Decremental Greedy Algorithm

Updated 6 July 2026

Decremental greedy algorithm is a deletion-based iterative method that starts from a full structure and removes elements based on local quality rules to achieve optimal bottleneck subsets.
It underpins diverse applications, from geometric and graph-theoretic problems to dynamic graph maintenance and chip-firing on metric graphs, ensuring efficient performance.
The method guarantees order-independence and exhibits an antimatroid structure, with extensions that analyze transfinite behavior under certain conditions.

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 (Eppstein, 6 Jul 2025). 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 (Dudeja, 2023, Nasre et al., 2014, Backman, 2012).

1. Formal bottleneck-subset framework

The most explicit arXiv formalization appears in "Decremental Greedy Polygons and Polyhedra Without Sharp Angles" (Eppstein, 6 Jul 2025). 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 (Eppstein, 6 Jul 2025).

For any nonempty subset $S$ , the bottleneck value is defined as

$Q(S)=\min_{x\in S} q(x,S),$

with

$q(x,S)$ 0

so the empty set cannot be optimal (Eppstein, 6 Jul 2025). The optimization goal is to find a nonempty subset maximizing $q(x,S)$ 1; such a subset is called a bottleneck subset, and an element $q(x,S)$ 2 with $q(x,S)$ 3 is a bottleneck element (Eppstein, 6 Jul 2025).

Within this framework, the decremental greedy algorithm is stated as follows (Eppstein, 6 Jul 2025):

Initialize two sets $q(x,S)$ 4 and $q(x,S)$ 5 to $q(x,S)$ 6.
While $q(x,S)$ $q (x, S)$ 7:
- If $q(x,S)$ 8, set $q(x,S)$ 9.
- Find any $x\in S\subset U$ 0 such that
$x\in S\subset U$ 1

Remove $x\in S\subset U$ 2 from $x\in S\subset U$ 3.

Return $x\in S\subset U$ 4.

The same paper also gives a known- $x\in S\subset U$ 5 variant: if the optimum bottleneck value $x\in S\subset U$ 6 is already known, repeatedly delete any element with

$x\in S\subset U$ 7

until all remaining elements have quality at least $x\in S\subset U$ 8 (Eppstein, 6 Jul 2025).

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 (Eppstein, 6 Jul 2025) 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- $x\in S\subset U$ 9 algorithm always find and return that maximal bottleneck subset (Eppstein, 6 Jul 2025).

The proof proceeds by letting $Q(S)=\min_{x\in S} q(x,S)$ 0 be the optimal bottleneck value and considering the union $Q(S)=\min_{x\in S} q(x,S)$ 1 of all bottleneck subsets. The paper proves that $Q(S)=\min_{x\in S} q(x,S)$ 2 is itself a bottleneck subset. For every element $Q(S)=\min_{x\in S} q(x,S)$ 3, there exists an optimal bottleneck subset $Q(S)=\min_{x\in S} q(x,S)$ 4 containing $Q(S)=\min_{x\in S} q(x,S)$ 5, and monotonicity yields

$Q(S)=\min_{x\in S} q(x,S)$ 6

Therefore $Q(S)=\min_{x\in S} q(x,S)$ 7, so $Q(S)=\min_{x\in S} q(x,S)$ 8 is optimal; because $Q(S)=\min_{x\in S} q(x,S)$ 9 contains every bottleneck subset, it is the unique maximal bottleneck subset (Eppstein, 6 Jul 2025).

The deletion argument is equally structural. If $U$ 0 is still larger than the maximal optimal subset, then $U$ 1 must contain some element with quality $U$ 2; by monotonicity, such an element cannot belong to $U$ 3. Hence the algorithms only remove elements outside $U$ 4 until $U$ 5 is reached (Eppstein, 6 Jul 2025). 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 (Eppstein, 6 Jul 2025). 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 (Eppstein, 6 Jul 2025). 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 (Eppstein, 6 Jul 2025).

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 (Eppstein, 6 Jul 2025) 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 $U$ 6 in the current subset $U$ 7, the quality is defined by

$U$ 8

if $U$ 9 is an interior point of the convex hull of $q(x,S)$ 0, and as the interior angle at $q(x,S)$ 1 if $q(x,S)$ 2 is a convex-hull vertex (Eppstein, 6 Jul 2025). 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 $q(x,S)$ 3-time algorithm (Eppstein, 6 Jul 2025).

For the max-min-solid-angle polyhedron in $q(x,S)$ 4, the paper again shows that an optimal solution can be taken to be the convex hull. Interior points are assigned quality $q(x,S)$ 5 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

$q(x,S)$ 6

using a decremental explicit 3D convex hull structure (Eppstein, 6 Jul 2025).

