Search for: All records

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. 

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. 

The A* algorithm is commonly used to solve NP-hard combinatorial optimization problems. When provided with a completely informed heuristic function, A* can solve such problems in time complexity that is polynomial in the solution cost and branching factor. In light of this fact, we examine a line of recent publications that propose fitting deep neural networks to the completely informed heuristic function. We assert that these works suffer from inherent scalability limitations since --- under the assumption of NP P/poly --- such approaches result in either (a) network sizes that scale super-polynomially in the instance sizes or (b) the accuracy of the fitted deep neural networks scales inversely with the instance sizes. Complementing our theoretical claims, we provide experimental results for three representative NP-hard search problems. The results suggest that fitting deep neural networks to informed heuristic functions requires network sizes that grow quickly with the problem instance size. We conclude by suggesting that the research community should focus on scalable methods for integrating heuristic search with machine learning, as opposed to methods relying on informed heuristic estimation.