are non-trivial only in highly restrictive graph classes (e.g., Min-Diameter). We initiate the problem of computing the (approximate) reachability diameter from a fine-grained complexity point of view. Under certain fine-grained assumptions, we prove that there is no algorithm in time $\mathcal{O}(n^{ω- \varepsilon}$) that gives any approximation of $\mathrm{ReachDiam}$ in weighted graphs. Similarly, there is no algorithm with better than $2$-approximation for unweighted graphs in this time. To supplement this, we provide algorithmic upper bounds that lead to additive approximation of $\mathrm{ReachDiam}$ for unweighted graphs. Hence, we establish a strong separation between the weighted and unweighted cases, which makes this type of diameter different in nature than other known notions. Considering the hardness in general weighted graphs, we also study special graph classes and get small constant approximations for DAGs with bounded width or graphs with bounded treewidth. Interestingly, our techniques also lead to exact hopsets with hopbound $2$ for bounded treewidth graphs. This and some of our upper bounds for general graphs show technical connections between approximating $\mathrm{ReachDiam}$ and computing shortcut sets and hopsets.

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arXiv2026

Greedy Algorithms for Shortcut Sets and Hopsets

For many popular graph metric sparsifiers, such as spanners, emulators, and preservers, simple and elegant greedy algorithms are known that achieve state-of-the-art or existentially optimal tradeoffs between size and quality. The goal of this paper is to develop and analyze comparable greedy algorithms for nearby objects in graph metric augmentation. We show the following: - A simple greedy algorithm for shortcut sets achieves the state-of-the-art size/hopbound tradeoff recently proved by Kogan and Parter (2022), up to $O(\log n)$ factors in the size. Moreover, with an additional preprocessing step, the greedy algorithm subpolynomially improves on the previous size bounds in some range of parameters. - The same greedy algorithm was already known to be existentially optimal for the size/hopbound tradeoff for hopsets, by an analysis of Berman, Raskhodnikova, and Ruan (2010) introduced for transitive-closure spanners. We provide a completely different analysis showing that the algorithm is also existentially optimal (up to $O(\log n)$ factors) for the matching hopset problem, in which one has a budget of roughly $O(m)$ additional edges (for an $m$-edge input graph).

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arXiv2026

Minimum Star Partitions of Simple Polygons in Polynomial Time

We devise a polynomial-time algorithm for partitioning a simple polygon $P$ into a minimum number of star-shaped polygons. The question of whether such an algorithm exists has been open for more than four decades [Avis and Toussaint, Pattern Recognit., 1981] and it has been repeated frequently, for example in O'Rourke's famous book [Art Gallery Theorems and Algorithms, 1987]. In addition to its strong theoretical motivation, the problem is also motivated by practical domains such as CNC pocket milling, motion planning, and shape parameterization. The only previously known algorithm for a non-trivial special case is for $P$ being both monotone and rectilinear [Liu and Ntafos, Algorithmica, 1991]. For general polygons, an algorithm was only known for the restricted version in which Steiner points are disallowed [Keil, SIAM J. Comput., 1985], meaning that each corner of a piece in the partition must also be a corner of $P$. Interestingly, the solution size for the restricted version may be linear for instances where the unrestricted solution has constant size. The covering variant in which the pieces are star-shaped but allowed to overlap--known as the Art Gallery Problem--was recently shown to be $\exists\mathbb R$-complete and is thus likely not in NP [Abrahamsen, Adamaszek and Miltzow, STOC 2018 & J. ACM 2022]; this is in stark contrast to our result. Arguably the most related work to ours is the polynomial-time algorithm to partition a simple polygon into a minimum number of convex pieces by Chazelle and Dobkin [STOC, 1979 & Comp. Geom., 1985].

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arXiv2025

Combinatorial Maximum Flow via Weighted Push-Relabel on Shortcut Graphs

We give a combinatorial algorithm for computing exact maximum flows in directed graphs with $n$ vertices and edge capacities from $\{1,\dots,U\}$ in $\tilde{O}(n^{2}\log U)$ time, which is near-optimal on dense graphs. This shaves an $n^{o(1)}$ factor from the recent result of [Bernstein-Blikstad-Saranurak-Tu FOCS'24] and, more importantly, greatly simplifies their algorithm. We believe that ours is by a significant margin the simplest of all algorithms that go beyond $\tilde{O}(m\sqrt{n})$ time in general graphs. To highlight this relative simplicity, we provide a full implementation of the algorithm in C++. The only randomized component of our work is the cut-matching game. Via existing tools, we show how to derandomize it for vertex-capacitated max flow and obtain a deterministic $\tilde{O}(n^2)$ time algorithm. This marks the first deterministic near-linear time algorithm for this problem (or even for the special case of bipartite matching) in any density regime.

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arXiv2025

Maximum Flow by Augmenting Paths in $n^{2+o(1)}$ Time

We present a combinatorial algorithm for computing exact maximum flows in directed graphs with $n$ vertices and edge capacities from $\{1,\dots,U\}$ in $n^{2+o(1)}\log U$ time, which is almost optimal in dense graphs. Our algorithm is a novel implementation of the classical augmenting-path framework; we list augmenting paths more efficiently using a new variant of the push-relabel algorithm that uses additional edge weights to guide the algorithm, and we derive the edge weights by constructing a directed expander hierarchy. Even in unit-capacity graphs, this breaks the long-standing $O(m\cdot\min\{\sqrt{m},n^{2/3}\})$ time bound of the previous combinatorial algorithms by Karzanov (1973) and Even and Tarjan (1975) when the graph has $m=ω(n^{4/3})$ edges. Notably, our approach does not rely on continuous optimization nor heavy dynamic graph data structures, both of which are crucial in the recent developments that led to the almost-linear time algorithm by Chen et al. (FOCS 2022). Our running time also matches the $n^{2+o(1)}$ time bound of the independent combinatorial algorithm by Chuzhoy and Khanna (STOC 2024) for computing the maximum bipartite matching, a special case of maximum flow.

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arXiv2025

Online Edge Coloring: Sharp Thresholds

Vizing's theorem guarantees that every graph with maximum degree $Δ$ admits an edge coloring using $Δ+ 1$ colors. In online settings - where edges arrive one at a time and must be colored immediately - a simple greedy algorithm uses at most $2Δ- 1$ colors. Over thirty years ago, Bar-Noy, Motwani, and Naor [IPL'92] proved that this guarantee is optimal among deterministic algorithms when $Δ= O(\log n)$, and among randomized algorithms when $Δ= O(\sqrt{\log n})$. While deterministic improvements seemed out of reach, they conjectured that for graphs with $Δ= ω(\log n)$, randomized algorithms can achieve $(1 + o(1))Δ$ edge coloring. This conjecture was recently resolved in the affirmative: a $(1 + o(1))Δ$-coloring is achievable online using randomization for all graphs with $Δ= ω(\log n)$ [BSVW STOC'24]. Our results go further, uncovering two findings not predicted by the original conjecture. First, we give a deterministic online algorithm achieving $(1 + o(1))Δ$-colorings for all $Δ= ω(\log n)$. Second, we give a randomized algorithm achieving $(1 + o(1))Δ$-colorings already when $Δ= ω(\sqrt{\log n})$. 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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arXiv2025

Global vs. s-t Vertex Connectivity Beyond Sequential: Almost-Perfect Reductions & Near-Optimal Separations

A recent breakthrough by [LNPSY STOC'21] showed that solving s-t vertex connectivity is sufficient (up to polylogarithmic factors) to solve (global) vertex connectivity in the sequential model. This raises a natural question: What is the relationship between s-t and global vertex connectivity in other computational models? In this paper, we demonstrate that the connection between global and s-t variants behaves very differently across computational models: 1.In parallel and distributed models, we obtain almost tight reductions from global to s-t vertex connectivity. In PRAM, this leads to a $n^{ω+o(1)}$-work and $n^{o(1)}$-depth algorithm for vertex connectivity, improving over the 35-year-old $\tilde O(n^{ω+1})$-work $O(\log^2n)$-depth algorithm by [LLW FOCS'86], where $ω$ is the matrix multiplication exponent and $n$ is the number of vertices. In CONGEST, the reduction implies the first sublinear-round (when the diameter is moderately small) vertex connectivity algorithm. This answers an open question in [JM STOC'23]. 2. In contrast, we show that global vertex connectivity is strictly harder than s-t vertex connectivity in the two-party communication setting, requiring $\tilde Θ(n^{1.5})$ bits of communication. The s-t variant was known to be solvable in $\tilde O(n)$ communication [BvdBEMN FOCS'22]. Our results resolve open problems raised by [MN STOC'20, BvdBEMN FOCS'22, AS SOSA'23]. At the heart of our results is a new graph decomposition framework we call \emph{common-neighborhood clustering}, which can be applied in multiple models. Finally, we observe that global vertex connectivity cannot be solved without using s-t vertex connectivity, by proving an s-t to global reduction in dense graphs, in the PRAM and communication models.

