Constant-factor approximation of maximum distance-2 independent set in graphs of bounded merge-width

Abstract: We give a constant-factor approximation algorithm for Max Dist-2 Independent Set in graphs of bounded radius-2 merge-width. The same result holds for Min Dominating Set from [Bonamy and Geniet, 2025], [Chan et al., SODA '12]. Both approximation algorithms are LP-based, showing that the domination-to-2-independence ratio is bounded in graphs of bounded radius-2 merge-width. Moreover, this result is tight in the sense that the ratio can be unbounded in graphs of bounded radius-1 merge-width.

• The paper introduces a polynomial-time LP-based greedy algorithm that approximates the Maximum Distance-2 Independent Set within an O(k·d²) factor.
• It leverages bounded radius-2 merge-width to extend approximation guarantees from specialized classes to a broader spectrum of sparse graphs.
• The work establishes tight integrality gap bounds and clarifies the domination-to-2-independence ratio, offering practical insights for related optimization problems.

Constant-Factor Approximation of Maximum Distance-2 Independent Set in Graphs of Bounded Merge-Width

Problem Setting and Background

The manuscript addresses the approximability of the Maximum Distance-2 Independent Set problem—also known as the Maximum Packing problem—in graphs characterized by bounded radius-2 merge-width, a parameter encompassing graph classes such as planar graphs, graphs of bounded degree, bounded expansion, and bounded twin-width.

A distance-2 independent set in a graph $G$ is a subset of vertices in which the pairwise distance between any two distinct elements exceeds 2. The size of the largest such set is the 2-independence number, $\alpha_2(G)$. The corresponding optimization problem is known to be NP-complete. Dually, the Minimum Dominating Set problem, with domination number $\gamma(G)$, seeks a minimal subset such that every vertex in the graph is either in the set or adjacent to at least one set member. While $\alpha_2(G) \leq \gamma(G)$ holds in any graph, both the optimization and approximability characteristics of these problems diverge widely across graph classes.

Prior to this work, constant-factor approximations for Maximum Packing were established for specialized graph classes, such as cubic, planar, bounded expansion, and graphs with bounded twin-width (assuming access to decompositions that are computationally hard to obtain). This work extends tight and efficiently computable guarantees to the broader class of graphs with bounded radius-2 merge-width by introducing an explicit, polynomial-time constant-factor approximation algorithm based on LP relaxations.

Merge-Width and Structural Constraints

Merge-width is a recently introduced structural graph parameter, defined via sequences of partitions over $V(G)$ and associated relations capturing homogeneity under merging operations. For a given radius $r$, the radius-$r$ merge-width, $\mathrm{mw}_r(G)$, quantifies the complexity of these merges in local neighborhoods. Graphs with bounded merge-width (for suitable $r$) form a rich class that strictly subsumes many familiar sparse graph classes.

The significance of bounded radius-2 merge-width in this context is threefold:

1. Neighborhood Complexity: As shown by Bonamy and Geniet, such graphs possess linear neighborhood complexity, which is central to efficient domination approximations.
2. VC-Dimension Control: The Sauer-Shelah-Perles lemma is leveraged to show containment of the shatter function in a polynomial regime, while merge-width bounds VC-dimension directly for radius-1.
3. Parameter Tightness: The results are shown to be essentially tight; with only radius-1 merge-width, neither approximation guarantees nor bounded domination-to-2-independence ratios are possible due to degenerate graph counterexamples.

Main Algorithmic Result

The principal contribution is a polynomial-time constant-factor approximation algorithm for Maximum Packing in graphs of bounded radius-2 merge-width. The algorithm employs a greedy strategy driven by the optimal solution to the LP relaxation of the problem, iteratively selecting vertices whose radius-2 neighborhood forms a ball of minimum fractional weight according to the current LP solution. At each step, the selected vertex and its radius-2 neighborhood are excluded from further consideration until the fractional weight is exhausted.

Key Theoretical Insights

• Ball Weight Control: The main technical lemma proves that, for any fractional distance-2 independent set weight function $w^*$ in a graph of duality order $\alpha_2(G)$0 and radius-2 merge-width $\alpha_2(G)$1, there exists a vertex $\alpha_2(G)$2 such that the total $\alpha_2(G)$3-weight of $\alpha_2(G)$4 is $\alpha_2(G)$5.
• Integrality Gap Bound: The integrality gap for Maximum Packing is bounded by $\alpha_2(G)$6, and the total number of greedy steps is controlled accordingly.
• LP Duality: The LP relaxations for Minimum Dominating Set and Maximum Packing are duals; thus, approximation results and integrality gap bounds for the former transfer to the latter in this structural setting.

These insights culminate in the following formal result:

Graph Class	Approximation Factor
$\alpha_2(G)$7, duality $\alpha_2(G)$8	$\alpha_2(G)$9
$\gamma(G)$0	$\gamma(G)$1
twin-width $\gamma(G)$2 ($\gamma(G)$3)	$\gamma(G)$4

The algorithm achieves a factor $\gamma(G)$5 in graphs with $\gamma(G)$6, $\gamma(G)$7, and duality order $\gamma(G)$8.

Implications and Further Developments

A significant theoretical corollary is that, in bounded radius-2 merge-width graphs, the domination-to-2-independence ratio is bounded by a polynomial in $\gamma(G)$9 and $\alpha_2(G) \leq \gamma(G)$0, resolving an open problem concerning the relationship between these invariants within large classes of sparse graphs. This result robustly extends previous work that depended on difficult-to-compute decompositions (e.g., twin-width) and provides practical bounds for optimization.

Furthermore, the greedy LP-based approach generalizes to natural extensions of the problem, including Min Dist-$\alpha_2(G) \leq \gamma(G)$1 Dominating Set and Max Dist-$\alpha_2(G) \leq \gamma(G)$2 Independent Set, owing to the closure of bounded merge-width classes under taking graph powers.

An important negative result is that these guarantees do not extend to graphs of bounded radius-1 merge-width, as witnessed by constructions in low-degeneracy bipartite or split graphs with unbounded domination-to-2-independence ratios.

Open Directions

The work raises fundamental questions regarding the boundaries of tractability for these packing and covering problems:

• Bounded Radius-1 Merge-Width: It remains unresolved whether Min Dominating Set admits a constant-factor approximation or bounded integrality gap within graphs of bounded radius-1 merge-width.
• Linear Neighborhood Complexity: Whether Maximum Packing admits constant-factor approximation in graph classes defined by linear neighborhood complexity remains an open problem, with significant implications for both theory and practical algorithm design.

Conclusion

By providing a polynomial-time, LP-driven, constant-factor approximation algorithm for Maximum Distance-2 Independent Set in graphs of bounded radius-2 merge-width, the paper advances the approximability landscape for packing and covering problems in sparse graphs. The theoretical guarantees derived are tight and readily extend to graph classes parameterized by bounded twin-width with improved approximation factors. These results clarify the interaction between structural graph parameters and optimization approximability, and suggest promising avenues for extension, particularly regarding broader classes defined by neighborhood complexity or weaker structural invariants.

Reference: "Constant-factor approximation of maximum distance-2 independent set in graphs of bounded merge-width" (2606.31369)