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Abstract We study the generalised Chvátal–Sankoff constant$$\gamma _{k,d}$$, which represents the normalised expected length of the longest common subsequence ofdindependent uniformly random strings over an alphabet of sizek. We derive asymptotically tight bounds for$$\gamma _{2,d}$$, establishing that$$\gamma _{2,d} = \tfrac 12 + \Theta ({1}/{\sqrt {d}})$$. We also derive asymptotically near-optimal bounds on$$\gamma _{k,d}$$for$$d\ge \Omega (\log k)$$. 

Abstract: We prove several results on linear codes achieving list-recovery capacity. We show that random linear codes achieve list-recovery capacity with constant output list size (independent of the alphabet size and length). That is, over alphabets of size at least 𝓁^Ω(1/ε), random linear codes of rate R are (1-R-ε, 𝓁, (𝓁/ε)^O(𝓁/ε))-list-recoverable for all R ∈ (0,1) and 𝓁. Together with a result of Levi, Mosheiff, and Shagrithaya, this implies that randomly punctured Reed-Solomon codes also achieve list-recovery capacity. We also prove that our output list size is near-optimal among all linear codes: all (1-R-ε, 𝓁, L)-list-recoverable linear codes must have L ≥ 𝓁^{Ω(R/ε)}. Our simple upper bound combines the Zyablov-Pinsker argument with recent bounds from Kopparty, Ron-Zewi, Saraf, Wootters, and Tamo on the maximum intersection of a "list-recovery ball" and a low-dimensional subspace with large distance. Our lower bound is inspired by a recent lower bound of Chen and Zhang. 

We study the tradeoffs between the locality and parameters of subsystem codes. We prove lower bounds on both the number and lengths of interactions in any D-dimensional embedding of a subsystem code. Specifically, we show that any embedding of a subsystem code with parameters [[n,k,d]] into R^D must have at least M^* interactions of length at least 𝓁^*, where M^* = Ω(max(k,d)), and 𝓁^* = Ω(max(d/(n^((D-1)/D)), ((kd^(1/(D-1))/n))^((D-1)/D))). We also give tradeoffs between the locality and parameters of commuting projector codes in D-dimensions, generalizing a result of Dai and Li [Dai and Li, 2025]. We provide explicit constructions of embedded codes that show our bounds are optimal in both the interaction count and interaction length. 

In this paper, we prove that with high probability, random Reed-Solomon codes approach the half-Singleton bound - the optimal rate versus error tradeoff for linear insdel codes - with linear-sized alphabets. More precisely, we prove that, for any ε > 0 and positive integers n and k, with high probability, random Reed-Solomon codes of length n and dimension k can correct (1-ε)n-2k+1 adversarial insdel errors over alphabets of size n+2^{poly(1/ε)}k. This significantly improves upon the alphabet size demonstrated in the work of Con, Shpilka, and Tamo (IEEE TIT, 2023), who showed the existence of Reed-Solomon codes with exponential alphabet size Õ(binom(n,2k-1)²) precisely achieving the half-Singleton bound. Our methods are inspired by recent works on list-decoding Reed-Solomon codes. Brakensiek-Gopi-Makam (STOC 2023) showed that random Reed-Solomon codes are list-decodable up to capacity with exponential-sized alphabets, and Guo-Zhang (FOCS 2023) and Alrabiah-Guruswami-Li (STOC 2024) improved the alphabet-size to linear. We achieve a similar alphabet-size reduction by similarly establishing strong bounds on the probability that certain random rectangular matrices are full rank. To accomplish this in our insdel context, our proof combines the random matrix techniques from list-decoding with structural properties of Longest Common Subsequences. 

Approximating the graph diameter is a basic task of both theoretical and practical interest. A simple folklore algorithm can output a 2-approximation to the diameter in linear time by running BFS from an arbitrary vertex. It has been open whether a better approximation is possible in near-linear time. A series of articles on fine-grained complexity have led to strong hardness results for diameter in directed graphs, culminating in a recent tradeoff curve independently discovered by [Li, STOC’21] and [Dalirrooyfard and Wein, STOC’21], showing that under the Strong Exponential Time Hypothesis (SETH), for any integerk≥ 2 and δ > 0, a\(2-\frac{1}{k}-\delta\)approximation for diameter in directedm-edge graphs requires\(m^{1+1/(k-1)-o(1)}\)time. In particular, the simple linear time 2-approximation algorithm is optimal for directed graphs. In this article, we prove that the same tradeoff lower bound curve is possible for undirected graphs as well, extending results of [Roditty and Vassilevska W., STOC’13], [Li’20] and [Bonnet, ICALP’21] who proved the first few cases of the curve,k=2,3, and 4, respectively. Our result shows in particular that the simple linear time 2-approximation algorithm is conditionally optimal for undirected graphs. To obtain our result, we extract the core ideas in known reductions and introduce a unification and generalization that could be useful for proving SETH-based hardness for other problems in undirected graphs related to distance computation. 

Computing the diameter of a graph, i.e. the largest distance, is a fundamental problem that is central in fine-grained complexity. In undirected graphs, the Strong Exponential Time Hypothesis (SETH) yields a lower bound on the time vs. approximation trade-off that is quite close to the upper bounds. In directed graphs, however, where only some of the upper bounds apply, much larger gaps remain. Since d(u,v) may not be the same as d(v,u), there are multiple ways to define the problem, the two most natural being the (one-way) diameter (max_(u,v) d(u,v)) and the roundtrip diameter (max_{u,v} d(u,v)+d(v,u)). In this paper we make progress on the outstanding open question for each of them. - We design the first algorithm for diameter in sparse directed graphs to achieve n^{1.5-ε} time with an approximation factor better than 2. The new upper bound trade-off makes the directed case appear more similar to the undirected case. Notably, this is the first algorithm for diameter in sparse graphs that benefits from fast matrix multiplication. - We design new hardness reductions separating roundtrip diameter from directed and undirected diameter. In particular, a 1.5-approximation in subquadratic time would refute the All-Nodes k-Cycle hypothesis, and any (2-ε)-approximation would imply a breakthrough algorithm for approximate 𝓁_∞-Closest-Pair. Notably, these are the first conditional lower bounds for diameter that are not based on SETH. 

Computing the diameter of a graph, i.e. the largest distance, is a fundamental problem that is central in fine-grained complexity. In undirected graphs, the Strong Exponential Time Hypothesis (SETH) yields a lower bound on the time vs. approximation trade-off that is quite close to the upper bounds. In directed graphs, however, where only some of the upper bounds apply, much larger gaps remain. Since d(u,v) may not be the same as d(v,u), there are multiple ways to define the problem, the two most natural being the (one-way) diameter (max_(u,v) d(u,v)) and the roundtrip diameter (max_{u,v} d(u,v)+d(v,u)). In this paper we make progress on the outstanding open question for each of them. - We design the first algorithm for diameter in sparse directed graphs to achieve n^{1.5-ε} time with an approximation factor better than 2. The new upper bound trade-off makes the directed case appear more similar to the undirected case. Notably, this is the first algorithm for diameter in sparse graphs that benefits from fast matrix multiplication. - We design new hardness reductions separating roundtrip diameter from directed and undirected diameter. In particular, a 1.5-approximation in subquadratic time would refute the All-Nodes k-Cycle hypothesis, and any (2-ε)-approximation would imply a breakthrough algorithm for approximate 𝓁_∞-Closest-Pair. Notably, these are the first conditional lower bounds for diameter that are not based on SETH.