treewidth parameterization

This work addresses efficient algorithms for bipartiteness-related graph problems—including $K_t$-subgraph covering, odd-cycle transversal, and maximum weighted cut—by introducing a novel graph width parameter: *bipartite treewidth* (btw), which unifies treewidth and odd-cycle transversal number. Building upon an enhanced tree decomposition framework, we develop the first dynamic programming paradigm tailored to btw, yielding fixed-parameter tractable (FPT) algorithms for multiple NP-hard problems. Furthermore, for any 2-connected graph $H$, we establish a complexity dichotomy theorem for $H$-odd-subgraph packing and related problems: under the btw parameterization, we precisely delineate the boundary between para-NP-completeness and XP-solvability. Integrating parameterized algorithmics, dynamic programming, and structural graph modeling, this work substantially extends the applicability of the treewidth framework to non-bipartite graph problems.

This paper studies the PATH COVER problem—covering all vertices of a graph with the minimum number of (not necessarily vertex-disjoint) paths. Historically overlooked, especially for trees and bounded-treewidth graphs, it lacked efficient exact algorithms. We present the first linear-time exact algorithm for trees (O(n)) and a polynomial-time algorithm for graphs of treewidth t running in n^{t^{O(t)}} time. For the related PATH PARTITION problem, we improve the randomized running time to 2^{O(t)}n. Our approach unifies these results under a tree-decomposition-based dynamic programming framework, enhanced by the Cut&Count technique to accelerate state merging. The framework naturally extends to variants imposing constraints such as induced paths or edge-disjointness. These contributions fill a theoretical gap in non-vertex-disjoint path covering, significantly advancing both computational efficiency and applicability to structured graph classes.

This work addresses the computational complexity and approximability of treewidth. Using a direct, self-contained reduction from 3-SAT to treewidth—bypassing indirect approaches via cutwidth or pathwidth—the authors achieve a technical breakthrough through refined analysis of graph expansion and tree decompositions. Their contributions are threefold: (1) They establish NP-hardness of computing a 1.00005-approximation to treewidth, thereby proving that no polynomial-time algorithm can achieve this approximation factor unless P = NP; (2) Assuming the Exponential Time Hypothesis (ETH), they prove a tight lower bound of $2^{Omega(n)}$ for exact treewidth computation on $n$-vertex graphs; (3) For any constant $delta > 1$, they show that $delta$-approximation requires $2^{Omega(n / log^c n)}$ time for some $c > 0$. This resolves long-standing open questions regarding the tightness of treewidth lower bounds and fills a critical gap in parameterized complexity theory.

This paper investigates the parameterized time lower bounds for three classical NP-complete graph problems—Metric Dimension, Strong Metric Dimension, and Geodetic Set—with respect to treewidth (tw) and vertex cover (vc). Under the Strong Exponential Time Hypothesis (SETH), we establish, for the first time, that none of these problems admits a $2^{2^{o(tw)}} cdot n^{O(1)}$-time algorithm, even when the input graph has bounded diameter; moreover, Strong Metric Dimension also lacks a $2^{2^{o(vc)}} cdot n^{O(1)}$-time algorithm. This yields the first double-exponential lower bounds for problems in NP, challenging the conventional belief that only problems beyond the polynomial hierarchy require double-exponential time. Technically, we introduce a generic construction based on Sperner families, integrating ETH-based reductions with precise parameterized complexity analysis. We complement our lower bounds with matching upper bounds, thereby fully characterizing the double-exponential hardness of these problems.

This work investigates the computational complexity and approximation algorithms for tree-partition-width. Regarding problem hardness, we establish that computing tree-partition-width exactly is XALP-complete—resolving a long-standing open question—and derive as a corollary the XALP-completeness of domino treewidth. We further characterize structural relationships between tree-partition-width and tree-cut width. Methodologically, we design the first polynomial-time O(k⁷)-approximation algorithm, running in kᴼ(¹)n² time, and prove that the problem is W[t]-hard for all t, revealing its intrinsic parameterized intractability. Collectively, our results precisely situate tree-partition-width within the sparse graph parameterization landscape: they resolve fundamental gaps concerning both exact solvability (via XALP-completeness) and efficient approximability (via the first nontrivial approximation guarantee), thereby providing a comprehensive theoretical foundation for this structural graph parameter.

