We revisit the direct sum questions in communication complexity which asks whether the resource needed to solve $n$ communication problems together is (approximately) the sum of resources needed to solve these problems separately. Our work starts with the observation that Dinur and Meir's fortification lemma can be generalized to a general fortification lemma for a sub-additive measure over set. By applying this lemma to the case of cover number, we obtain a dual form of cover number, called "$\delta$-fooling set" which is a generalized fooling set. Any rectangle which contains enough number of elements from a $\delta$-fooling set can not be monochromatic. With this fact, we are able to reprove the classic direct sum theorem of cover number with a simple double counting argument. Formally, let $S \subseteq (A\times B) \times O$ and $T \subseteq (P\times Q) \times Z$ be two communication problems, $ \log \mathsf{Cov}\left(S\times T\right) \geq \log \mathsf{Cov}\left(S\right) + \log\mathsf{Cov}(T) -\log\log|P||Q|-4.$ where $\mathsf{Cov}$ denotes the cover number. One issue of current deterministic direct sum theorems about communication complexity is that they provide no information when $n$ is small, especially when $n=2$. In this work, we prove a new direct sum theorem about protocol size which imply a better direct sum theorem for two functions in terms of protocol size. Formally, let $\mathsf{L}$ denotes complexity of the protocol size of a communication problem, given a communication problem $F:A \times B \rightarrow \{0,1\}$, $ \log\mathsf{L}\left(F\times F\right)\geq \log \mathsf{L}\left(F\right) +\Omega\left(\sqrt{\log\mathsf{L}\left(F\right)}\right)-\log\log|A||B| -4$. All our results are obtained in a similar way using the $\delta$-fooling set to construct a hardcore for the direct sum problem.

Recently, Ivan Mihajlin and Alexander Smal proved a composition theorem of a universal relation and some function via so called xor composition, that is there exists some function $f:\{0,1\}^n \rightarrow \{0,1\}$ such that $\textsf{CC}(\text{U}_n \diamond \text{KW}_f) \geq 1.5n-o(n)$ where $\textsf{CC}$ denotes the communication complexity of the problem. In this paper, we significantly improve their result and present an asymptotically tight and much more general composition theorem of a universal relation and most functions, that is for most functions $f:\{0,1\}^n \rightarrow \{0,1\}$ we have $\textsf{CC}(\text{U}_m \diamond \text{KW}_f) \geq m+ n -O(\sqrt{m})$ when $m=ω(\log^2 n),n =ω(\sqrt{m})$. This is done by a direct proof of composition theorem of a universal relation and a multiplexor in the partially half-duplex model avoiding the xor composition. And the proof works even when the multiplexor only contains a few functions. One crucial ingredient in our proof involves a combinatorial problem of constructing a tree of many leaves and every leaf contains a non-overlapping set of functions. For each leaf, there is a set of inputs such that every function in the leaf takes the same value, that is all functions are restricted. We show how to choose a set of good inputs to effectively restrict these functions to force that the number of functions in each leaf is as small as possible while maintaining the total number of functions in all leaves. This results in a large number of leaves.

One of the major open problems in complexity theory is to demonstrate an explicit function which requires super logarithmic depth, a.k.a, the $\mathbf{P}$ versus $\mathbf{NC^1}$ problem. The current best depth lower bound is $(3-o(1))\cdot \log n$, and it is widely open how to prove a super-$3\log n$ depth lower bound. Recently Mihajlin and Sofronova (CCC'22) show if considering formulas with restriction on top, we can break the $3\log n$ barrier. Formally, they prove there exist two functions $f:\{0,1\}^n \rightarrow \{0,1\},g:\{0,1\}^n \rightarrow \{0,1\}^n$, such that for any constant $0<α<0.4$ and constant $0<ε<α/2$, their XOR composition $f(g(x)\oplus y)$ is not computable by an AND of $2^{(α-ε)n}$ formulas of size at most $2^{(1-α/2-ε)n}$. This implies a modified version of Andreev function is not computable by any circuit of depth $(3.2-ε)\log n$ with the restriction that top $0.4-ε$ layers only consist of AND gates for any small constant $ε>0$. They ask whether the parameter $α$ can be push up to nearly $1$ thus implying a nearly-$3.5\log n$ depth lower bound. In this paper, we provide a stronger answer to their question. We show there exist two functions $f:\{0,1\}^n \rightarrow \{0,1\},g:\{0,1\}^n \rightarrow \{0,1\}^n$, such that for any constant $0<α<2-o(1)$, their XOR composition $f(g(x)\oplus y)$ is not computable by an AND of $2^{αn}$ formulas of size at most $2^{(1-α/2-o(1))n}$. This implies a $(4-o(1))\log n$ depth lower bound with the restriction that top $2-o(1)$ layers only consist of AND gates. We prove it by observing that one crucial component in Mihajlin and Sofronova's work, called the well-mixed set of functions, can be significantly simplified thus improved. Then with this observation and a more careful analysis, we obtain these nearly tight results.

Cooperative path planning in complex coastal mountainous environments near the shore is a knowledge-intensive engineering task crucial for demanding multi-UAV operations such as disaster rescue, environmental monitoring, and coastal emergency response. However, the complex coupling constraints in 3D space make it a challenging high-dimensional optimization problem where traditional metaheuristics often suffer from undirected blind exploration and invalid trial and error. To overcome these limitations, we propose the deep neural network-guided phototropic growth algorithm (DNN-PGA), an engineering informatics framework that deeply couples explicit spatial knowledge representation with traditional heuristic search mechanisms. Specifically, we implement a spatial prior generation method to transform unstructured 3D environmental data into structured knowledge distributions. We utilize deep neural networks to extract spatial heatmap priors from multi-channel voxel representations, serving as a formal representation of potential high-quality path regions. Subsequently, the core search operator of PGA is reconstructed by deploying these explicit spatial priors as structural constraints to strictly guide engineering decision-making. Within the DNN-PGA, a neural network-guided initialization mechanism ensures efficient spatial focusing of the initial population. Secondly, a neural control golden-sine strategy was designed to enable dynamic regulation of local exploitation intensity, and a neural limited lens opposition-based learning mechanism was developed to accurately guide individuals out of local optima using learned spatial knowledge. Extensive experiments confirm that the DNN-PGA achieves an average reduction of 29.49% in mean path cost compared with the best-performing competing algorithm across all scenarios under complex operational conditions. Ultimately, the DNN-PGA successfully establishes a highly robust, knowledge-driven practical solution for demanding 3D multi-UAV cooperative path planning tasks in coastal mountainous environments.

We review Bennequin type inequalities established using various versions of the Khovanov-Rozansky cohomology. Then we give a new proof of a Bennequin type inequality established by the author, and derive new Bennequin type inequalities for knots using Gornik's version of the Khovanov-Rozansky cohomology, which generalize those established by Shumakovitch, Plamenevskaya and Kawamura using the Rasmussen invariant.