Improved Hardness of Approximating k-Clique under ETH

In this paper, we prove that assuming the exponential time hypothesis (ETH), there is no $f(k)\cdot n^{k^{o(1/\log\log k)}}$-time algorithm that can decide whether an $n$-vertex graph contains a clique of size $k$ or contains no clique of size $k/2$, and no FPT algorithm can decide whether an input graph has a clique of size $k$ or no clique of size $k/f(k)$, where $f(k)$ is some function in $k^{1-o(1)}$. Our results significantly improve the previous works [Lin21, LRSW22]. The crux of our proof is a framework to construct gap-producing reductions for the $k$-Clique problem. More precisely, we show that given an error-correcting code $C:\Sigma_1^k\to\Sigma_2^{k'}$ that is locally testable and smooth locally decodable in the parallel setting, one can construct a reduction which on input a graph $G$ outputs a graph $G'$ in $(k')^{O(1)}\cdot n^{O(\log|\Sigma_2|/\log|\Sigma_1|)}$ time such that: $\bullet$ If $G$ has a clique of size $k$, then $G'$ has a clique of size $K$, where $K = (k')^{O(1)}$. $\bullet$ If $G$ has no clique of size $k$, then $G'$ has no clique of size $(1-\varepsilon)\cdot K$ for some constant $\varepsilon\in(0,1)$. We then construct such a code with $k'=k^{\Theta(\log\log k)}$ and $|\Sigma_2|=|\Sigma_1|^{k^{0.54}}$, establishing the hardness results above. Our code generalizes the derivative code [WY07] into the case with a super constant order of derivatives.Comment: 48 page

Automated Tail Bound Analysis for Probabilistic Recurrence Relations

Probabilistic recurrence relations (PRRs) are a standard formalism for describing the runtime of a randomized algorithm. Given a PRR and a time limit $\kappa$, we consider the classical concept of tail probability $\Pr[T \ge \kappa]$, i.e., the probability that the randomized runtime $T$ of the PRR exceeds the time limit $\kappa$. Our focus is the formal analysis of tail bounds that aims at finding a tight asymptotic upper bound $u \geq \Pr[T\ge\kappa]$ in the time limit $\kappa$. To address this problem, the classical and most well-known approach is the cookbook method by Karp (JACM 1994), while other approaches are mostly limited to deriving tail bounds of specific PRRs via involved custom analysis. In this work, we propose a novel approach for deriving exponentially-decreasing tail bounds (a common type of tail bounds) for PRRs whose preprocessing time and random passed sizes observe discrete or (piecewise) uniform distribution and whose recursive call is either a single procedure call or a divide-and-conquer. We first establish a theoretical approach via Markov's inequality, and then instantiate the theoretical approach with a template-based algorithmic approach via a refined treatment of exponentiation. Experimental evaluation shows that our algorithmic approach is capable of deriving tail bounds that are (i) asymptotically tighter than Karp's method, (ii) match the best-known manually-derived asymptotic tail bound for QuickSelect, and (iii) is only slightly worse (with a $\log\log n$ factor) than the manually-proven optimal asymptotic tail bound for QuickSort. Moreover, our algorithmic approach handles all examples (including realistic PRRs such as QuickSort, QuickSelect, DiameterComputation, etc.) in less than 0.1 seconds, showing that our approach is efficient in practice.Comment: 46 pages, 15 figure

Parameterized Inapproximability Hypothesis under ETH

The Parameterized Inapproximability Hypothesis (PIH) asserts that no fixed parameter tractable (FPT) algorithm can distinguish a satisfiable CSP instance, parameterized by the number of variables, from one where every assignment fails to satisfy an $\varepsilon$ fraction of constraints for some absolute constant $\varepsilon > 0$. PIH plays the role of the PCP theorem in parameterized complexity. However, PIH has only been established under Gap-ETH, a very strong assumption with an inherent gap. In this work, we prove PIH under the Exponential Time Hypothesis (ETH). This is the first proof of PIH from a gap-free assumption. Our proof is self-contained and elementary. We identify an ETH-hard CSP whose variables take vector values, and constraints are either linear or of a special parallel structure. Both kinds of constraints can be checked with constant soundness via a "parallel PCP of proximity" based on the Walsh-Hadamard code

