[Paper Review] Efficient polynomial-time approximation scheme for the genus of dense graphs

Yifan Jing, Bojan Mohar|arXiv (Cornell University)|Nov 16, 2020

Advanced Graph Theory Research37 references3 citations

TL;DR

This paper presents an Efficient Polynomial-Time Approximation Scheme (EPTAS) and an Efficient Polynomial-Time Randomized Approximation Scheme (EPRAS) for approximating the genus of dense graphs—graphs with edge density at least α|V|² for some α > 0. The algorithm uses quasirandom graph decompositions, edge-disjoint triangle and quadrilateral packing via linear programming, and random greedy matching on hypergraphs to achieve (1+ε)-approximation of the minimum genus in O(f(ε)n²) time, with embedding construction in the EPRAS variant.

ABSTRACT

The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that $|E(G)|\ge α|V(G)|^2$ for some fixed $α>0$. While a constant factor approximation is trivial for this class of graphs, approximations with factor arbitrarily close to 1 need a sophisticated algorithm and complicated mathematical justification. More precisely, we provide an algorithm that for a given (dense) graph $G$ of order $n$ and given $\varepsilon>0$, returns an integer $g$ such that $G$ has an embedding into a surface of genus $g$, and this is $\varepsilon$-close to a minimum genus embedding in the sense that the minimum genus $\mathsf{g}(G)$ of $G$ satisfies: $\mathsf{g}(G)\le g\le (1+\varepsilon)\mathsf{g}(G)$. The running time of the algorithm is $O(f(\varepsilon)\,n^2)$, where $f(\cdot)$ is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) whose genus is $g$. This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time $O(f_1(\varepsilon)\,n^2)$.

Motivation & Objective

• To develop a polynomial-time approximation scheme with efficient dependence on ε for the genus of dense graphs, where |E(G)| ≥ α|V(G)|² for fixed α > 0.
• To achieve an approximation factor of (1+ε) for the minimum genus, improving upon trivial constant-factor approximations.
• To construct an algorithm that not only approximates the genus but also outputs a genus-minimal rotation system (embedding) for the graph.
• To provide a mathematically rigorous framework based on quasirandom graph theory and the regularity lemma for topological graph theory applications.

Proposed method

• Apply an algorithmic version of the Szemerédi Regularity Lemma to partition the input graph into a bounded number of parts with quasirandom edge distributions.
• Decompose the graph into tripartite and bipartite subgraphs (T_ijk and B_ij) with controlled edge densities, ensuring quasirandomness with high probability.
• Partition the edge sets of these subgraphs into t₁ and t₂ parts via random partitioning to generate multiple quasirandom digraphs and associated 3- and 4-uniform hypergraphs.
• Use the random greedy algorithm on 3-uniform and 4-uniform hypergraphs to find τ-near-perfect matchings corresponding to edge-disjoint triangles and quadrilaterals.
• Construct a rotation system Π from the matched cycles, ensuring arc-disjointness and resolving blossoms via a pruning step to yield a valid embedding.
• Leverage linear programming to compute ν(G), an ε-approximation of the maximum number of edge-disjoint triangles, with size bounded by f₂(ε), enabling genus approximation via Corollary 1.7.

Figure 1. A $3$ -blossom and a $4$ -blossom of length $6$ . The cycle $C_{1}$ is shown by bold edges.

Experimental results

Research questions

• RQ1 Can a (1+ε)-approximation of the genus of dense graphs be computed in polynomial time with efficient dependence on ε?
• RQ2 How can quasirandom graph decompositions be used to construct near-optimal embeddings in surfaces of bounded genus?
• RQ3 What is the role of edge-disjoint triangle and quadrilateral packing in approximating the genus of dense graphs?
• RQ4 Can a randomized algorithm output both a genus approximation and a valid rotation system (embedding) in near-quadratic time?
• RQ5 What is the theoretical connection between quasirandomness, hypergraph matching, and topological graph invariants like genus?

Key findings

• The paper achieves an EPTAS with running time O(f(ε)n²) that returns a genus estimate g satisfying g ≤ (1+ε)g(G), where g(G) is the true genus of the graph.
• An EPRAS variant is developed that not only approximates the genus within (1+ε) factor but also outputs a rotation system realizing the genus estimate g.
• The algorithm relies on a linear program of size f₂(ε) to approximate the maximum number of edge-disjoint triangles ν(G), which is crucial for genus estimation.
• With high probability, at least (1−r(ε)−τ) of the generated hypergraphs (for triangles and quadrilaterals) admit τ-near-perfect matchings, enabling robust cycle packing.
• The construction ensures arc-disjointness of the selected cycles in the digraph and its inverse, allowing the formation of a valid rotation system after blossom resolution.
• The final genus estimate satisfies (1+ε)g(G) ≥ g(G,Π), proving the (1+ε)-approximation guarantee for the embedding genus.