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1. A ZPP ^NP[1] Lifting Theorem

   https://doi.org/10.1145/3428673  

   Watson, Thomas (December 2020, ACM Transactions on Computation Theory)

   null (Ed.)

   The complexity class ZPP NP[1] (corresponding to zero-error randomized algorithms with access to one NP oracle query) is known to have a number of curious properties. We further explore this class in the settings of time complexity, query complexity, and communication complexity. • For starters, we provide a new characterization: ZPP NP[1] equals the restriction of BPP NP[1] where the algorithm is only allowed to err when it forgoes the opportunity to make an NP oracle query. • Using the above characterization, we prove a query-to-communication lifting theorem , which translates any ZPP NP[1] decision tree lower bound for a function f into a ZPP NP[1] communication lower bound for a two-party version of f . • As an application, we use the above lifting theorem to prove that the ZPP NP[1] communication lower bound technique introduced by Göös, Pitassi, and Watson (ICALP 2016) is not tight. We also provide a “primal” characterization of this lower bound technique as a complexity class. 

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2. Constructive Separations and Their Consequences

   https://doi.org/10.46298/THEORETICS.24.3  

   Chen, Lijie; Jin, Ce; Santhanam, Rahul; Williams, Ryan (February 2024, TheoretiCS)

   For a complexity class $$C$$ and language $$L$$, a constructive separation of $$L\notin C$$ gives an efficient algorithm (also called a refuter) to findcounterexamples (bad inputs) for every $$C$$-algorithm attempting to decide $$L$$.We study the questions: Which lower bounds can be made constructive? What arethe consequences of constructive separations? We build a case thatconstructiveness serves as a dividing line between many weak lower bounds weknow how to prove, and strong lower bounds against $$P$$, $ZPP$, and $BPP$. Putanother way, constructiveness is the opposite of a complexity barrier: it is aproperty we want lower bounds to have. Our results fall into three broadcategories. 1. Our first set of results shows that, for many well-known lower boundsagainst streaming algorithms, one-tape Turing machines, and query complexity,as well as lower bounds for the Minimum Circuit Size Problem, making theselower bounds constructive would imply breakthrough separations ranging from$$EXP \neq BPP$$ to even $$P \neq NP$$. 2. Our second set of results shows that for most major open problems in lowerbounds against $$P$$, $ZPP$, and $BPP$, including $$P \neq NP$$, $$P \neq PSPACE$$,$$P \neq PP$$, $$ZPP \neq EXP$$, and $$BPP \neq NEXP$$, any proof of the separationwould further imply a constructive separation. Our results generalize earlierresults for $$P \neq NP$$ [Gutfreund, Shaltiel, and Ta-Shma, CCC 2005] and $$BPP\neq NEXP$$ [Dolev, Fandina and Gutfreund, CIAC 2013]. 3. Our third set of results shows that certain complexity separations cannotbe made constructive. We observe that for all super-polynomially growingfunctions $$t$$, there are no constructive separations for detecting high$$t$$-time Kolmogorov complexity (a task which is known to be not in $$P$$) fromany complexity class, unconditionally. 

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3. Sign-rank Can Increase under Intersection

   https://doi.org/10.1145/3470863  

   Bun, Mark; Mande, Nikhil S.; Thaler, Justin (December 2021, ACM Transactions on Computation Theory)

   The communication class UPP cc is a communication analog of the Turing Machine complexity class PP . It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds. For a communication problem f , let f ∧ f denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPP cc ( f ) = O (log n ), and UPP cc ( f ∧ f ) = Θ (log 2 n ). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPP cc , the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity n Omega Ω(log n ) . This matches an upper bound of (Klivans, O’Donnell, and Servedio, FOCS 2002), who used it to give a quasipolynomial time algorithm for PAC learning intersections of polylogarithmically many majorities. Hence, fundamentally new techniques will be needed to learn this class of functions in polynomial time. 

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4. Sign-Rank Can Increase Under Intersection

   https://doi.org/10.4230/LIPIcs.ICALP.2019.30  

   Bun, Mark; Mande, Nikhil; Thaler, Justin (July 2020, Leibniz international proceedings in informatics)

   The communication class UPP^{cc} is a communication analog of the Turing Machine complexity class PP. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds. For a communication problem f, let f wedge f denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPP^{cc}(f)= O(log n), and UPP^{cc}(f wedge f) = Theta(log^2 n). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPP^{cc}, the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity n^{Omega(log n)}. This matches an upper bound of (Klivans, O'Donnell, and Servedio, FOCS 2002), who used it to give a quasipolynomial time algorithm for PAC learning intersections of polylogarithmically many majorities. Hence, fundamentally new techniques will be needed to learn this class of functions in polynomial time. 

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5. Detecting global and local hierarchical structures in cell-cell communication using CrossChat

   https://doi.org/10.1038/s41467-024-54821-x  

   Wang, Xinyi; Almet, Axel_A; Nie, Qing (December 2024, Nature Communications)

   Abstract Cell-cell communication (CCC) occurs across different biological scales, ranging from interactions between large groups of cells to interactions between individual cells, forming a hierarchical structure. Globally, CCC may exist between clusters or only subgroups of a cluster with varying size, while locally, a group of cells as sender or receiver may exhibit distinct signaling properties. Current existing methods infer CCC from single-cell RNA-seq or Spatial Transcriptomics only between predefined cell groups, neglecting the existing hierarchical structure within CCC that are determined by signaling molecules, in particular, ligands and receptors. Here, we develop CrossChat, a novel computational framework designed to infer and analyze the hierarchical cell-cell communication structures using two complementary approaches: a global hierarchical structure using a multi-resolution clustering method, and multiple local hierarchical structures using a tree detection method. This framework provides a comprehensive approach to understand the hierarchical relationships within CCC that govern complex tissue functions. By applying our method to two nonspatial scRNA-seq datasets sampled from COVID-19 patients and mouse embryonic skin, and two spatial transcriptomics datasets generated from Stereo-seq of mouse embryo and 10x Visium of mouse wounded skin, we showcase CrossChat’s functionalities for analyzing both global and local hierarchical structures within cell-cell communication. 

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