fixed-parameter algorithms

This work addresses the limited generalization of existing methods in complex scenarios by proposing a novel framework based on adaptive feature fusion and dynamic inference. By integrating multi-level semantic alignment with an uncertainty-aware module, the approach significantly enhances model robustness under distribution shifts. Extensive experiments demonstrate that the proposed method consistently outperforms state-of-the-art approaches across multiple benchmark datasets, achieving an average accuracy improvement of 3.2% while maintaining low computational overhead. The primary contribution lies in the design of a plug-and-play dynamic fusion architecture, offering a promising direction toward reliable artificial intelligence in open-world environments.

This study addresses the practical applicability of parameterized algorithms by investigating how theoretical graph parameters correlate with real-world instances. Method: We conduct a large-scale empirical analysis—first of its kind across diverse real-world graph benchmarks—measuring distributions of over ten structural parameters, including degeneracy, neighborhood diversity, modular width, vertex cover number, feedback vertex set size, and treewidth. Contribution/Results: We reveal critical discrepancies between theoretical assumptions and empirical behavior: treewidth is typically ≈ *n*/9 (well below the worst-case *n*/3), rendering *O*(2^tw) algorithms practically viable; in contrast, vertex cover number often approaches *n*/2, substantially eroding its fixed-parameter tractability advantage. We publicly release the first unified computational framework supporting emerging parameters (e.g., 4-path vertex cover number) and a comprehensive experimental dataset, establishing a data-driven foundation for the design, selection, and optimization of parameterized algorithms.

This work investigates the FPT inapproximability of the parameterized $k$-clique problem. Assuming $W[1] eq ext{FPT}$, it proves that no FPT algorithm can achieve an approximation factor of $k^{1/H(k)}$, where $H(k)$ is any increasing computable function (e.g., $log^* k$), thereby ruling out—for the first time—FPT approximations with super-slowly growing factors such as $k^{1/log^* k}$. The proof introduces, for the first time in parameterized inapproximability, list decoding of Hadamard codes over large prime fields into the hardness framework. This significantly strengthens Lin’s (STOC 2021) constant-factor lower bound. Technically, the result integrates gap-ETH-based hardness, $W[1]$-hardness reductions, combinatorial coding theory, and fine-grained parameterized complexity analysis. It establishes near-polynomial FPT inapproximability for $k$-clique—i.e., no $k^{o(1)}$-factor FPT approximation unless $W[1] = ext{FPT}$—and introduces a novel paradigm for parameterized approximation theory.

This paper addresses the single-machine scheduling problem with release times and deadlines, aiming to minimize maximum lateness (L_max)—a strongly NP-hard problem. To overcome its exponential complexity bottleneck, we introduce, for the first time in exact scheduling algorithms, the fixed-parameter tractability paradigm, proposing a Variable-Parameter (VP) analytical framework that strictly confines exponential dependence to the “number of emerging jobs”—a dynamic parameter significantly smaller than n. We prove the algorithm’s time complexity is O*(c^k), where k is the number of emerging jobs and c is a constant, with the dominant component being polynomial-time. Probabilistic analysis and empirical evaluation confirm that k/n asymptotically tends to zero. Our approach enables efficient exact resolution of this strongly NP-hard scheduling problem in practical scenarios. It constitutes the first fixed-parameter exact algorithm for scheduling that dynamically parameterizes based on intrinsic problem structure.

This work addresses automatic runtime and variable-size bound analysis for integer programs, focusing on the decidable subclass of periodic rational-solvable loops (PRS-loops). The proposed method introduces a modular analysis framework: it first derives local bounds for PRS-loops, then lifts them to global bounds via program transformation and inductive reasoning. Crucially, it extends the decidability of PRS-loop analysis to arbitrary integer programs by designing a synergistic synthesis mechanism combining abstract interpretation with rational linear algebra. The approach is fully automated in the tool KoAT, supporting precise derivation of polynomial and exponential complexity bounds as well as variable growth bounds. Experimental evaluation demonstrates effectiveness on diverse nontrivial integer programs, significantly improving the completeness, precision, and practicality of automated complexity analysis.

该研究解决了4-块整数规划问题，通过使用固定参数算法，并基于矩阵最大维度k和绝对值最大的元素Δ作为参数，证实了Eisenbrand和Rothvoss的猜想。

This study presents the first investigation into the parameterized complexity of the multiplicative α-spanner problem for undirected graphs with independent weights and lengths. Methodologically, it proposes two algorithmic frameworks—exclusion and inclusion—and optimizes the exclusion approach by introducing a compactness parameter to improve results on basic instances. The primary contributions are twofold. First, it proves that the problem is W[2]-hard when parameterized solely by total weight, yet becomes fixed-parameter tractable (FPT) when combined with auxiliary parameters. Second, it establishes FPT algorithm bounds for arbitrary weights and lengths, significantly overcoming the limitations of existing methods restricted to unit weights.

This study addresses the multi-bottleneck matching problem, which involves determining and optimizing perfect matchings where edge weights are vectors and costs are defined as the sum of component-wise maxima. From a parameterized complexity perspective and motivated by scheduling applications, the theoretical analysis integrates reduction techniques with combinatorial optimization methods. The primary contributions include establishing fixed-parameter tractability (FPT) when jointly parameterized by k and Z, while revealing W[1]-hardness under single-parameter settings. Furthermore, this work proposes an efficient approximation scheme based on k and establishes a super-logarithmic lower bound for approximability. Overall, this research systematically completes the complexity classification framework for multi-bottleneck matching.

This work resolves three open problems posed by Bumpus et al. concerning treewidth-parameterized decision problems. We introduce a unified framework that yields truly linear fixed-parameter tractable (TLFPT) algorithms, deciding in time $O(n + m) + f(k, \varphi)$ whether an $n$-vertex, $m$-edge graph satisfies a given CMSO₂ formula $\varphi$, where $k$ bounds the treewidth. Our main contributions include the first linear-time algorithm realizing Courcelle’s theorem within the TLFPT paradigm, a linear-time approximation algorithm for treewidth with approximation ratio $2^{O(k)}$, and a TLFPT algorithm for exact treewidth computation. By integrating CMSO₂ logic, tree decompositions, and parameterized algorithmic techniques, our approach significantly advances the efficiency of solving treewidth-related problems.

研究通过图的treewidth、k值及特定类型图参数化方法，解决k-core问题在次多项式复杂度类中的定位，提出并证明了若干算法及其复杂度界限。