The same paper also treats bottleneck cycles in graphs. For an edge in a graph $q(x,S)$ 7, the quality is its weight if it still belongs to some cycle in the remaining graph, and $q(x,S)$ 8 if it cannot lie on any cycle (Eppstein, 6 Jul 2025). In directed graphs, the decremental greedy algorithm repeatedly removes any vertex with in-degree $q(x,S)$ 9 or out-degree $x\in S\subset U$ 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 (Eppstein, 6 Jul 2025). For presorted or integer weights, the algorithm is linear time, and with the Gabow–Tarjan clustering idea the paper obtains

$x\in S\subset U$ 1

for comparison-based directed bottleneck cycles; the same bound is given for mixed graphs after a reduction eliminating u-turns (Eppstein, 6 Jul 2025).

The paper identifies graph degeneracy as a canonical special case. Here $x\in S\subset U$ 2 is the set of vertices and

$x\in S\subset U$ 3

the degree of $x\in S\subset U$ 4 in the induced subgraph on $x\in S\subset U$ 5. Then the bottleneck value of $x\in S\subset U$ 6 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 (Eppstein, 6 Jul 2025). 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 (Eppstein, 6 Jul 2025). An important nuance noted in (Eppstein, 6 Jul 2025) 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" (Dudeja, 2023), the problem is to maintain a $x\in S\subset U$ 7-approximate maximum weight matching under adversarial edge deletions. If $x\in S\subset U$ 8 is the current graph and $x\in S\subset U$ 9 denotes the weight of a maximum weight matching in $x\in S\subset T\subset U \implies q(x,S)\le q(x,T).$ 0, the maintained integral matching $x\in S\subset T\subset U \implies q(x,S)\le q(x,T).$ 1 satisfies

$x\in S\subset T\subset U \implies q(x,S)\le q(x,T).$ 2

with high probability, against an adaptive adversary (Dudeja, 2023). The main theorem gives a decremental algorithm with total update time

$x\in S\subset T\subset U \implies q(x,S)\le q(x,T).$ 3

where $x\in S\subset T\subset U \implies q(x,S)\le q(x,T).$ 4 is the ratio between the maximum and minimum edge weight (Dudeja, 2023).

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 (Dudeja, 2023). It maintains a robust fractional matching $x\in S\subset T\subset U \implies q(x,S)\le q(x,T).$ 5, and when deletions destroy part of this structure, it either repairs it by increasing capacities on a carefully chosen set of bottleneck edges $x\in S\subset T\subset U \implies q(x,S)\le q(x,T).$ 6, or extracts a sparse subgraph supporting a near-optimal matching (Dudeja, 2023). A key subroutine, WeightedM-or- $x\in S\subset T\subset U \implies q(x,S)\le q(x,T).$ 7, either returns a fractional matching $x\in S\subset T\subset U \implies q(x,S)\le q(x,T).$ 8 with

$x\in S\subset T\subset U \implies q(x,S)\le q(x,T).$ 9

or a bottleneck set $S$ 0 such that

$S$ 1

and every near-optimal matching must route a nontrivial amount of weight through $S$ 2 (Dudeja, 2023).

Its amortized analysis is driven by the potential

$S$ 3

where $S$ 4 is the family of matchings of weight at least $S$ 5 (Dudeja, 2023). If WeightedM-or- $S$ 6 returns a bottleneck set, increasing capacities on $S$ 7 raises the potential by $S$ 8, while the potential is always $S$ 9 in a phase, so the number of bottleneck updates is only $Q(S)=\min_{x\in S} q(x,S),$ 0 per phase (Dudeja, 2023). 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" (Nasre et al., 2014) 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

$Q(S)=\min_{x\in S} q(x,S),$ 1

time per update, where $Q(S)=\min_{x\in S} q(x,S),$ 2 bounds the number of edges that lie on shortest paths through any given vertex (Nasre et al., 2014). 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 $Q(S)=\min_{x\in S} q(x,S),$ 3 and $Q(S)=\min_{x\in S} q(x,S),$ 4 during cleanup and fixup (Nasre et al., 2014). The core invariant is that the first extracted set from $Q(S)=\min_{x\in S} q(x,S),$ 5 for a pair $Q(S)=\min_{x\in S} q(x,S),$ 6 has weight $Q(S)=\min_{x\in S} q(x,S),$ 7, the new shortest distance (Nasre et al., 2014). 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" (Backman, 2012). 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 (Backman, 2012).

