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1. Efficient Interactive Proofs for Non-Deterministic Bounded Space

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.47  

   Cook, Joshua; Rothblum, Ron D. (September 2023, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023))

   Megow, Nicole; Smith, Adam (Ed.)

   The celebrated IP = PSPACE Theorem gives an efficient interactive proof for any bounded-space algorithm. In this work we study interactive proofs for non-deterministic bounded space computations. While Savitch’s Theorem shows that nondeterministic bounded-space algorithms can be simulated by deterministic bounded-space algorithms, this simulation has a quadratic overhead. We give interactive protocols for nondeterministic algorithms directly to get faster verifiers. More specifically, for any non-deterministic space S algorithm, we construct an interactive proof in which the verifier runs in time Õ(n+S²). This improves on the best previous bound of Õ(n+S³) and matches the result for deterministic space bounded algorithms, up to polylog(S) factors. We further generalize to alternating bounded space algorithms. For any language L decided by a time T, space S algorithm that uses d alternations, we construct an interactive proof in which the verifier runs in time Õ(n + S log(T) + S d) and the prover runs in time 2^O(S). For d = O(log(T)), this matches the best known interactive proofs for deterministic algorithms, up to polylog(S) factors, and improves on the previous best verifier time for nondeterministic algorithms by a factor of log(T). We also improve the best prior verifier time for unbounded alternations by a factor of S. Using known connections of bounded alternation algorithms to bounded depth circuits, we also obtain faster verifiers for bounded depth circuits with unbounded fan-in. 

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2. On the fine-grained complexity of small-size geometric set cover and discrete k-center for small k

   https://doi.org/10.4230/LIPIcs.ICALP.2023.34  

   Chan, Timothy M.; He, Qizheng; Yu, Yuancheng (July 2023, Proc. 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023))

   Etessami, Kousha; Feige, Uriel; Puppis, Gabriele (Ed.)

   We study the time complexity of the discrete k-center problem and related (exact) geometric set cover problems when k or the size of the cover is small. We obtain a plethora of new results: - We give the first subquadratic algorithm for rectilinear discrete 3-center in 2D, running in Õ(n^{3/2}) time. - We prove a lower bound of Ω(n^{4/3-δ}) for rectilinear discrete 3-center in 4D, for any constant δ > 0, under a standard hypothesis about triangle detection in sparse graphs. - Given n points and n weighted axis-aligned unit squares in 2D, we give the first subquadratic algorithm for finding a minimum-weight cover of the points by 3 unit squares, running in Õ(n^{8/5}) time. We also prove a lower bound of Ω(n^{3/2-δ}) for the same problem in 2D, under the well-known APSP Hypothesis. For arbitrary axis-aligned rectangles in 2D, our upper bound is Õ(n^{7/4}). - We prove a lower bound of Ω(n^{2-δ}) for Euclidean discrete 2-center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of Õ(n^ω), if the matrix multiplication exponent ω is equal to 2. - We similarly prove an Ω(n^{k-δ}) lower bound for Euclidean discrete k-center in O(k) dimensions for any constant k ≥ 3, under the Hyperclique Hypothesis. This lower bound again nearly matches known upper bounds if ω = 2. - We also prove an Ω(n^{2-δ}) lower bound for the problem of finding 2 boxes to cover the largest number of points, given n points and n boxes in 12D . This matches the straightforward near-quadratic upper bound. 

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3. Witness-Sensitive Detection of Induced Diamonds

   https://doi.org/10.4230/lipics.icalp.2026.52  

   Censor-Hillel, Keren; Even, Tomer; Vassilevska_Williams, Virginia; Wallheimer, Nathan (July 2026, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bhattacharya, Sayan; Nanongkai, Danupon; Benedikt, Michael; Puppis, Gabriele (Ed.)