TreeGen: A Tree-Based Transformer Architecture for Code Generation

A code generation system generates programming language code based on an input natural language description. State-of-the-art approaches rely on neural networks for code generation. However, these code generators suffer from two problems. One is the long dependency problem, where a code element often depends on another far-away code element. A variable reference, for example, depends on its definition, which may appear quite a few lines before. The other problem is structure modeling, as programs contain rich structural information. In this paper, we propose a novel tree-based neural architecture, TreeGen, for code generation. TreeGen uses the attention mechanism of Transformers to alleviate the long-dependency problem, and introduces a novel AST reader (encoder) to incorporate grammar rules and AST structures into the network. We evaluated TreeGen on a Python benchmark, HearthStone, and two semantic parsing benchmarks, ATIS and GEO. TreeGen outperformed the previous state-of-the-art approach by 4.5 percentage points on HearthStone, and achieved the best accuracy among neural network-based approaches on ATIS (89.1%) and GEO (89.6%). We also conducted an ablation test to better understand each component of our model

Microwave-based preparation and characterization of Fe-cored carbon nanocapsules with novel stability and super electromagnetic wave absorption performance

Microwave-metal discharge was proposed as a facile methodology to prepare unique Fe-cored carbon nanocapsules (Fe@CNCs) with high purity, novel stability and extraordinary electromagnetic wave (EMW) absorption performance. The effect of microwave power, irradiation time and cyclohexane/ferrocene ratio on the production of Fe@CNCs was examined and the properties of the nanocapsules, such as their Fe content, phase, yield, degree of graphitization and associated microstructures were investigated in detail. It was found that the prepared Fe@CNCs, which can easily be separated from the reaction system, displayed exceedingly high electromagnetic wave (EMW) absorption performance over the 2–18 GHz range. At the minimal reflection loss (RL) values over −10 dB, the EMW absorption bandwidth can reach up to 13.8 GHz with an absorber thickness of 1.5–5 mm. In addition, novel thermo-oxidative stability and super anti-corrosion property were also obtained for the Fe@CNCs as no signs of any corrosion or oxidative degradation loss were observed from the accelerated degradation tests in air and acid at temperatures up to 420 °C. The exceedingly high EMW absorption performance coupled with the superior anti-degradation and anti-corrosion properties of the prepared nanocomposite microcapsules highlights the novel capability of microwave-metal discharge in synthesizing advanced metal-cored nanocarbon microcapsules with promising application potentials in diverse fields, such as but not limited to microwave absorption, EM shielding and advanced separations etc

Signature splitting inversion and backbending in 80Rb

High spin states of 80Rb are studied via the fusion-evaporation reactions 65Cu+19F, 66Zn+18O and 68Zn+16O with the beam energies of 75 MeV, 76 MeV and 80 MeV, respectively. Twenty-three new states with twenty-eight new \gamma transitions were added to the previously proposed level scheme, where the second negative-parity band is significantly pushed up to spins of 22^{-} and 15^{-} and two new sidebands are built on the known first negative-parity band. Two successive band crossings with frequencies 0.51 MeV and 0.61 MeV in the \alpha=0 branch as well as another one in the \alpha=1 branch of the second negative-parity band are observed for the first time. Signature inversions occur in the positive- and first negative-parity bands at the spins of 11\hbar and 15\hbar, respectively. The signature splitting is seen obviously in the second negative-parity band, but the signature inversion is not observed. It is also found that the structure of the two negative-parity bands is similar to that of its isotone ^{82}Y. Signature inversion in the positive-parity yrast band with configuration \pi g_{9/2} \otimes \nu g_{9/2} in this nucleus is discussed using the projected shell model (PSM)