arXiv subjects
Publications and source records attributed to Masataka Yoneda.
We show that every planar graph with $n \geq 3$ vertices admits a 4-coloring in which each color is used on fewer than $n/2$ vertices. This bound is the best possible. Moreover, such a coloring can be found in $O(n \log n)$ time. We also extend these results to five or more colors and to graphs on general surfaces.
We present SoftMatcha 2, an ultra-fast and flexible search algorithm that enables search over trillion-scale natural language corpora in under 0.3 seconds while allowing semantic variations in the form of substitution, insertion, and deletion. Our approach employs string matching based on suffix arrays that scales well with corpus size, and represents words as vectors, which underpin its semantic flexibility. To mitigate the combinatorial explosion induced by the semantic relaxation of queries, our method is built on two key algorithmic ideas: dynamic corpus-aware pruning and fast exact lookup enabled by a disk-aware design. We theoretically analyze the efficiency of the proposed method, indicating that it can mitigate exponential growth in the search space. Empirically, on FineWeb-Edu (Lozhkov et al., 2024) (1.4T tokens), it attains substantially lower search latency than existing methods: infini-gram (Liu et al., 2024), infini-gram mini (Xu et al., 2025), and SoftMatcha (Deguchi et al., 2025). As a practical application, our method uncovers benchmark contamination in training corpora that existing approaches miss, and it also benefits information retrieval and paraphrase detection. We also provide an online demo of fast, soft search across corpora in seven languages.
We study the problem of online coloring for graphs with large odd girth. The best previously known algorithm uses $O(n^{1/2})$ colors, which was discovered by Kierstead in 1998. This algorithm works when the odd girth is 7 or more. In this paper, we provide the following: for every $\varepsilon > 0$, there exists a constant $g' \in \{3, 5, 7, \dots\}$ such that graphs with odd girth at least $g'$ can be deterministically colored online using $O(n^{\varepsilon})$ colors.
We study the problem of online graph coloring for $k$-colorable graphs. The best previously known deterministic algorithm uses $\widetilde{O}(n^{1-\frac{1}{k!}})$ colors for general $k$ and $\widetilde{O}(n^{5/6})$ colors for $k = 4$, both given by Kierstead in 1998. In this paper, we finally break this barrier, achieving the first major improvement in nearly three decades. Our results are summarized as follows: (1) $k \geq 5$ case. We provide a deterministic online algorithm to color $k$-colorable graphs with $\widetilde{O}(n^{1-\frac{1}{k(k-1)/2}})$ colors, significantly improving the current upper bound of $\widetilde{O}(n^{1-\frac{1}{k!}})$ colors. Our algorithm also matches the best-known bound for $k = 4$ ($\widetilde{O}(n^{5/6})$ colors). (2) $k = 4$ case. We provide a deterministic online algorithm to color $4$-colorable graphs with $\widetilde{O}(n^{14/17})$ colors, improving the current upper bound of $\widetilde{O}(n^{5/6})$ colors. (3) $k = 2$ case. We show that for randomized algorithms, the upper bound is $1.034 \log_2 n + O(1)$ colors and the lower bound is $\frac{91}{96} \log_2 n - O(1)$ colors. This means that we close the gap to a factor of $1.09$. With our algorithm for the $k \geq 5$ case, we also obtain a deterministic online algorithm for graph coloring that achieves a competitive ratio of $O(\frac{n}{\log \log n})$, which improves the best-known result of $O(\frac{n \log \log \log n}{\log \log n})$ by Kierstead. For the bipartite graph case ($k = 2$), the limit of online deterministic algorithms is known: any deterministic algorithm requires $2 \log_2 n - O(1)$ colors. Our results imply that randomized algorithms can perform slightly better but still have a limit.
We study the fair division of indivisible goods with conflicts between pairs of goods, represented by a graph $G = (V, E)$. We consider ``soft'' conflicts: assigning two adjacent goods to the same agent is allowed, but we seek allocations that are envy-free up to one good (EF1) while keeping the number of such conflict violations small. We propose a linear-time algorithm for general additive valuations that finds an EF1 allocation with at most $|E|/n + O(|E|^{1-1/(2n-2)})$ violations, for any constant number of agents $n$. The leading term $|E|/n$ matches the worst-case bound on the number of violations. We use a novel approach that combines an algorithm for fair division with cardinality constraints from Biswas \& Barman (2018) and a geometric ``closest points'' argument. For identical additive valuations, we also propose a simple round-robin-based algorithm that finds an EF1 allocation with at most $|E|/n$ violations.