최대 가중치 k와 l 희소 서브그래프를 찾는 2차 시간 알고리즘

📝 Abstract

The family of $(k, \ell) $-sparse graphs, introduced by Lorea, plays a central role in combinatorial optimization and has a wide range of applications, particularly in rigidity theory. A key algorithmic challenge is to compute a maximum-weight $(k, \ell) $-sparse subgraph of a given edge-weighted graph. Although prior approaches have long provided an $O(nm) $-time solution, a previously proposed $O(n^2 + m)$ method was based on an incorrect analysis, leaving open whether this bound is achievable. We answer this question affirmatively by presenting the first $O(n^2 + m) $-time algorithm for computing a maximum-weight $(k, \ell) $-sparse subgraph, which combines an efficient data structure with a refined analysis. This quadratic-time algorithm enables faster solutions to key problems in rigidity theory, including computing minimum-weight redundantly rigid and globally rigid subgraphs. Further applications include enumerating non-crossing minimally rigid frameworks and recognizing kinematic joints. Our implementation of the proposed algorithm is publicly available online.

💡 Analysis

The family of $(k, \ell) $-sparse graphs, introduced by Lorea, plays a central role in combinatorial optimization and has a wide range of applications, particularly in rigidity theory. A key algorithmic challenge is to compute a maximum-weight $(k, \ell) $-sparse subgraph of a given edge-weighted graph. Although prior approaches have long provided an $O(nm) $-time solution, a previously proposed $O(n^2 + m)$ method was based on an incorrect analysis, leaving open whether this bound is achievable. We answer this question affirmatively by presenting the first $O(n^2 + m) $-time algorithm for computing a maximum-weight $(k, \ell) $-sparse subgraph, which combines an efficient data structure with a refined analysis. This quadratic-time algorithm enables faster solutions to key problems in rigidity theory, including computing minimum-weight redundantly rigid and globally rigid subgraphs. Further applications include enumerating non-crossing minimally rigid frameworks and recognizing kinematic joints. Our implementation of the proposed algorithm is publicly available online.

📄 Content

Quadratic-Time Algorithm for the Maximum-Weight (k, ℓ)-Sparse Subgraph Problem Bence De´ak1 P´eter Madarasi2 Abstract The family of (k, ℓ)-sparse graphs, introduced by Lorea, plays a central role in combinatorial optimization and has a wide range of applications, particularly in rigidity theory. A key algorithmic challenge is to compute a maximum-weight (k, ℓ)-sparse subgraph of a given edge-weighted graph. Although prior approaches have long provided an O(nm)-time solution, a previously proposed O(n2 + m) method was based on an incorrect analysis, leaving open whether this bound is achievable. We answer this question affirmatively by presenting the first O(n2 + m)-time algorithm for computing a maximum-weight (k, ℓ)-sparse subgraph, which combines an efficient data structure with a refined analysis. This quadratic-time algorithm enables faster solutions to key problems in rigidity theory, including computing minimum-weight redundantly rigid and globally rigid subgraphs. Further applications include enumerating non- crossing minimally rigid frameworks and recognizing kinematic joints. Our implementation of the proposed algorithm is publicly available online. Keywords: (k, ℓ)-sparse graphs, pebble game algorithm, rigidity theory, efficient data structures 1 Introduction Throughout this paper, let k and ℓbe non-negative integers with ℓ< 2k, let G = (V, E) be a loopless multigraph with n = |V | vertices and m = |E| edges, and assume that the edges e1, . . . , em are sorted in non-increasing order with respect to a given weight function. The graph G = (V, E) is called (k, ℓ)-sparse if, for each subset X ⊆V , the number i(X) of edges induced by X is at most max{k|X| −ℓ, 0}. Furthermore, if G is (k, ℓ)-sparse and has exactly max{k|V | −ℓ, 0} edges, then we say that G is (k, ℓ)-tight. A graph is (k, ℓ)-spanning if it contains a (k, ℓ)-tight subgraph that spans the entire vertex set. A (k, ℓ)-block of a (k, ℓ)-sparse graph is a subset X ⊆V that induces a (k, ℓ)-tight subgraph. A (k, ℓ)-component is an inclusion-wise maximal (k, ℓ)-block. These definitions form the foundation of a rich theory, closely connected to matroids and rigidity, and they naturally lead to fundamental optimization problems that are the focus of this paper. The concept of (k, ℓ)-sparse graphs was first introduced by Lorea [1] as part of his work on matroidal families. Since then, these graphs have been the subject of extensive research, with numerous applications in various areas of mathematics and computer science. For example, (k, k)-tight graphs appeared in the work of Nash-Williams [2] and Tutte [3] as a characterization of graphs that can be decomposed into k edge-disjoint spanning trees. A classical result in rigidity theory, due to Laman [4], characterizes generic minimally rigid bar-joint frameworks in the plane as (2, 3)-tight graphs, while the rigid graphs correspond to the (2, 3)-spanning graphs. For a comprehensive overview of rigidity theory, see [5, 6]. A wide range of optimization and decision problems can be reduced to the maximum-size (k, ℓ)-sparse subgraph problem, where k and ℓare fixed non-negative integers determined by the specific task. The goal is to find a (k, ℓ)- sparse subgraph of a given graph G = (V, E) that contains the maximum number of edges. A natural generalization is the maximum-weight (k, ℓ)-sparse subgraph problem, where the goal is to find a (k, ℓ)-sparse subgraph with maximum total weight under a given weight function on the edges. Many classical problems in graph theory can be viewed as special cases of this framework — for example, finding a maximum-weight spanning tree, a maximum- weight subgraph that can be decomposed into k forests, or a maximum-weight rigid spanning subgraph in the plane. 1Department of Operations Research, E¨otv¨os Lor´and University, Budapest, Hungary. E-mail: deakbence2002@gmail.com 2HUN-REN Alfr´ed R´enyi Institute of Mathematics, and Department of Operations Research, E¨otv¨os Lor´and University, Budapest, Hungary. E-mail: madarasip@staff.elte.hu 1 arXiv:2511.20882v1 [cs.DS] 25 Nov 2025 Both the maximum-size and the maximum-weight (k, ℓ)-sparse subgraph problems can be solved using the pebble game algorithms introduced in [7, 8, 9, 10, 11]. The formulation in [8] provides a more convenient orientation-based view of these algorithms, which we adopt throughout this paper. The naive implementation of these algorithms runs in O(nm) time, where n and m denote the number of vertices and edges, respectively. A widely accepted approach in the literature claimed to improve this to O(n2 + m) by employing a more sophisticated data structure [10, 11], building on ideas in [8]. In recent years, however, several researchers have pointed out that the running time analysis of the improved method is flawed [12, 13], and the question of whether such a time bound is truly achievable has remained open. In our paper, we provide a positive answer to this question: we present an algorithm that solv

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