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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, $Δ=O(\log n)$ [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 $Δ=Ω(\log n)$, and in particular has competitive ratio $\frac{e}{e-1}+o(1)$ for all $Δ=ω(\log n)$. 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

Efficient Matroid Intersection via a Batch-Update Auction Algorithm

Given two matroids $\mathcal{M}_1$ and $\mathcal{M}_2$ over the same $n$-element ground set, the matroid intersection problem is to find a largest common independent set, whose size we denote by $r$. We present a simple and generic auction algorithm that reduces $(1-\varepsilon)$-approximate matroid intersection to roughly $1/\varepsilon^2$ rounds of the easier problem of finding a maximum-weight basis of a single matroid. Plugging in known primitives for this subproblem, we obtain both simpler and improved algorithms in two models of computation, including: * The first near-linear time/independence-query $(1-\varepsilon)$-approximation algorithm for matroid intersection. Our randomized algorithm uses $\tilde{O}(n/\varepsilon + r/\varepsilon^5)$ independence queries, improving upon the previous $\tilde{O}(n/\varepsilon + r\sqrt{r}/{\varepsilon^3})$ bound of Quanrud (2024). * The first sublinear exact parallel algorithms for weighted matroid intersection, using $O(n^{2/3})$ rounds of rank queries or $O(n^{5/6})$ rounds of independence queries. For the unweighted case, our results improve upon the previous $O(n^{3/4})$-round rank-query and $O(n^{7/8})$-round independence-query algorithms of Blikstad (2022).

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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 $Δ$ can be edge colored (offline) using no more than $Δ+1$ colors (with $Δ$ being a trivial lower bound). In the online setting, Bar-Noy, Motwani and Naor (IPL'92) conjectured that a $(1+o(1))Δ$-edge-coloring can be computed online in $n$-vertex graphs of maximum degree $Δ=ω(\log n)$. 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

Simple and Asymptotically Optimal Online Bipartite Edge Coloring

We provide a simple online $Δ(1+o(1))$-edge-coloring algorithm for bipartite graphs of maximum degree $Δ=ω(\log n)$ 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 $Δ(1+o(1))$ 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

Incremental $(1-ε)$-approximate dynamic matching in $O(poly(1/ε))$ update time

In the dynamic approximate maximum bipartite matching problem we are given bipartite graph $G$ undergoing updates and our goal is to maintain a matching of $G$ which is large compared the maximum matching size $μ(G)$. We define a dynamic matching algorithm to be $α$ (respectively $(α, β)$)-approximate if it maintains matching $M$ such that at all times $|M | \geq μ(G) \cdot α$ (respectively $|M| \geq μ(G) \cdot α- β$). We present the first deterministic $(1-ε)$-approximate dynamic matching algorithm with $O(poly(ε^{-1}))$ amortized update time for graphs undergoing edge insertions. Previous solutions either required super-constant [Gupta FSTTCS'14, Bhattacharya-Kiss-Saranurak SODA'23] or exponential in $1/ε$ [Grandoni-Leonardi-Sankowski-Schwiegelshohn-Solomon SODA'19] update time. Our implementation is arguably simpler than the mentioned algorithms and its description is self contained. Moreover, we show that if we allow for additive $(1, ε\cdot n)$-approximation our algorithm seamlessly extends to also handle vertex deletions, on top of edge insertions. This makes our algorithm one of the few small update time algorithms for $(1-ε)$-approximate dynamic matching allowing for updates both increasing and decreasing the maximum matching size of $G$ in a fully dynamic manner.

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arXiv2023

Fast Algorithms via Dynamic-Oracle Matroids

We initiate the study of matroid problems in a new oracle model called dynamic oracle. Our algorithms in this model lead to new bounds for some classic problems, and a "unified" algorithm whose performance matches previous results developed in various papers. We also show a lower bound that answers some open problems from a few decades ago. Concretely, our results are as follows. * We show an algorithm with $\tilde{O}_k(n+r\sqrt{r})$ dynamic-rank-query and time complexities for the matroid union problem over $k$ matroids. This implies the following consequences. (i) An improvement over the $\tilde{O}_k(n\sqrt{r})$ bound implied by [Chakrabarty-Lee-Sidford-Singla-Wong FOCS'19] for matroid union in the traditional rank-query model. (ii) An $\tilde{O}_k(|E|+|V|\sqrt{|V|})$-time algorithm for the $k$-disjoint spanning tree problem. This improves the $\tilde{O}_k(|V|\sqrt{|E|})$ bounds of Gabow-Westermann [STOC'88] and Gabow [STOC'91]. * We show a matroid intersection algorithm with $\tilde{O}(n\sqrt{r})$ dynamic-rank-query and time complexities. This implies new bounds for some problems and bounds that match the classic ones obtained in various papers, e.g. colorful spanning tree [Gabow-Stallmann ICALP'85], graphic matroid intersection [Gabow-Xu FOCS'89], simple scheduling matroid intersection [Xu-Gabow ISAAC'94], and Hopcroft-Karp combinatorial bipartite matching. More importantly, this is done via a "unified" algorithm in the sense that an improvement over our dynamic-rank-query algorithm would imply improved bounds for all the above problems simultaneously. * We show simple super-linear ($Ω(n\log n)$) query lower bounds for matroid intersection in our dynamic-rank-oracle and the traditional independence-query models; the latter improves the previous $\log_2(3)n - o(n)$ bound by Harvey [SODA'08] and answers an open problem raised by, e.g., Welsh [1976] and CLSSW [FOCS'19].

