TSP is a classic and extensively studied problem with numerous real-world applications in artificial intelligence and operations research. It is well-known that TSP admits a constant approximation ratio on metric graphs but becomes NP-hard to approximate within any computable function f(n) on general graphs. This disparity highlights a significant gap between the results on metric graphs and general graphs. Recent research has introduced some parameters to measure the ``distance'' of general graphs from being metric and explored FPT approximation algorithms parameterized by these parameters. Two commonly studied parameters are p, the number of vertices in triangles violating the triangle inequality, and q, the minimum number of vertices whose removal results in a metric graph. In this paper, we present improved FPT approximation algorithms with respect to these two parameters. For p, we propose an FPT algorithm with a 1.5-approximation ratio, improving upon the previous ratio of 2.5. For q, we significantly enhance the approximation ratio from 11 to 3, advancing the state of the art in both cases.

With the frequent occurrences of natural disasters and the rising risk of malicious attacks, improving network survivability and guaranteeing connectivity in the presence of large-scale failures have emerged as a critical research challenge. Traditional studies on improving edge/node connectivity assume that failures occur at random and fail to capture the locality of large-scale failures. Although studies on region-based connectivity can address this limitation, they fail to consider how local failures affect the communication between certain key source-destination (SD) pairs. In this paper, we first extend the definition of region-based connectivity to include SD pairs. Given ℓ failure regions andkSD pairs, we study the problem of shielding edges with minimum cost to improve region-based connectivity between thekSD pairs. Second, we systematically analyze the computational complexity of the problem under different settings of ℓ,kand topologies of failure regions. Third, we design an ILP-based formulation to solve the general problem and propose two polynomial-time algorithms for two special cases based on the matroid technique and the biconnected component decomposition, respectively. Experimental results show that our algorithms are much faster than previously known algorithms.

The #2-SAT and #3-SAT problems involve counting the number of satisfying assignments (also called models) for instances of 2-SAT and 3-SAT, respectively. In 2010, Zhou et al. (https://doi.org/10.1609/aaai.v24i1.7537) proposed an O*(1.1892^m)-time algorithm for #2-SAT and an efficient approach for #3-SAT, where m denotes the number of clauses. In this paper, we show that the weighted versions of #2-SAT and #3-SAT can be solved in O*(1.1082^m) and O*(1.4423^m) time, respectively. These results directly apply to the unweighted cases and achieve substantial improvements over the previous results. These advancements are enabled by the introduction of novel reduction rules, a refined analysis of branching operations, and the application of path decompositions on the primal and dual graphs of the formula.