Greedy Strategy in Optimization

• Greedy strategy is an iterative local optimization paradigm that selects at each step the feasible action offering the highest immediate gain without backtracking.
• It applies across various domains such as set and string submodular maximization, spectral optimization, and graph clustering, with performance sensitive to factors like curvature and feasibility constraints.
• Variants like batched, repeated, and selective greedy methods adapt the basic strategy to address issues such as non-monotonicity, sparsity, and cycling in complex optimization problems.

Searching arXiv for recent and foundational papers relevant to greedy strategies across optimization, submodular maximization, spectral optimization, sequential decision-making, and graph methods. Greedy strategy denotes an iterative optimization paradigm in which each step selects the action that is locally best with respect to the current state and objective, typically without backtracking. In the literature represented here, the same idea appears in markedly different forms: choosing the feasible element with largest marginal gain in submodular optimization, updating each matrix row against a current Perron vector in spectral-radius optimization, retaining the currently best processes in a multistage stochastic elimination problem, selecting the largest admissible coin in change-making, or merging the pair of graph clusters that most decreases a cut objective. Taken together, these works suggest that “greedy strategy” is not a single algorithm but a recurrent design pattern whose guarantees depend sharply on structure: submodularity, matroid feasibility, product families, independent increments, or special recurrence properties can make greedy near-optimal or exact, while non-monotonicity, sparsity, or combinatorial interactions can invalidate stronger claims (Liu et al., 2019, Protasov et al., 2018).

1. Canonical forms of the greedy rule

Across the cited literature, greedy strategy is always state-dependent and incremental. The update is chosen from the currently feasible moves, and the criterion is immediate objective improvement rather than global lookahead. This local character is explicit in set optimization, where one adds the feasible element with maximal marginal gain; in spectral-radius optimization over product families, where one replaces each row by the row maximizing its dot product with the current leading eigenvector; in change-making, where one repeatedly takes the largest denomination not exceeding the remaining amount; and in graph cut, where one merges the pair of clusters that most decreases the global cut objective (Liu et al., 2019, Protasov et al., 2018, Gupta et al., 2024, Nie et al., 2024).

Setting	Greedy move	Local criterion
Set submodular maximization	Add one feasible element	Maximize marginal gain
String optimization	Append one action	Maximize current string value
Perron eigenvalue optimization	Replace each row independently	Maximize or minimize $r^\top v_k$
Multistage stochastic selection	Retain current leaders	Highest observed values at time $t_j$
Coin change	Take one denomination repeatedly	Largest coin not exceeding the remainder
Graph cut	Merge two clusters	Largest decrease of the objective

In submodular set optimization, the canonical greedy rule under a cardinality or matroid constraint chooses the feasible element with maximum marginal gain. In string optimization, the analogous rule appends the action maximizing the current string objective. In the product-family Perron-eigenvalue problem, the greedy rule is row-separable: for maximization, given the current Perron vector $v_k$, each row is updated by

$$a_i^{(k+1)} \in \arg\max_{r\in\mathcal{R}_i} r^\top v_k.$$

In the toy multistage selection model with i.i.d. discrete-time processes and independent increments, the greedy rule keeps the top $n_j$ processes by their observed values at each observation time $t_j$. In the Greedy Coin Change problem, the greedy decomposition uses

$$x_m=\left\lfloor \frac{W}{c_m}\right\rfloor,\quad x_{m-1}=\left\lfloor \frac{W-x_m c_m}{c_{m-1}}\right\rfloor,$$

and so on in descending denomination order. In graph cut, GGC starts from singleton clusters and merges the neighboring pair with maximal decrease in the normalized-cut objective (Yang, 22 Jun 2025, Gupta et al., 2024, Nie et al., 2024).

A plausible implication is that greedy strategy is best understood through the geometry of the feasible family and the locality of the update map, not through a single universal performance theorem.

2. Greedy strategy in set and string submodular optimization

The most systematic theory in the supplied corpus concerns submodular optimization. A set function $f:2^V\to\mathbb{R}$ is submodular when it satisfies diminishing returns, and a polymatroid function is normalized, monotone, and submodular. Under a uniform matroid of rank $K$, the standard greedy set construction selects

$$j_t \in \arg\max_{j\in V\setminus G_{t-1}} f(j\mid G_{t-1}),$$

and yields the classical guarantee

$t_j$0

For the intersection of $t_j$1 matroids, the survey records the bound

$t_j$2

and for additive objectives under a matroid, greedy is optimal (Liu et al., 2019).

The same pattern extends to ordered actions. In finite-horizon string optimization, the greedy string

$t_j$3

is analyzed under $t_j$4-submodularity and $t_j$5-GO-concavity. These conditions are weaker than requiring full string submodularity beyond the horizon. Under them, greedy attains

$t_j$6

and the bound improves to

$t_j$7

when the curvature parameter $t_j$8 is introduced (Liu et al., 2015).