   Abstract: We provide a fast witness-sensitive algorithm for detecting an induced diamond (a K₄ minus an edge) in an n-vertex graph containing t induced diamonds. Our algorithm runs in time Õ(min(n^2.425/t^0.25 + n², n^ω)) with high probability, improving upon the prior state of the art (witness-oblivious) algorithm that runs in time O(n^ω log n) [Vassilevska Williams, Wang, Williams, Yu, SODA 2014] whenever t ≥ n^{(3-ω)/3}, where ω < 2.372 is the matrix multiplication exponent. Our key insight is that the size of a clique containing one of the triangles of an induced diamond plays a crucial role in detecting such a diamond. We say that a diamond is r-heavy if this size is at least r, and we provide a fast detection algorithm for r-heavy diamonds in Õ(r⋅(n/r)^ω + (n/r)³+ nr) time. When there are no r-heavy diamonds, we provide a different fast detection algorithm in Õ(MM(n,n,n√{r/t})) time, where MM(a,b,c) denotes the time to multiply an a × b matrix by a b × c matrix, which is conditionally optimal for r = Õ(1). Our main technical contribution is in designing a refinement framework for sampling vectors, which allows sampling vertices for detecting diamonds in a manner that is adaptive to the structure of graphs with no r-heavy diamonds. We establish that our technique is of a wide applicability, by showing how it also allows for faster witness-sensitive algorithms for 4-SUM and for a special case of 4-cycles. 

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4. Regularization of Low Error PCPs and an Application to MCSP

   https://doi.org/10.4230/LIPIcs.ISAAC.2023.39  

   Hirahara, Shuichi; Moshkovitz, Dana (November 2023, 34th International Symposium on Algorithms and Computation (ISAAC 2023))

   Iwata, Satoru; Kakimura, Naonori (Ed.)

   In a regular PCP the verifier queries each proof symbol in the same number of tests. This number is called the degree of the proof, and it is at least 1/(sq) where s is the soundness error and q is the number of queries. It is incredibly useful to have regularity and reduced degree in PCP. There is an expander-based transformation by Papadimitriou and Yannakakis that transforms any PCP with a constant number of queries and constant soundness error to a regular PCP with constant degree. There are also transformations for low error projection and unique PCPs. Other PCPs are constructed especially to be regular. In this work we show how to regularize and reduce degree of PCPs with a possibly large number of queries and low soundness error. As an application, we prove NP-hardness of an unweighted variant of the collective minimum monotone satisfying assignment problem, which was introduced by Hirahara (FOCS'22) to prove NP-hardness of MCSP^* (the partial function variant of the Minimum Circuit Size Problem) under randomized reductions. We present a simplified proof and sufficient conditions under which MCSP^* is NP-hard under the standard notion of reduction: MCSP^* is NP-hard under deterministic polynomial-time many-one reductions if there exists a function in E that satisfies certain direct sum properties. 

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5. Hardness of approximation in p via short cycle removal: cycle detection, distance oracles, and beyond

   https://doi.org/10.1145/3519935.3520066  

   Abboud, Amir; Bringmann, Karl; Khoury, Seri; Zamir, Or (June 2022, STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing)

   We present a new technique for efficiently removing almost all short cycles in a graph without unintentionally removing its triangles. Consequently, triangle finding problems do not become easy even in almost k-cycle free graphs, for any constant k≥ 4. Triangle finding is at the base of many conditional lower bounds in P, mainly for distance computation problems, and the existence of many 4- or 5-cycles in a worst-case instance had been the obstacle towards resolving major open questions. Hardness of approximation: Are there distance oracles with m1+o(1) preprocessing time and mo(1) query time that achieve a constant approximation? Existing algorithms with such desirable time bounds only achieve super-constant approximation factors, while only 3− factors were conditionally ruled out (Pătraşcu, Roditty, and Thorup; FOCS 2012). We prove that no O(1) approximations are possible, assuming the 3-SUM or APSP conjectures. In particular, we prove that k-approximations require Ω(m1+1/ck) time, which is tight up to the constant c. The lower bound holds even for the offline version where we are given the queries in advance, and extends to other problems such as dynamic shortest paths. The 4-Cycle problem: An infamous open question in fine-grained complexity is to establish any surprising consequences from a subquadratic or even linear-time algorithm for detecting a 4-cycle in a graph. This is arguably one of the simplest problems without a near-linear time algorithm nor a conditional lower bound. We prove that Ω(m1.1194) time is needed for k-cycle detection for all k≥ 4, unless we can detect a triangle in √n-degree graphs in O(n2−δ) time; a breakthrough that is not known to follow even from optimal matrix multiplication algorithms. 

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