Dynamic programming on bipartite tree decompositions

TL;DR

This paper introduces bipartite treewidth (btw), a width parameter that generalizes treewidth and the odd-cycle transversal number, and develops a dynamic programming framework for solving problems on graphs with small btw. The authors design a general, annotation-driven DP scheme on 1-$ ext{H}$-tree decompositions, leveraging boundaried graphs and gadgets to replace interactions with a bounded structure, enabling FPT algorithms for problems such as $K_t$-Subgraph-Cover, Weighted Vertex Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut. They also provide XP algorithms for packing variants (H-Subgraph/Induced-Subgraph/Odd-Minor/Scattered-Packing) and establish a dichotomy for 2-connected $H$: if $H$ is bipartite, the problems are para-NP-complete parameterized by btw, otherwise they admit XP-time solutions. The work highlights the potential and limitations of the btw framework, clarifying the role of “nice” reductions and the gluing property, and points to connections with recent results on odd-minors and bipartite decompositions as well as several open questions for future research. Overall, the study advances algorithmic techniques for odd-minor related graph classes and provides a structured pathway to extend FPT methods beyond classical treewidth.

Abstract

We revisit a graph width parameter that we dub bipartite treewidth (btw). Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number, and is closely related to odd-minors. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one "bipartite" vertex, while the width of such decomposition measures the number of "non-bipartite" vertices in a bag. We provide para-NP-completeness results and develop dynamic programming techniques to solve problems on graphs of small btw. In particular, we show that $K_t$-Subgraph-Cover, Weighted Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are $FPT$ parameterized by btw. We also provide the following dichotomy when $H$ is a 2-connected graph: if $H$ is bipartite, then $H$-Subgraph/Induced-Subgraph/Odd-Minor/Scattered-Packing is para-NP-complete parameterized by btw while, if $H$ is non-bipartite, then the problem is solvable in XP-time.

Figures (15)

• Figure 1: Illustration of a bipartite tree decomposition. The blue part induces a bipartite graph, while the additional vertices are depicted in red. This bipartite tree decomposition has width at most $k$ if there are at most $k$ additional vertices in each bag.
• Figure 2: An odd $H$-expansion $\eta$ in a graph $G$. The nodes of $\eta$ are the subgraphs in the blue disks, and the edges of $\eta$ are the blue edges in $G$.
• Figure 3: A graph of bipartite treewidth one. A corresponding bipartite tree decomposition of width one is depicted, with two bags (one blue and one pink). The apex vertex of each bag is the squared vertex of the same color.
• Figure 4: Examples of use of the operations $\oplus$, $\boxplus$, and $\triangleright$.
• Figure 5: Illustration of ${\bf G}={\bf X}\boxplus(\boxplus_{i\in[d]}{\bf G}_i)$.

Theorems & Definitions (75)

• Proposition 1.1: Tazari12fast, adapted from DemaineHK10deco
• Lemma 3.1
• proof
• Proposition 3.1: Campbell23odd-
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof