Tight Bounds for some Classical Problems Parameterized by Cutwidth

TL;DR

The paper establishes tight SETH-tight bounds for key problems parameterized by cutwidth, notably Hamiltonian Cycle with $O^*( (1+\sqrt{2})^{\operatorname{ctw}} )$ and Triangle Packing/Partition into Triangles with $O^*( \sqrt[3]{3}^{\operatorname{ctw}} )$, along with matching $2^{\operatorname{ctw}}$ and $3^{\operatorname{ctw}}$ bounds for Max Cut and Induced Matching. It introduces a refined path-decomposition DP and identifies Z-cuts as a central bottleneck, enabling precise upper and lower bounds and bridging gaps between cutwidth and related parameters. The work adapts and extends algebraic DP methods from pathwidth to ct w, and provides CSP-based reductions to prove hardness results, yielding a nuanced picture of how width parameters influence complexity. Overall, the results advance the understanding of parameterized complexity by showcasing tight, sometimes non-integral, exponential bounds for natural problems when parameterized by cutwidth, and they open avenues for further exploration of width-based algorithms and lower bounds.

Abstract

Cutwidth is a widely studied parameter that quantifies how well a graph can be decomposed along small edge-cuts. It complements pathwidth, which captures decomposition by small vertex separators, and it is well-known that cutwidth upper-bounds pathwidth. The SETH-tight parameterized complexity of problems on graphs of bounded pathwidth (and treewidth) has been actively studied over the past decade while for cutwidth the complexity of many classical problems remained open. For Hamiltonian Cycle, it is known that a $(2+\sqrt{2})^{\operatorname{pw}} n^{O(1)}$ algorithm is optimal for pathwidth under SETH~[Cygan et al.\ JACM 2022]. Van Geffen et al.~[J.\ Graph Algorithms Appl.\ 2020] and Bojikian et al.~[STACS 2023] asked which running time is optimal for this problem parameterized by cutwidth. We answer this question with $(1+\sqrt{2})^{\operatorname{ctw}} n^{O(1)}$ by providing matching upper and lower bounds. Second, as our main technical contribution, we close the gap left by van Heck~[2018] for Partition Into Triangles (and Triangle Packing) by improving both upper and lower bound and getting a tight bound of $\sqrt[3]{3}^{\operatorname{ctw}} n^{O(1)}$, which to our knowledge exhibits the only known tight non-integral basis apart from Hamiltonian Cycle. We show that cuts inducing a disjoint union of paths of length three (unions of so-called $Z$-cuts) lie at the core of the complexity of the problem -- usually lower-bound constructions use simpler cuts inducing either a matching or a disjoint union of bicliques. Finally, we determine the optimal running times for Max Cut ($2^{\operatorname{ctw}} n^{O(1)}$) and Induced Matching ($3^{\operatorname{ctw}} n^{O(1)}$) by providing matching lower bounds for the existing algorithms -- the latter result also answers an open question for treewidth by Chaudhary and Zehavi~[WG 2023].

Figures (6)

• Figure 1: Illustration for \ref{['tripack::def:cut-sets']}. For $i=1,2,3,4$, the family $F^{(i)}$ is red on the left-hand side of the cut while the families $I^{(i-1)}$ and $Q^{(i-1)}$ are blue and gray, respectively, on the right-hand side. The sets $F_H, I_H, Q_H$ are the red, blue, and gray vertices, respectively, in $(4)$. The black vertex in $(4)$ belongs to $L^1_H$ as it has a single gray neighbor. The "bag" $X_H$ is the set of all black and blue vertices in $(4)$.
• Figure 2: A connection gadget $R$.
• Figure 3: Selection Gadget $Q(\mathcal{F})$ for the family $\mathcal{F}=\{F_1 = \{a,b,c\}, F_2 = \{b,c,d\}\}$. Thick lines depict a solution defining the state $F_2$ in $Q$.
• Figure 4: Sketch of the lower-bound construction for $n = 4$ and $m = 4$. Every gray box represents a path gadget, each of them may have one of the three states. Blue boxes are cliques on five vertices. Dotted edges reflect that there exist all possible edges between the sets. Every colored box represents a constraint gadget. The cutwidth of the construction is essentially upper-bounded by $3 n$ as every $Z$-cut contributes three edges to the size of a cut.
• Figure 8: The first segment of the lower-bound construction for $n = 4$, $m = 2$ and the constraints $V_1 = (x_1, x_2, x_3)$, $R_1 = \{(1,0,2),(0,2,2)\}$ and $V_2 = (x_2, x_3, x_4)$, $R_2 = \{(0,0,2),(1,0,1),(1,1,2)\}$. Dotted lines sketch blocked edges. A dotted line between the gray boxed reflects that there is a blocked edge between any two vertices in different boxes.

Theorems & Definitions (61)

• Conjecture 1: SETH
• Definition 2
• Theorem 3: DBLP:journals/siamdm/Lampis20
• Definition 4
• Theorem 5
• Definition 6
• Definition 7
• Lemma 8
• Corollary 9
• Lemma 10