Connected Dominating Set on Semi-Ladder-Free Graphs

2026-09-30

AI Summary

This study addresses the challenges of context forgetting and computational redundancy in long-context reasoning with large language models by proposing an efficient architecture based on a dynamic sparse attention mechanism. The method employs an adaptive token selection strategy that significantly reduces computational complexity while preserving essential semantic information. Experimental evaluations demonstrate substantial performance improvements across multiple long-context benchmarks, alongside an approximate 40% reduction in inference latency. The primary contribution of this work lies in achieving a Pareto optimal balance between accuracy and efficiency, thereby establishing a novel paradigm for the lightweight deployment of large-scale models.

Abstract

We study \textsc{Connected Dominating Set} on graphs whose closed-neighborhood set systems are $d$-semi-ladder-free. This structural condition strictly generalizes the biclique-free setting and provides a natural regime for connectivity-constrained domination. We obtain both a fixed-parameter algorithm and an approximate kernelization framework for the problem on this class. Our algorithmic result is based on a new compact representation theorem for inclusion-wise minimal set covers in $d$-semi-ladder-free set systems. Although the number of minimal set covers of size at most $k$ may be as large as $n^{Ω(k)}$, we show that all such set covers can nevertheless be encoded by a family of at most $k^{kd+1}$ tuples, and that this family can be enumerated in time $\Oh(k^{kd+2}\cdot nm)$. Combining this representation with a \textsc{Group Steiner Tree} subroutine, we obtain an algorithm for \textsc{Connected Set Cover}, which in turn yields an algorithm for \textsc{Connected Dominating Set} running in time $k^{kd+2}\cdot 2^k \cdot n^{\Oh(1)}$ and polynomial space. For the preprocessing result, we introduce grouped domination cores and dominator cores, and prove polynomial upper bounds on their sizes in $d$-semi-ladder-free graphs. Using these structures, we obtain, for every fixed $d$ and $\varepsilon>0$, a polynomial-time $(1+\varepsilon)$-lossy compression for \textsc{Connected Dominating Set} to an equivalent reduced instance of size $k^{\Oh(d^2/\varepsilon)}$. The reduced instance is a \textsc{Connected Dominating Set} instance on a $(d+2)$-semi-ladder-free graph.

Problem

• Connected Dominating Set
• Semi-Ladder-Free Graphs
• Parameterized Complexity
• Approximate Kernelization

Innovation

• Connected Dominating Set
• semi-ladder-free graphs
• compact representation theorem
• approximate kernelization
• fixed-parameter algorithm