The survey also records parallel developments for string submodularity under string matroids and postfix-monotonicity. For polymatroid string functions, greedy again achieves a $t_j$9 bound under a uniform string matroid, and curvature refinements use backward-curvature quantities $v_k$0 or $v_k$1 to replace the classical $v_k$2 form by

$v_k$3

This establishes a close formal analogy between greedy strategies on sets and on sequences, while keeping the distinction that the latter depends on order and concatenation rather than inclusion alone (Liu et al., 2019).

3. Curvature, non-monotonicity, and batched greedy variants

A major refinement of greedy analysis replaces uniform worst-case constants by curvature-sensitive guarantees. For monotone submodular maximization under a matroid, the total curvature

$v_k$4

yields the tight bound

$v_k$5

and under a uniform matroid of rank $v_k$6,

$v_k$7

When the objective is only given on the matroid domain, the paper on improved curvature bounds defines partial curvature

$v_k$8

and shows that if an extension $v_k$9 satisfies $$a_i^{(k+1)} \in \arg\max_{r\in\mathcal{R}_i} r^\top v_k.$$0, then the same greedy guarantees hold with $$a_i^{(k+1)} \in \arg\max_{r\in\mathcal{R}_i} r^\top v_k.$$1 in place of $$a_i^{(k+1)} \in \arg\max_{r\in\mathcal{R}_i} r^\top v_k.$$2; this removes dependence on infeasible supersets from the bound statement (Liu et al., 2017).

Greedy behavior changes substantially in the non-monotone regime. “Greed is Good” emphasizes that classical greedy performs well for monotone submodular functions but poorly for non-monotone ones. To address this, the paper introduces Repeated Greedy for $$a_i^{(k+1)} \in \arg\max_{r\in\mathcal{R}_i} r^\top v_k.$$3-systems and Sample Greedy for $$a_i^{(k+1)} \in \arg\max_{r\in\mathcal{R}_i} r^\top v_k.$$4-extendible systems. Repeated Greedy invokes greedy $$a_i^{(k+1)} \in \arg\max_{r\in\mathcal{R}_i} r^\top v_k.$$5 times and achieves a $$a_i^{(k+1)} \in \arg\max_{r\in\mathcal{R}_i} r^\top v_k.$$6 approximation using $$a_i^{(k+1)} \in \arg\max_{r\in\mathcal{R}_i} r^\top v_k.$$7 function evaluations. Sample Greedy samples each element with probability $$a_i^{(k+1)} \in \arg\max_{r\in\mathcal{R}_i} r^\top v_k.$$8, runs one greedy pass, and achieves a $$a_i^{(k+1)} \in \arg\max_{r\in\mathcal{R}_i} r^\top v_k.$$9 approximation with only $n_j$0 function evaluations. The same paper proves lower bounds showing that no polynomial-time algorithm can achieve approximation ratio smaller than $n_j$1 for monotone submodular maximization over $n_j$2-extendible systems, and smaller than $n_j$3 for the linear case (Feldman et al., 2017).

Batching modifies both the objective geometry and the curvature parameter. In the $n_j$4-batch greedy strategy, one adds $n_j$5 elements jointly at each step. For a general matroid, the paper on $n_j$6-batch curvature defines

$n_j$7

and proves the harmonic bound

$n_j$8

For a uniform matroid with $n_j$9,

$t_j$0

Moreover, if $t_j$1 divides $t_j$2, then $t_j$3, so larger batches have better curvature-based bounds (Liu et al., 2015).

Taken together, these results show that greedy strategy does not admit a single approximation constant even within submodular optimization. The decisive quantities are monotonicity, curvature, the feasible system, and whether the algorithm is elementwise, repeated, sampled, or batched.

4. Exact optimality and model-specific greedy phenomena

Several papers establish exact greedy optimality, but only under highly structured assumptions. In the toy model of multistage elimination among $t_j$4 i.i.d. discrete-time processes with independent increments, the greedy policy retains the top $t_j$5 currently observed processes at each observation time $t_j$6. The main theorem states that this greedy selection algorithm maximizes

$t_j$7

over all admissible policies. The same source is explicit that the theorem concerns expected terminal value, not the success probability of identifying the globally best process at time $t_j$8 (Yang, 22 Jun 2025).

In product-family optimization of the Perron eigenvalue, the greedy rule is exact for finite and polyhedral row uncertainty sets when termination occurs with a positive Perron vector. The basic method can cycle on sparse families because multiple Perron eigenvectors and ties can create non-strict relaxations. The selective greedy method replaces the arbitrary leading eigenvector by the selected leading eigenvector

$t_j$9

which can be computed by the power method started from $$x_m=\left\lfloor \frac{W}{c_m}\right\rfloor,\quad x_{m-1}=\left\lfloor \frac{W-x_m c_m}{c_{m-1}}\right\rfloor,$$0. Theorem 2 states that selective greedy does not cycle, and the paper proves global linear convergence and local quadratic convergence. In numerical examples, matrices with minimal or maximal spectral radii in product families were found within a few iterations for dimensions up to $$x_m=\left\lfloor \frac{W}{c_m}\right\rfloor,\quad x_{m-1}=\left\lfloor \frac{W-x_m c_m}{c_{m-1}}\right\rfloor,$$1 (Protasov et al., 2018).

