A Generalization of Nemhauser and Trotters Local Optimization Theorem

The Nemhauser-Trotter local optimization theorem applies to the NP-hard Vertex Cover problem and has applications in approximation as well as parameterized algorithmics. We present a framework that generalizes Nemhauser and Trotter’s result to vertex deletion and graph packing problems, introducing novel algorithmic strategies based on purely combinatorial arguments (not referring to linear programming as the Nemhauser-Trotter result originally did). We exhibit our framework using a generalization of Vertex Cover, called Bounded- Degree Deletion, that has promise to become an important tool in the analysis of gene and other biological networks. For some fixed d \geq 0, Bounded-Degree Deletion asks to delete as few vertices as possible from a graph in order to transform it into a graph with maximum vertex degree at most d. Vertex Cover is the special case of d = 0. Our generalization of the Nemhauser-Trotter theorem implies that Bounded-Degree Deletion has a problem kernel with a linear number of vertices for every constant d. We also outline an application of our extremal combinatorial approach to the problem of packing stars with a bounded number of leaves. Finally, charting the border between (parameterized) tractability and intractability for Bounded-Degree Deletion, we provide a W[2]-hardness result for Bounded-Degree Deletion in case of unbounded d-values.

💡 Research Summary

The paper presents a broad generalization of the classic Nemhauser‑Trotter (NT) local optimization theorem, moving beyond its original linear‑programming‑based formulation for Vertex Cover to a purely combinatorial framework that applies to a wide class of graph‑editing and packing problems. The authors first motivate the need for an LP‑free approach, noting that the NT theorem’s reliance on LP duality limits its applicability and obscures the underlying structural reasons for its powerful kernelization results. They then introduce a new “local reduction” scheme that partitions a graph into three disjoint vertex sets: (i) a safe set A whose vertices already satisfy the target property, (ii) a forced‑deletion set B that must be removed in any optimal solution, and (iii) a residual set C whose status is undecided. The reduction rules are defined solely in terms of vertex degrees and adjacency, and they are applied iteratively until a fixed point is reached.

The core application of this framework is to the Bounded‑Degree Deletion (BDD) problem. For a fixed integer d ≥ 0, BDD asks for the smallest vertex set whose removal leaves a graph of maximum degree at most d; Vertex Cover corresponds to the special case d = 0. The authors prove a “degree‑bounded kernel theorem”: for any constant d, the iterative reduction yields a kernel with at most O(d·OPT) vertices, where OPT denotes the size of a minimum solution. The proof proceeds by constructing a bipartite graph between B and C, computing a maximum matching M, and showing that |B| ≤ 2·OPT and |C| ≤ d·|B|. Consequently the total number of vertices after reduction is bounded linearly in the optimum, establishing a linear‑size kernel for every constant d. This result matches the classic NT kernel for Vertex Cover but is obtained without any linear‑programming arguments.

Beyond BDD, the authors demonstrate that the same combinatorial machinery extends to a star‑packing problem. Given a parameter k, the task is to pack as many vertex‑disjoint stars each having at most k leaves. By interpreting each star’s center as a candidate for deletion and its leaves as matched partners, they reduce the packing instance to a BDD‑like structure. The resulting kernel size is O(k·OPT), showing that star packing also admits efficient preprocessing when k is constant.

The paper then turns to the parameterized complexity landscape when the degree bound d is part of the input. Using a parameter‑preserving reduction from SET‑COVER, the authors prove that BDD is W