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Legal Judgment Prediction (LJP) has become an increasingly crucial task in Legal AI, i.e., predicting the judgment of the case in terms of case fact description. Precedents are the previous legal cases with similar facts, which are the basis for the judgment of the subsequent case in national legal systems. Thus, it is worthwhile to explore the utilization of precedents in the LJP. Recent advances in deep learning have enabled a variety of techniques to be used to solve the LJP task. These can be broken down into two categories: large language models (LLMs) and domain-specific models. LLMs are capable of interpreting and generating complex natural language, while domain models are efficient in learning task-specific information. In this paper, we propose the precedent-enhanced LJP framework (PLJP), a system that leverages the strength of both LLM and domain models in the context of precedents. Specifically, the domain models are designed to provide candidate labels and find the proper precedents efficiently, and the large models will make the final prediction with an in-context precedents comprehension. Experiments on the real-world dataset demonstrate the effectiveness of our PLJP. Moreover, our work shows a promising direction for LLM and domain-model collaboration that can be generalized to other vertical domains.

The Graphical House Allocation problem asks: how can $n$ houses (each with a fixed non-negative value) be assigned to the vertices of an undirected graph $G$, so as to minimize the "aggregate local envy", i.e., the sum of absolute differences along the edges of $G$? This problem generalizes the classical Minimum Linear Arrangement problem, as well as the well-known House Allocation Problem from Economics, the latter of which has notable practical applications in organ exchanges. Recent work has studied the computational aspects of Graphical House Allocation and observed that the problem is NP-hard and inapproximable even on particularly simple classes of graphs, such as vertex disjoint unions of paths. However, the dependence of any approximations on the structural properties of the underlying graph had not been studied. In this work, we give a complete characterization of the approximability of the Graphical House Allocation problem. We present algorithms to approximate the optimal envy on general graphs, trees, planar graphs, bounded-degree graphs, bounded-degree planar graphs, and bounded-degree trees. For each of these graph classes, we then prove matching lower bounds, showing that in each case, no significant improvement can be attained unless P = NP. We also present general approximation ratios as a function of structural parameters of the underlying graph, such as treewidth; these match the aforementioned tight upper bounds in general, and are significantly better approximations for many natural subclasses of graphs. Finally, we present constant factor approximation schemes for the special classes of complete binary trees and random graphs.

In this paper, we propose two new classes of binary array codes, termed V-ETBR and V-ESIP codes, by reformulating and generalizing the variant technique of deriving the well-known generalized row-diagonal parity~(RDP) codes from shortened independent parity~(IP) codes. The V-ETBR and V-ESIP codes are both based on binary parity-check matrices and are essentially variants of two classes of codes over a special polynomial ring (termed ETBR and ESIP codes in this paper). To explore the conditions that make the variant codes binary Maximum Distance Separable~(MDS) array codes that achieve optimal storage efficiency, this paper derives the connections between V-ETBR/V-ESIP codes and ETBR/ESIP codes. These connections are beneficial for constructing various forms of the variant codes. By utilizing these connections, this paper also explicitly presents the constructions of V-ETBR and V-ESIP MDS array codes with any number of parity columns $r$, along with their fast syndrome computations. In terms of construction, all proposed MDS array codes have an exponentially growing total number of data columns with respect to the column size, while alternative codes have that only with linear order. In terms of computation, the proposed syndrome computations make the corresponding encoding/decoding asymptotically require $\lfloor \lg r \rfloor+1$ XOR~(exclusive OR) operations per data bit, when the total number of data columns approaches infinity. This is also the lowest known asymptotic complexity in MDS codes.

The Self-Sovereign Identity (SSI) is a decentralized paradigm enabling full control over the data used to build and prove the identity. In Internet of Things networks with security requirements, the Self-Sovereign Identity can play a key role and bring benefits with respect to centralized identity solutions. The challenge is to make the SSI compatible with resource-constraint IoT networks. In line with this objective, the paper proposes and discusses an alternative (mutual) authentication process for IoT nodes under the same administration domain. The main idea is to combine the Decentralized IDentifier (DID)-based verification of private key ownership with the verification of a proof that the DID belongs to an evolving trusted set. The solution is built around the proof of membership notion. The paper analyzes two membership solutions, a novel solution designed by the Authors based on Merkle trees and a second one based on the adaptation of Boneh, Boyen and Shacham (BBS) group signature scheme. The paper concludes with a performance estimation and a comparative analysis.

AI in Math deals with mathematics in a constructive manner so that reasoning becomes automated, less laborious, and less error-prone. For algorithms, the question becomes how to automate analyses for specific problems. For the first time, this work provides an automatic method for approximation analysis on a well-studied problem in theoretical computer science: computing approximate Nash equilibria in two-player games. We observe that such algorithms can be reformulated into a search-and-mix paradigm, which involves a search phase followed by a mixing phase. By doing so, we are able to fully automate the procedure of designing and analyzing the mixing phase. For example, we illustrate how to perform our method with a program to analyze the approximation bounds of all the algorithms in the literature. Same approximation bounds are computed without any hand-written proof. Our automatic method heavily relies on the LP-relaxation structure in approximate Nash equilibria. Since many approximation algorithms and online algorithms adopt the LP relaxation, our approach may be extended to automate the analysis of other algorithms.

