--- title: Decremental Greedy Algorithm url: https://www.emergentmind.com/topics/decremental-greedy-algorithm type: topic --- # Decremental Greedy Algorithm Searching arXiv for recent and foundational papers on decremental greedy algorithms and closely related decremental greedy-style methods. A decremental greedy algorithm is an iterative method that starts from a full structure and repeatedly deletes elements according to a local rule, while tracking a global objective over the surviving set. In the literature represented here, the term has both a narrow and a broad usage. In its most explicit formalization, it denotes a deletion-based algorithm for a monotone bottleneck subset problem: given a finite universe \(U\), a quality function \(q(x,S)\) defined for \(x\in S\subset U\), and a bottleneck objective \(Q(S)=\min_{x\in S} q(x,S)\), the algorithm removes elements whose current quality is no better than the best bottleneck value already seen, and is guaranteed to return the unique maximal optimal subset [2507.04538]. More broadly, several decremental dynamic algorithms are “greedy-like” in the sense that they react to deletions by preserving only high-value or low-weight structure, processing affected objects in best-first order, or taking maximal legal local moves instead of rebuilding from scratch [2312.08996], [1411.4073], [1204.3647]. ## 1. Formal bottleneck-subset framework The most explicit arXiv formalization appears in "Decremental Greedy Polygons and Polyhedra Without Sharp Angles" [2507.04538]. The paper defines a monotone bottleneck subset problem by specifying a finite universe \(U\) and a quality function \[ q(x,S) \] for \(x\in S\subset U\), subject to the monotonicity condition \[ x\in S\subset T\subset U \implies q(x,S)\le q(x,T). \] Thus, when elements are removed from the ambient subset, the quality of any remaining element can only stay the same or get worse [2507.04538]. For any nonempty subset \(S\), the bottleneck value is defined as \[ Q(S)=\min_{x\in S} q(x,S), \] with \[ Q(\emptyset)=-\infty, \] so the empty set cannot be optimal [2507.04538]. The optimization goal is to find a nonempty subset maximizing \(Q(S)\); such a subset is called a bottleneck subset, and an element \(x\in S\) with \(q(x,S)=Q(S)\) is a bottleneck element [2507.04538]. Within this framework, the decremental greedy algorithm is stated as follows [2507.04538]: 1. Initialize two sets \(S\) and \(T\) to \(U\). 2. While \(S\neq\emptyset\): - If \(Q(S)>Q(T)\), set \(T=S\). - Find any \(x\) such that \[ q(x,S)\le Q(T). \] - Remove \(x\) from \(S\). 3. Return \(T\). The same paper also gives a known-\(\beta\) variant: if the optimum bottleneck value \(\beta\) is already known, repeatedly delete any element with \[ q(x,S)<\beta \] until all remaining elements have quality at least \(\beta\) [2507.04538]. This formulation makes the decremental character precise. The algorithm does not build a solution by accretion. It starts from the full universe and prunes elements that are provably irrelevant to any better solution than the best one already recorded. A plausible implication is that decremental greedy algorithms are most naturally associated with objectives whose value is controlled by the weakest surviving component rather than by additive accumulation. ## 2. Optimality theorem and antimatroid structure The central theorem in [2507.04538] states that for every monotone bottleneck subset problem, there is a unique maximal bottleneck subset, and regardless of tie-breaking or deletion choices, both the decremental greedy algorithm and the known-\(\beta\) algorithm always find and return that maximal bottleneck subset [2507.04538]. The proof proceeds by letting \(\beta\) be the optimal bottleneck value and considering the union \(M\) of all bottleneck subsets. The paper proves that \(M\) is itself a bottleneck subset. For every element \(x\in M\), there exists an optimal bottleneck subset \(X\subseteq M\) containing \(x\), and monotonicity yields \[ q(x,M)\ge q(x,X)\ge \beta. \] Therefore \(Q(M)\ge \beta\), so \(M\) is optimal; because \(M\) contains every bottleneck subset, it is the unique maximal bottleneck subset [2507.04538]. The deletion argument is equally structural. If \(V\supset M\) is still larger than the maximal optimal subset, then \(V\) must contain some element with quality \(<\beta\); by monotonicity, such an element cannot belong to \(M\). Hence the algorithms only remove elements outside \(M\) until \(M\) is reached [2507.04538]. In this sense, greediness is