This work addresses the challenge of verifying graph-theoretic properties on graph classes with bounded treewidth or pathwidth by proposing a unified framework that integrates tree-decomposition-based dynamic programming with formal reductions of graph properties. The framework enables automatic verification of atomic properties and their Boolean combinations, achieving for the first time a modular composition of dynamic programming algorithms coupled with parameterized automated theorem proving in treewidth. The developed TreeWidzard engine automatically checks whether all graphs of treewidth at most \(k\) satisfy a given Boolean expression \(P\) over graph properties, significantly enhancing the scalability and automation of complex graph property verification.

Traditional tree decomposition–based dynamic programming struggles to scale to large graphs, limiting its applicability to NP-hard graph optimization problems. This work proposes a general-purpose enhancement framework that overcomes this limitation by efficiently constructing treewidth modulators to partition the original graph into an easily solvable component and a residual subgraph of small treewidth. The framework seamlessly integrates heuristic methods—such as greedy algorithms, evolutionary search, and graph neural networks—with tree decomposition–based dynamic programming in a synergistic manner. Notably, it introduces tree decomposition–based dynamic programming as a universal booster applicable across diverse heuristic paradigms. Empirical evaluations on the maximum independent set, minimum vertex cover, and maximum cut problems demonstrate substantial performance gains over baseline algorithms; remarkably, the enhanced greedy approach matches or even surpasses state-of-the-art commercial solvers.

This work proposes a new graph parameter, the maximum leaf number of contracted connected components (denoted cml↓), which strictly lies between clique-width and reduced bandwidth. The parameter is introduced to unify the algorithmic tractability of bounded-clique-width graphs and unit interval graphs. Within the framework of contraction sequences, the authors show that cml↓ is bounded on unit interval graphs but unbounded on planar graphs. They establish a connection between maximum degree and treewidth in sparse graphs with bounded cml↓ and leverage balanced separators together with first-order transductions to design polynomial-time algorithms for NP-hard problems such as Maximum Induced d-Regular Subgraph. A key contribution is the proof that bounded maximum degree in sparse graphs of bounded cml↓ implies bounded treewidth, along with the demonstration that three-dimensional grids have unbounded reduced bandwidth and thus are not first-order transductions of planar graphs.

This paper establishes lower bounds for pure dynamic programming (DP) algorithms solving connectivity problems—such as the Traveling Salesman Problem (TSP)—on graphs of bounded pathwidth. It addresses the open question of whether algebraic techniques (e.g., convolution, determinant computation) are inherently necessary to achieve optimal worst-case time complexity. Method: The authors forge a novel connection between tropical circuit complexity and nondeterministic communication complexity, integrating compatibility matrix construction with structural characterizations of pathwidth-k graphs. Contribution/Results: They prove that any pure DP algorithm for such problems requires at least $2^{Omega(k log log k)}$ state transition units on graphs of pathwidth $k$. This is the first exponential lower bound for canonical problems like TSP within the pure DP framework, breaking the prior dominance of algebraic-method-based analyses. The result rigorously confirms the indispensability of algebraic techniques for achieving optimal time complexity in this setting.

This work investigates the fine-grained complexity of Max Cut, Hamiltonian Cycle, and Edge Dominating Set parameterized by modular treewidth. Assuming the Exponential Time Hypothesis (ETH), it establishes that Max Cut admits no algorithm running in time \(n^{2^{o(k)}} \cdot f(k)\), highlighting a fundamental distinction from its behavior under standard treewidth. In contrast, the study presents \(n^{O(k)}\)-time algorithms for both Hamiltonian Cycle and Edge Dominating Set, matching known conditional lower bounds and thereby resolving three long-standing open problems. By integrating conditional lower bounds, parameterized algorithm design, and structural analysis of graphs of bounded modular treewidth, this research provides a complete characterization of the complexity landscape for these problems under this parameterization.