merge-width parameterized reductions

This work addresses compact integer linear programming (ILP) modeling for parameterized problems—specifically, polynomial-time reduction of an instance ((I, k)) to an ILP with only ( ext{poly}(k)) constraints. For problems admitting no polynomial kernel, establishing theoretical connections between WK[1]-hardness and compact ILP modeling remains open. Method: We introduce a novel preprocessing framework based on fast witness verification protocols, circumventing classical kernelization limitations and enabling a new compression paradigm. Integrating data structure optimizations with protocol design, we construct explicit ILP and mixed-integer linear programming (MILP) formulations for classic problems—including (r)-Way Cut, Steiner Tree, and Weighted Vertex Cover—whose constraint counts depend solely on the parameter (k). Results: Our models achieve provably compact formulations with ( ext{poly}(k)) constraints, and empirical evaluation demonstrates substantial improvements in solver efficiency. The approach provides a theoretically grounded yet practically effective modeling pathway for computationally hard parameterized problems.

This study addresses the parameterized complexity of the $T_{h+1}$-Free Edge Deletion problem, which asks whether at most $k$ edges can be removed from a graph so that every connected component has size at most $h$. By integrating tools from parameterized complexity theory, integer linear programming, bicriteria approximation algorithms, and specialized algorithms for restricted graph classes such as split graphs and interval graphs, the work systematically delineates the tractability frontier of the problem under various structural parameters. Key contributions include establishing new W[1]-hardness results for parameters like treewidth and pathwidth, proving fixed-parameter tractability for the first time when parameterized by cluster vertex deletion number plus $h$ and neighborhood diversity plus $h$, and presenting the first parameterized bicriteria approximation algorithm. Additionally, the paper shows that the natural directed extension of the problem remains intractable even on DAGs.

This paper studies the Maximum Common Subgraph (MCS) problem for labeled graphs: finding the minimum number of edge contractions to make two graphs isomorphic. It systematically characterizes the parameterized complexity of MCS with respect to maximum degree, degeneracy, clique-width, treewidth, and the number of contractions, and contrasts it with the Contractibility problem (deciding whether one labeled graph can be contracted to another). The analysis reveals, for the first time, that MCS and Contractibility exhibit nearly identical complexity spectra across almost all parameters—except for the joint parameter (degenericity, number of contractions), where MCS is W[1]-hard while Contractibility is FPT. Through novel parameterized algorithms, W[1]-hardness reductions, and structural graph-theoretic arguments, the work establishes the first complete parameterized complexity map for contraction problems on labeled graphs. This uncovers a counterintuitive high degree of alignment between the two problems and precisely delineates their tractability boundaries.

Existing theoretical frameworks for unified graph parameters—such as treewidth, clique-width, and twin-width—lack robustness and algorithmic support. This work focuses on merge-width and its variants, establishing deep connections to sparse graph theory, neighborhood covers, and bounded expansion graph classes through characterizations based on vertex orderings and logical definability. We present the first set of equivalent definitions for merge-width, design the first non-trivial approximation algorithm running in $n^{O(1)} \cdot 2^n$ time, and demonstrate its existence in structures with constant-overlap neighborhood covers and quasi-isometric embeddings. These contributions collectively establish the robustness and algorithmic applicability of merge-width as a graph parameter.

This work addresses parameterized approximation algorithms for Vertex Cover and 3-Hitting Set. Methodologically, it introduces a novel randomized branching paradigm grounded in an equivalence between the algorithm’s recursive structure and a binary stochastic process. Leveraging a type-theoretic adaptation of Sanov’s theorem, the framework performs large-deviation analysis on bivariate recurrence relations, yielding an analytically tractable master theorem for asymptotic running time. Contribution-wise, this is the first unified theoretical framework providing rigorous approximation-ratio–dependent guarantees across multiple approximation factors. It substantially improves worst-case time complexity over prior deterministic branching approaches and overcomes fundamental analytical limitations inherent in traditional branching analysis. The framework establishes a general methodology for characterizing the asymptotic performance of parameterized approximation algorithms, bridging stochastic analysis and combinatorial optimization.

This study addresses the problem of verifying correctness preservation under natural reductions in parameterized concurrent programs: given a program template and (semi-)commutativity relations, it determines whether the reduction maintains correctness. The work proposes the first systematic framework that characterizes the semantics of natural reductions by introducing atomic blocks and global convergence points to simplify verification. The main contributions include a polynomial-time complete decision algorithm for the synchronization-free setting, and a proof that the problem becomes coNP-hard in the presence of synchronization primitives such as locks. Furthermore, the paper establishes general complexity lower bounds dependent on the synchronization mechanism, revealing that even simple forms of synchronization induce substantial computational hardness.

This work establishes that determining whether a graph has twin-width at most 4 is NP-hard and demonstrates the absence of fixed-parameter approximation algorithms parameterized solely by twin-width. To overcome this limitation, the paper introduces two novel structural parameters—treedepth and vertex integrity—and presents the first fixed-parameter tractable algorithms for approximating and exactly computing twin-width. Specifically, it develops the first fixed-parameter approximation algorithm for twin-width based on treedepth that does not rely on deletion distance, and provides an optimal contraction sequence via a fixed-parameter exact algorithm parameterized by vertex integrity. A key technical contribution is the introduction of directed twin-width as an intermediate tool, which, combined with the structural properties of the two parameters, enables the design of efficient algorithms.

This work addresses the limitation of existing exact solvers for large-scale Maximum k-Cut problems (k > 2), which stems from the absence of effective preprocessing techniques. The paper introduces, for the first time, optimality-preserving data reduction rules tailored to this problem, leveraging structured cutset identification and graph decomposition strategies to partition the input graph into independently solvable connected components. A novel proof framework based on weighted graph superposition is developed to underpin these reductions. By engineering an integration of established MaxCut preprocessing methods into a unified system, the authors present the first efficient preprocessing pipeline specifically designed for k > 2. Experimental results demonstrate that the proposed approach substantially reduces instance sizes, significantly accelerates exact solvers when integrated, and enables solving more instances to optimality than previously possible.

This work addresses the challenge of verifying safety properties for infinite-state parameterized programs under complex topologies by introducing a novel proof system called the “parameterized proof space.” Leveraging local symmetries inherent in program topologies, the approach enables efficient verification of entire families of parameterized programs through the reuse of proof arguments across isomorphic neighborhoods. The key contributions include the development of a relatively complete proof system that operates without requiring explicit axiomatization of the underlying topology, integration of the model-theoretic notion of limit programs to support automatic construction and verification of universally quantified invariants, and the establishment of decidability guarantees for the verification process under certain conditions.

This study addresses the Bounded-Density Vertex Deletion problem: determining whether at most $k$ vertices can be removed from a graph so that the density of its densest subgraph falls below a given threshold. By leveraging parameterized complexity theory, structural graph properties, and refined reduction techniques, the authors establish for the first time that the problem is W[1]-hard when parameterized by treedepth or feedback vertex set size. Conversely, they show that when the target density is a constant, the problem admits a fixed-parameter tractable algorithm parameterized by clique-width. These results comprehensively delineate the parameterized complexity landscape of the problem across prominent graph parameters, precisely demarcating the boundary between tractable and intractable parameter regimes.