We study a natural complexity measure of Boolean functions known as the (exact) rational degree. For total functions $f$, it is conjectured that $\mathrm{rdeg}(f)$ is polynomially related to $\mathrm{deg}(f)$, where $\mathrm{deg}(f)$ is the Fourier degree. Towards this conjecture, we show that symmetric functions have rational degree at least $\mathrm{deg}(f)/2$ and monotone functions have rational degree at least $\sqrt{\mathrm{deg}(f)}$. We observe that both of these lower bounds are tight. In addition, we show that all read-once depth-$d$ Boolean formulae have rational degree at least $\Omega(\mathrm{deg}(f)^{1/d})$. Furthermore, we show that almost every Boolean function on $n$ variables has rational degree at least $n/2 - O(\sqrt{n})$. In contrast to total functions, we exhibit partial functions that witness unbounded separations between rational and approximate degree, in both directions. As a consequence, we show that for quantum computers, post-selection and bounded-error are incomparable resources in the black-box model.

We present a comprehensive analysis of the coupled scheme introduced in [Springer Proceedings in Mathematics \& Statistics, vol 237. Springer, Cham 2018 \cite{S2018}] for linear and Hamilton-Jacobi equations. This method merges two distinct schemes, each tailored to handle specific solution characteristics. It offers a versatile framework for coupling various schemes, enabling the integration of accurate methods for smooth solutions and the treatment of discontinuities and gradient jumps. In \cite{S2018}, the emphasis was on coupling an anti-dissipative scheme designed for discontinuous solutions with a semi-Lagrangian scheme developed for smooth solutions. In this paper, we rigorously establish the essential properties of the resulting coupled scheme, especially in the linear case. To illustrate the effectiveness of this coupled approach, we present a series of one-dimensional examples.

Synthesis consists in deciding whether a given labeled transition system (TS) $A$ can be implemented by a net $N$ of type $\tau$. In case of a negative decision, it may be possible to convert $A$ into an implementable TS $B$ by applying various modification techniques, like relabeling edges that previously had the same label, suppressing edges/states/events, etc. It may however be useful to limit the number of such modifications to stay close to the original problem, or optimize the technique. In this paper, we show that most of the corresponding problems are NP-complete if $\tau$ corresponds to the type of flip-flop nets or some flip-flop net derivatives.

Subset Sum Ratio is the following optimization problem: Given a set of $n$ positive numbers $I$, find disjoint subsets $X,Y \subseteq I$ minimizing the ratio $\max\{\Sigma(X)/\Sigma(Y),\Sigma(Y)/\Sigma(X)\}$, where $\Sigma(Z)$ denotes the sum of all elements of $Z$. Subset Sum Ratio is an optimization variant of the Equal Subset Sum problem. It was introduced by Woeginger and Yu in '92 and is known to admit an FPTAS [Bazgan, Santha, Tuza '98]. The best approximation schemes before this work had running time $O(n^4/\varepsilon)$ [Melissinos, Pagourtzis '18], $\tilde O(n^{2.3}/\varepsilon^{2.6})$ and $\tilde O(n^2/\varepsilon^3)$ [Alonistiotis et al. '22]. In this work, we present an improved approximation scheme for Subset Sum Ratio running in time $O(n / \varepsilon^{0.9386})$. Here we assume that the items are given in sorted order, otherwise we need an additional running time of $O(n \log n)$ for sorting. Our improved running time simultaneously improves the dependence on $n$ to linear and the dependence on $1/\varepsilon$ to sublinear. For comparison, the related Subset Sum problem admits an approximation scheme running in time $O(n/\varepsilon)$ [Gens, Levner '79]. If one would achieve an approximation scheme with running time $\tilde O(n / \varepsilon^{0.99})$ for Subset Sum, then one would falsify the Strong Exponential Time Hypothesis [Abboud, Bringmann, Hermelin, Shabtay '19] as well as the Min-Plus-Convolution Hypothesis [Bringmann, Nakos '21]. We thus establish that Subset Sum Ratio admits faster approximation schemes than Subset Sum. This comes as a surprise, since at any point in time before this work the best known approximation scheme for Subset Sum Ratio had a worse running time than the best known approximation scheme for Subset Sum.

Pre-trained Language Models (PLMs) which are trained on large text corpus via self-supervised learning method, have yielded promising performance on various tasks in Natural Language Processing (NLP). However, though PLMs with huge parameters can effectively possess rich knowledge learned from massive training text and benefit downstream tasks at the fine-tuning stage, they still have some limitations such as poor reasoning ability due to the lack of external knowledge. Research has been dedicated to incorporating knowledge into PLMs to tackle these issues. In this paper, we present a comprehensive review of Knowledge-Enhanced Pre-trained Language Models (KE-PLMs) to provide a clear insight into this thriving field. We introduce appropriate taxonomies respectively for Natural Language Understanding (NLU) and Natural Language Generation (NLG) to highlight these two main tasks of NLP. For NLU, we divide the types of knowledge into four categories: linguistic knowledge, text knowledge, knowledge graph (KG), and rule knowledge. The KE-PLMs for NLG are categorized into KG-based and retrieval-based methods. Finally, we point out some promising future directions of KE-PLMs.