not justified by exchange arguments typical of matroid algorithms, but by a one-sided persistence property: once an element is bad enough to be deletable, further deletions cannot rehabilitate it. The paper emphasizes that the set of possible deletion sequences forms an antimatroid [2507.04538]. Once an element becomes removable, it stays removable until it is removed, and removability depends only on the set of removed elements, not on their order [2507.04538]. It also remarks that this decremental greedy framework is not the same as matroid-greedy algorithms or Dijkstra-like algorithms, because in those settings element priority typically improves as the algorithm progresses, whereas here quality only gets worse [2507.04538]. This antimatroid interpretation is one of the clearest conceptual characterizations of decremental greedy behavior in the available literature. It suggests that the essential combinatorial invariant is monotone availability of deletions rather than monotone growth of feasible partial solutions. ## 3. Geometric and graph-theoretic instances The framework of [2507.04538] is instantiated in several geometric and graph problems. For the max-min-angle polygon in the plane, the paper proves that there exists an optimal polygon that is convex. For a point \(x\) in the current subset \(S\subset\mathbb{R}^2\), the quality is defined by \[ q(x,S)=2\pi \] if \(x\) is an interior point of the convex hull of \(S\), and as the interior angle at \(x\) if \(x\) is a convex-hull vertex [2507.04538]. The decremental greedy algorithm starts with the convex hull of all points, repeatedly deletes the hull vertex with the sharpest angle, tracks the best hull angle seen so far, and returns the hull corresponding to the best recorded stage; the paper gives an \(O(n\log n)\)-time algorithm [2507.04538]. For the max-min-solid-angle polyhedron in \(\mathbb{R}^3\), the paper again shows that an optimal solution can be taken to be the convex hull. Interior points are assigned quality \(2\pi\) conceptually as a best possible placeholder, while hull vertices are assigned their solid angle. The algorithm maintains a dynamic 3D convex hull, repeatedly deletes the hull vertex with the smallest solid angle, records the best hull encountered, and returns the hull at the best stage, with total time \[ O(n^2) \] using a decremental explicit 3D convex hull structure [2507.04538]. The same paper also treats bottleneck cycles in graphs. For an edge in a graph \(G\), the quality is its weight if it still belongs to some cycle in the remaining graph, and \(-\infty\) if it cannot lie on any cycle [2507.04538]. In directed graphs, the decremental greedy algorithm repeatedly removes any vertex with in-degree \(0\) or out-degree \(0\), if one exists; otherwise it removes a minimum-weight edge and remembers its weight. The last minimum-weight edge removed in this way is guaranteed to lie on a cycle of heavier edges, yielding a bottleneck cycle [2507.04538]. For presorted or integer weights, the algorithm is linear time, and with the Gabow–Tarjan clustering idea the paper obtains \[ O(m\log^* m) \] for comparison-based directed bottleneck cycles; the same bound is given for mixed graphs after a reduction eliminating u-turns [2507.04538]. The paper identifies graph degeneracy as a canonical special case. Here \(U\) is the set of vertices and \[ q(x,S)=\deg_{G[S]}(x), \] the degree of \(x\) in the induced subgraph on \(S\). Then the bottleneck value of \(S\) is its minimum degree, the optimal bottleneck subset is the induced subgraph with maximum minimum degree, and the bottleneck value is the graph’s degeneracy [2507.04538]. The Matula–Beck algorithm repeatedly removes a minimum-degree vertex until the graph is empty, and the degeneracy is the maximum minimum degree encountered during this process [2507.04538]. An important nuance noted in [2507.04538] is that the general framework allows removal of any vertex with degree at most the current quality bound, not just the current minimum-degree vertex. Together, these instances show that decremental greedy algorithms are especially natural when feasible solutions are themselves subsets of a full structure and the objective is governed by a worst local feature such as minimum angle, minimum solid angle, minimum cycle edge weight, or minimum induced degree. ## 4. Decremental greedy-like maintenance in dynamic graphs Several dynamic graph papers in the supplied corpus do not present a decremental greedy algorithm in the formal bottleneck-subset