A divisor on a metric graph $Q(S)=\min_{x\in S} q(x,S),$ 8 is a formal integer combination

$Q(S)=\min_{x\in S} q(x,S),$ 9

and chip-firing replaces $q(x,S)$ 00 by

$q(x,S)$ 01

for a piecewise affine function $q(x,S)$ 02 with integer slopes (Backman, 2012). A divisor is $q(x,S)$ 03-reduced if it is nonnegative away from $q(x,S)$ 04, and every firing toward $q(x,S)$ 05 causes some point to go negative (Backman, 2012). The greedy reduction is defined as a sequence

$q(x,S)$ 06

where each $q(x,S)$ 07 is a maximal legal firing for $q(x,S)$ 08 (Backman, 2012). “Legal” means the firing keeps the divisor nonnegative away from $q(x,S)$ 09, and “maximal” means the firing is extended as far as possible toward $q(x,S)$ 10 until some chip hits a combinatorial vertex and can no longer continue along that cut (Backman, 2012).

The paper’s central construction simulates the Euclidean algorithm by encoding two lengths $q(x,S)$ 11 in chip positions. Repeated maximal legal firings effectively replace $q(x,S)$ 12 by $q(x,S)$ 13, and if $q(x,S)$ 14, the process never terminates (Backman, 2012). At the same time, the total firing lengths satisfy

$q(x,S)$ 15

so the infinite process is geometrically well-defined (Backman, 2012).

A key lemma states that if $q(x,S)$ 16 is an infinite sequence of maximal legal firings reducing $q(x,S)$ 17, then the series $q(x,S)$ 18 converges and the greedy reduction has a well defined limit (Backman, 2012). This allows the greedy reduction algorithm to be treated as a transfinite algorithm, with running time analyzed via ordinals. The paper proves that for every $q(x,S)$ 19, there exists a metric graph and divisor whose greedy reduction takes time at least $q(x,S)$ 20, while any greedy reduction takes at most

$q(x,S)$ 21

steps; therefore the worst-case running time is

$q(x,S)$ 22

(Backman, 2012)

This is a notable counterpoint to finite antimatroid-style decremental greedy algorithms. The local rule “take the maximal legal move” is still greedily natural, but greediness here does not imply finite termination. Instead, infinite descent is controlled through convergence and ordinal complexity.

6. Scope, variants, and conceptual boundaries

The literature here indicates that “decremental greedy algorithm” is not a single universally standardized object. At least three distinct uses occur.

First, in the strict formal sense of (Eppstein, 6 Jul 2025), it denotes an optimal deletion process for monotone bottleneck subset problems. This is the clearest theorem-driven definition, and it comes with a unique maximal optimal subset, deletion-order independence, and an antimatroid interpretation (Eppstein, 6 Jul 2025).

Second, in decremental dynamic graph algorithms, the term can describe greedy-like maintenance rather than literal subset deletion. The weighted matching algorithm of (Dudeja, 2023) preserves a robust fractional structure, reacts only when enough weight has been lost, and either repairs a small set of bottleneck edges or rounds a sparse certificate. The APASP algorithm of (Nasre et al., 2014) processes affected path tuples in increasing weight order. These are decremental because the update sequence is deletion-only or weight-increase-only, and greedy-like because they preserve only the most valuable or lightest-first structure rather than recomputing globally.

Third, in metric-graph divisor reduction, decremental greediness means repeatedly applying the maximal legal move toward a target vertex (Backman, 2012). The same local principle can generate nontermination and transfinite behavior.

The boundaries of the notion are also visible in nearby papers. "Incremental Submodular Maximization: Better Than Greedy" (Bienkowski et al., 26 Jun 2026) explicitly concerns increasing cardinality constraints rather than decreasing ones, and states that it does not discuss decremental algorithms directly (Bienkowski et al., 26 Jun 2026). "A Greedy Version of the Frame Algorithm" (Johnson, 23 Jun 2025) is greedy in an adaptive step-size sense, not decremental in a deletion sense (Johnson, 23 Jun 2025). "Greedy Selection under Independent Increments: A Toy Model Analysis" (Yang, 22 Jun 2025) studies iterative elimination and proves that retaining the top observed values at each stage is optimal under i.i.d. independent-increment assumptions, but this is a staged filtering model rather than a monotone bottleneck subset framework (Yang, 22 Jun 2025). These contrasts help delimit the core use of the term.

A plausible synthesis is that decremental greedy algorithms are characterized less by any single proof technique than by an orientation: begin with a rich or complete structure, remove or discard elements over time, and exploit monotonicity so that deletions, once justified, remain justified. In bottleneck subset problems, that monotonicity yields exact optimality (Eppstein, 6 Jul 2025). In dynamic maintenance problems, it yields efficient repair and amortization under deletions (Dudeja, 2023, Nasre et al., 2014). In metric chip-firing, it yields a maximal-move process whose complexity is ordinal rather than finite (Backman, 2012).