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1. Multiparty Communication Complexity of Collision-Finding and Cutting Planes Proofs of Concise Pigeonhole Principles

   Beame, Paul; Whitmeyer, MIchael (April 2025, Electronic colloquium on computational complexity)

   We prove several results concerning the communication complexity of a collision-finding problem, each of which has applications to the complexity of cutting-plane proofs, which make inferences based on integer linear inequalities. In particular, we prove an Ω(n^{1−1/k} log k /2^k) lower bound on the k-party number-in-hand communication complexity of collision-finding. This implies a 2^{n^{1−o(1)}} lower bound on the size of tree-like cutting-planes refutations of the bit pigeonhole principle CNFs, which are compact and natural propositional encodings of the negation of the pigeonhole principle, improving on the best previous lower bound of 2^{Ω(√n)}. Using the method of density-restoring partitions, we also extend that previous lower bound to the full range of pigeonhole parameters. Finally, using a refinement of a bottleneck-counting framework of Haken and Cook and Sokolov for DAG-like communication protocols, we give a 2^{Ω(n^{1/4})} lower bound on the size of fully general (not necessarily tree-like) cutting planes refutations of the same bit pigeonhole principle formulas, improving on the best previous lower bound of 2^{Ω(n^{1/8})}. 

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2. Multiparty Communication Complexity of Collision-Finding

   Beame, Paul; Whitmeyer, Michael (November 2024, arXivorg)

   We prove an Omega(n^{1−1/k} log k /2^k) lower bound on the k-party number-in-hand communication complexity of collision-finding. This implies a 2^{n^{1−o(1)}} lower bound on the size of tree-like cutting-planes proofs of the bit pigeonhole principle, a compact and natural propositional encoding of the pigeonhole principle, improving on the best previous lower bound of 2^{Omega(sqrt{n})}. 

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3. A Min-Entropy Approach to Multi-Party Communication Lower Bounds

   https://doi.org/10.4230/lipics.ccc.2025.33  

   Huang, Mi-Ying Miryam; Mao, Xinyu; Wang, Shuo; Yang, Guangxu; Zhang, Jiapeng (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Srinivasan, Srikanth (Ed.)

   Information complexity is one of the most powerful techniques to prove information-theoretical lower bounds, in which Shannon entropy plays a central role. Though Shannon entropy has some convenient properties, such as the chain rule, it still has inherent limitations. One of the most notable barriers is the square-root loss, which appears in the square-root gap between entropy gaps and statistical distances, e.g., Pinsker’s inequality. To bypass this barrier, we introduce a new method based on min-entropy analysis. Building on this new method, we prove the following results. - An Ω(N^{∑_i α_i - max_i {α_i}}/k) randomized communication lower bound of the k-party set-intersection problem where the i-th party holds a random set of size ≈ N^{1-α_i}. - A tight Ω(n/k) randomized lower bound of the k-party Tree Pointer Jumping problems, improving an Ω(n/k²) lower bound by Chakrabarti, Cormode, and McGregor (STOC 08). - An Ω(n/k+√n) lower bound of the Chained Index problem, improving an Ω(n/k²) lower bound by Cormode, Dark, and Konrad (ICALP 19). Since these problems served as hard problems for numerous applications in streaming lower bounds and cryptography, our new lower bounds directly improve these streaming lower bounds and cryptography lower bounds. On the technical side, min-entropy does not have nice properties such as the chain rule. To address this issue, we enhance the structure-vs-pseudorandomness decomposition used by Göös, Pitassi, and Watson (FOCS 17) and Yang and Zhang (STOC 24); both papers used this decomposition to prove communication lower bounds. In this paper, we give a new breath to this method in the multi-party setting, presenting a new toolkit for proving multi-party communication lower bounds. 

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4. Gadgetless Lifting Beats Round Elimination: Improved Lower Bounds for Pointer Chasing

   https://doi.org/10.4230/lipics.itcs.2025.75  

   Mao, Xinyu; Yang, Guangxu; Zhang, Jiapeng (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   We prove an Ω(n / k + k) communication lower bound on (k - 1)-round distributional complexity of the k-step pointer chasing problem under uniform input distribution, improving the Ω(n/k - klog n) lower bound due to Yehudayoff (Combinatorics Probability and Computing, 2020). Our lower bound almost matches the upper bound of Õ(n/k + k) communication by Nisan and Wigderson (STOC 91). As part of our approach, we put forth gadgetless lifting, a new framework that lifts lower bounds for a family of restricted protocols into lower bounds for general protocols. A key step in gadgetless lifting is choosing the appropriate definition of restricted protocols. In this paper, our definition of restricted protocols is inspired by the structure-vs-pseudorandomness decomposition by Göös, Pitassi, and Watson (FOCS 17) and Yang and Zhang (STOC 24). Previously, round-communication trade-offs were mainly obtained by round elimination and information complexity. Both methods have some barriers in some situations, and we believe gadgetless lifting could potentially address these barriers. 

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5. 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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