A Polynomial-Time Approximation Algorithm for Complete Interval Minors

TL;DR

The paper addresses detecting $K_t$ as an interval minor in ordered graphs by introducing delayed decompositions and a computable notion of delayed rank. It proves that $K_t$-interval-minor-free ordered graphs have bounded rank, enabling a polynomial-time $f(t)$-approximation where $f(t)$ is triply exponential in $t$ and independent of the input size $n$, with a running time of $O(t \cdot m n^2)$. The approach combines a structural decomposition into simpler quotient graphs, a Ramsey-type bound to control the approximation factor, and a heavy-leaf criterion in delayed decompositions to certify the presence of $K_t$ when possible. As a practical outcome, the authors provide a linear-time $K_3$-interval-minor detector and establish that graphs avoiding large interval minors have bounded chromatic number, highlighting the structural tightness of the method. Overall, the work connects graph minor theory concepts to the ordered-graph interval-minor setting, delivering a scalable algorithmic framework for approximate interval-minor detection.

Abstract

As shown by Robertson and Seymour, deciding whether the complete graph $K_t$ is a minor of an input graph $G$ is a fixed parameter tractable problem when parameterized by $t$. From the approximation viewpoint, the gap to fill is quite large, as there is no PTAS for finding the largest complete minor unless $P = NP$, whereas a polytime $O(\sqrt n)$-approximation algorithm was given by Alon, Lingas and Wahlén. We investigate the complexity of finding $K_t$ as interval minor in ordered graphs (i.e. graphs with a linear order on the vertices, in which intervals are contracted to form minors). Our main result is a polytime $f(t)$-approximation algorithm, where $f$ is triply exponential in $t$ but independent of $n$. The algorithm is based on delayed decompositions and shows that ordered graphs without a $K_t$ interval minor can be constructed via a bounded number of three operations: closure under substitutions, edge union, and concatenation of a stable set. As a byproduct, graphs avoiding $K_t$ as an interval minor have bounded chromatic number.

Figures (7)

• Figure 1: A $K_4$ interval minor: the red zones represent the intervals we contract and the red edges are the remaining edges. Observe that this interval minor model is not a minor model since we contracted non-connected subsets of vertices.
• Figure 2: Construction showing that the ordered $P_6$ has rank at most $4$. The edge graph $K_2$ has rank $2$, which is why we go from rank $1$ to rank $\le3$.
• Figure 3: A graph $G$ drawn with an arbitrary order and the corresponding ordered graph $\hat{G}$. The fresh vertices $u_1$ and $u_2$ are depicted in red, as well as the edges incident to them.
• Figure 4: A delayed structured tree and its realization. The edges in the realization are colored according to their corresponding quotient graph.
• Figure 5: A 4-branching vertex $x$, with $\mathcal{I}'$ being the set of red intervals, and a 4-interval path $I_1,I_2,I_3,I_4$.

Theorems & Definitions (64)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Claim
• proof : Proof of the Claim
• Theorem 6
• proof