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arXiv2022

Nearly Optimal Communication and Query Complexity of Bipartite Matching

We settle the complexities of the maximum-cardinality bipartite matching problem (BMM) up to poly-logarithmic factors in five models of computation: the two-party communication, AND query, OR query, XOR query, and quantum edge query models. Our results answer open problems that have been raised repeatedly since at least three decades ago [Hajnal, Maass, and Turan STOC'88; Ivanyos, Klauck, Lee, Santha, and de Wolf FSTTCS'12; Dobzinski, Nisan, and Oren STOC'14; Nisan SODA'21] and tighten the lower bounds shown by Beniamini and Nisan [STOC'21] and Zhang [ICALP'04]. We also settle the communication complexity of the generalizations of BMM, such as maximum-cost bipartite $b$-matching and transshipment; and the query complexity of unique bipartite perfect matching (answering an open question by Beniamini [2022]). Our algorithms and lower bounds follow from simple applications of known techniques such as cutting planes methods and set disjointness.

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arXiv2021

Breaking O(nr) for Matroid Intersection

We present algorithms that break the $\tilde O(nr)$-independence-query bound for the Matroid Intersection problem for the full range of $r$; where $n$ is the size of the ground set and $r\leq n$ is the size of the largest common independent set. The $\tilde O(nr)$ bound was due to the efficient implementations [CLSSW FOCS'19; Nguyen 2019] of the classic algorithm of Cunningham [SICOMP'86]. It was recently broken for large $r$ ($r=ω(\sqrt{n})$), first by the $\tilde O(n^{1.5}/ε^{1.5})$-query $(1-ε)$-approximation algorithm of CLSSW [FOCS'19], and subsequently by the $\tilde O(n^{6/5}r^{3/5})$-query exact algorithm of BvdBMN [STOC'21]. No algorithm, even an approximation one, was known to break the $\tilde O(nr)$ bound for the full range of $r$. We present an $\tilde O(n\sqrt{r}/ε)$-query $(1-ε)$-approximation algorithm and an $\tilde O(nr^{3/4})$-query exact algorithm. Our algorithms improve the $\tilde O(nr)$ bound and also the bounds by CLSSW and BvdBMN for the full range of $r$.

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arXiv2021

Breaking the Quadratic Barrier for Matroid Intersection

The matroid intersection problem is a fundamental problem that has been extensively studied for half a century. In the classic version of this problem, we are given two matroids $\mathcal{M}_1 = (V, \mathcal{I}_1)$ and $\mathcal{M}_2 = (V, \mathcal{I}_2)$ on a comment ground set $V$ of $n$ elements, and then we have to find the largest common independent set $S \in \mathcal{I}_1 \cap \mathcal{I}_2$ by making independence oracle queries of the form "Is $S \in \mathcal{I}_1$?" or "Is $S \in \mathcal{I}_2$?" for $S \subseteq V$. The goal is to minimize the number of queries. Beating the existing $\tilde O(n^2)$ bound, known as the quadratic barrier, is an open problem that captures the limits of techniques from two lines of work. The first one is the classic Cunningham's algorithm [SICOMP 1986], whose $\tilde O(n^2)$-query implementations were shown by CLS+ [FOCS 2019] and Nguyen [2019]. The other one is the general cutting plane method of Lee, Sidford, and Wong [FOCS 2015]. The only progress towards breaking the quadratic barrier requires either approximation algorithms or a more powerful rank oracle query [CLS+ FOCS 2019]. No exact algorithm with $o(n^2)$ independence queries was known. In this work, we break the quadratic barrier with a randomized algorithm guaranteeing $\tilde O(n^{9/5})$ independence queries with high probability, and a deterministic algorithm guaranteeing $\tilde O(n^{11/6})$ independence queries. Our key insight is simple and fast algorithms to solve a graph reachability problem that arose in the standard augmenting path framework [Edmonds 1968]. Combining this with previous exact and approximation algorithms leads to our results.

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