sense, but they use deletion-only dynamics together with greedy-like maintenance rules. In "Decremental Matching in General Weighted Graphs" [2312.08996], the problem is to maintain a \((1-\varepsilon)\)-approximate maximum weight matching under adversarial edge deletions. If \(G\) is the current graph and \(\mathrm{mwm}(G)\) denotes the weight of a maximum weight matching in \(G\), the maintained integral matching \(M\) satisfies \[ w(M)\ge (1-\varepsilon)\cdot \mathrm{mwm}(G) \] with high probability, against an adaptive adversary [2312.08996]. The main theorem gives a decremental algorithm with total update time \[ O_{\varepsilon}(m\cdot (\log n)\cdot \log R), \] where \(R\) is the ratio between the maximum and minimum edge weight [2312.08996]. The paper states that the algorithm is not a single greedy matching algorithm in the usual static sense, but uses a greedy-like maintenance structure inside a decremental framework [2312.08996]. It maintains a robust fractional matching \(\vec{x}\), and when deletions destroy part of this structure, it either repairs it by increasing capacities on a carefully chosen set of bottleneck edges \(E^*\), or extracts a sparse subgraph supporting a near-optimal matching [2312.08996]. A key subroutine, WeightedM-or-\(E^*\), either returns a fractional matching \(\vec{x}\) with \[ \sum_{e\in E} w(e)\,x(e)\ge (1-2\varepsilon)\cdot \mathrm{mwm}(G), \] or a bottleneck set \(E^*\) such that \[ \sum_{e\in E^*} w(e)\,\kappa(e)=O(\mathrm{mwm}(G)\log n), \] and every near-optimal matching must route a nontrivial amount of weight through \(E^*\) [2312.08996]. Its amortized analysis is driven by the potential \[ \Pi(G,\kappa)=\min_{M\in\mathcal M}\sum_{e\in M} c(e)\,w(e), \qquad c(e)=\log(n\cdot \kappa(e)), \] where \(\mathcal M\) is the family of matchings of weight at least \((1-\varepsilon)\mathrm{mwm}(G)\) [2312.08996]. If WeightedM-or-\(E^*\) returns a bottleneck set, increasing capacities on \(E^*\) raises the potential by \(\Omega(\mu)\), while the potential is always \(O(\mu\log n)\) in a phase, so the number of bottleneck updates is only \(O(\log n)\) per phase [2312.08996]. This suggests a broader notion of decremental greediness: expensive repair is postponed until enough structure has been lost, and then only a compact high-impact part of the graph is updated. "Decremental All-Pairs ALL Shortest Paths and Betweenness Centrality" [1411.4073] gives another related pattern. The update model allows deletion of a vertex, or weight increases on edges incident to a vertex, and the algorithm runs in amortized \[ O(\vstar^2 \cdot \log n) \] time per update, where \(\vstar\) bounds the number of edges that lie on shortest paths through any given vertex [1411.4073]. The paper explicitly says that the algorithm is not greedy in the usual combinatorial sense of choosing a locally best edge or path and committing to it, but is greedy in a best-first maintenance sense: it processes shortest-path-related structures in nondecreasing weight order, using heaps \(H_c\) and \(H_f\) during cleanup and fixup [1411.4073]. The core invariant is that the first extracted set from \(H_f\) for a pair \((x,y)\) has weight \(d'(x,y)\), the new shortest distance [1411.4073]. Here the greedy principle is best-first reconstruction after monotone damage. ## 5. Maximal legal moves and transfinite behavior A different decremental greedy phenomenon appears in "Infinite Reduction of Divisors on Metric Graphs" [1204.3647]. The paper studies the greedy reduction algorithm for divisors on metric graphs and shows that, unlike the discrete-graph case, it need not terminate after finitely many steps [1204.3647]. A divisor on a metric graph \(\Gamma\) is a formal integer combination \[ D = \sum_{p \in \Gamma} D(p)\,p, \] and chip-firing replaces \(D\) by \[ D - Q(f) \] for a piecewise affine function \(f\) with integer slopes [1204.3647]. A divisor is \(q\)-reduced if it is nonnegative away from \(q\), and every firing toward \(q\) causes some point to go negative [1204.3647]. The greedy reduction is defined as a sequence \[ D = D_0,\qquad D_i = D_{i-1} - Q(f_{i-1}), \] where each \(Q(f_{i-1})\) is a maximal legal firing for \(D_{i-1}\) [1204.3647]. “Legal” means the firing keeps the divisor nonnegative away from \(q\), and “maximal” means the firing is extended as far as possible toward \(q\) until some chip hits a combinatorial vertex and can no longer continue along that cut [1204.3647]. The paper’s central construction simulates the Euclidean algorithm by encoding two lengths \(a