More Like this

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

   https://doi.org/10.4230/LIPIcs.ICALP.2025.21  

   Beame, Paul; Whitmeyer, Michael (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Censor-Hillel, Keren; Grandoni, Fabrizio; Ouaknine, Joel; Puppis, Gabriele (Ed.)

   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})}. 

   more » « less
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})}. 

   more » « less
3. 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. 

   more » « less
4. Extremely Deep Proofs

   https://doi.org/10.4230/LIPIcs.ITCS.2022.70  

   Fleming, Noah and (January 2022, Leibniz international proceedings in informatics)

   Braverman, Mark (Ed.)

   We further the study of supercritical tradeoffs in proof and circuit complexity, which is a type of tradeoff between complexity parameters where restricting one complexity parameter forces another to exceed its worst-case upper bound. In particular, we prove a new family of supercritical tradeoffs between depth and size for Resolution, Res(k), and Cutting Planes proofs. For each of these proof systems we construct, for each c ≤ n^{1-ε}, a formula with n^{O(c)} clauses and n variables that has a proof of size n^{O(c)} but in which any proof of size no more than roughly exponential in n^{1-ε}/c must necessarily have depth ≈ n^c. By setting c = o(n^{1-ε}) we therefore obtain exponential lower bounds on proof depth; this far exceeds the trivial worst-case upper bound of n. In doing so we give a simplified proof of a supercritical depth/width tradeoff for tree-like Resolution from [Alexander A. Razborov, 2016]. Finally, we outline several conjectures that would imply similar supercritical tradeoffs between size and depth in circuit complexity via lifting theorems. 

   more » « less
5. Stabbing Planes

   https://doi.org/10.48550/arXiv.1710.03219  

   Beame, Paul; Fleming, Noah; Impagliazzo, Russell; Pankratov, Denis; Pitassi, Toniann; Robere, Robert (May 2022, ArXivorg)

   We develop a new semi-algebraic proof system called Stabbing Planes which formalizes modern branch-and-cut algorithms for integer programming and is in the style of DPLL-based modern SAT solvers. As with DPLL there is only a single rule: the current polytope can be subdivided by branching on an inequality and its “integer negation.” That is, we can (non-deterministically choose) a hyperplane ax ≥ b with integer coefficients, which partitions the polytope into three pieces: the points in the polytope satisfying ax ≥ b, the points satisfying ax ≤ b, and the middle slab b − 1 < ax < b. Since the middle slab contains no integer points it can be safely discarded, and the algorithm proceeds recursively on the other two branches. Each path terminates when the current polytope is empty, which is polynomial-time checkable. Among our results, we show that Stabbing Planes can efficiently simulate the Cutting Planes proof system, and is equivalent to a tree-like variant of the R(CP) system of Krajicek [54]. As well, we show that it possesses short proofs of the canonical family of systems of F_2-linear equations known as the Tseitin formulas. Finally, we prove linear lower bounds on the rank of Stabbing Planes refutations by adapting lower bounds in communication complexity and use these bounds in order to show that Stabbing Planes proofs cannot be balanced. 

   more » « less