Change-making gives the classical example where greedy is intuitive but not universally correct. The Greedy Coin Change paper defines the decision problem GCC$$x_m=\left\lfloor \frac{W}{c_m}\right\rfloor,\quad x_{m-1}=\left\lfloor \frac{W-x_m c_m}{c_{m-1}}\right\rfloor,$$2: given target amount $$x_m=\left\lfloor \frac{W}{c_m}\right\rfloor,\quad x_{m-1}=\left\lfloor \frac{W-x_m c_m}{c_{m-1}}\right\rfloor,$$3, denominations $$x_m=\left\lfloor \frac{W}{c_m}\right\rfloor,\quad x_{m-1}=\left\lfloor \frac{W-x_m c_m}{c_{m-1}}\right\rfloor,$$4, and a queried coin $$x_m=\left\lfloor \frac{W}{c_m}\right\rfloor,\quad x_{m-1}=\left\lfloor \frac{W-x_m c_m}{c_{m-1}}\right\rfloor,$$5, determine whether $$x_m=\left\lfloor \frac{W}{c_m}\right\rfloor,\quad x_{m-1}=\left\lfloor \frac{W-x_m c_m}{c_{m-1}}\right\rfloor,$$6 belongs to the greedy decomposition. It proves that this problem is $$x_m=\left\lfloor \frac{W}{c_m}\right\rfloor,\quad x_{m-1}=\left\lfloor \frac{W-x_m c_m}{c_{m-1}}\right\rfloor,$$7-complete under log-space reductions, which implies that the greedy behavior itself is unlikely to be efficiently parallelizable in full generality. At the same time, the same source notes that greedy is optimal for many real-world canonical coin systems, whereas examples such as $$x_m=\left\lfloor \frac{W}{c_m}\right\rfloor,\quad x_{m-1}=\left\lfloor \frac{W-x_m c_m}{c_{m-1}}\right\rfloor,$$8 and $$x_m=\left\lfloor \frac{W}{c_m}\right\rfloor,\quad x_{m-1}=\left\lfloor \frac{W-x_m c_m}{c_{m-1}}\right\rfloor,$$9 show failure of optimality because greedy returns $f:2^V\to\mathbb{R}$0 instead of $f:2^V\to\mathbb{R}$1 (Gupta et al., 2024).

The paper on totally greedy sequences strengthens the positive side of this story. For second-order linear recurrences

$f:2^V\to\mathbb{R}$2

and for

$f:2^V\to\mathbb{R}$3

the resulting denomination sequences are totally greedy, meaning every finite prefix is a greedy coin system. This gives constructive infinite families where greedy change-making is exact for every prefix and every target amount (Pérez-Rosés, 2024).

A related robustness result appears in decision-making under complete uncertainty. In a zero-sum game between a decision-maker and adversarial Nature, where each product has $f:2^V\to\mathbb{R}$4 sampled ratings and the strategy chooses a product after observing the sample matrix $f:2^V\to\mathbb{R}$5, the greedy rule selects the product with highest observed average rating. For the binary case $f:2^V\to\mathbb{R}$6 and $f:2^V\to\mathbb{R}$7, the paper proves

$f:2^V\to\mathbb{R}$8

so greedy is minimax-optimal for worst-case regret in that setting; more generally, its regret converges to zero as $f:2^V\to\mathbb{R}$9 (Atanasov et al., 11 Feb 2025).

5. Learning, control, and experimental sciences

In reinforcement learning, greedy behavior commonly appears through $K$0-greedy exploration. The MARL paper on Semantic Epsilon Greedy keeps the classical rule

$K$1

but modifies exploration by clustering actions into semantic groups and applying a two-level $K$2-greedy policy: with probability $K$3, act greedily over the full action set; with probability $K$4, first sample a group uniformly and then apply lower-level $K$5-greedy within that group. Integrated into QMIX, this strategy largely outperformed vanilla QMIX on SMAC, with examples such as $K$6 versus $K$7 on MMM2, $K$8 versus $K$9 on 3s5z_vs_3s6z, and $$j_t \in \arg\max_{j\in V\setminus G_{t-1}} f(j\mid G_{t-1}),$$0 versus $$j_t \in \arg\max_{j\in V\setminus G_{t-1}} f(j\mid G_{t-1}),$$1 on 6h_vs_8z (Tse et al., 2022).

In Bayesian quantum parameter estimation with finitely many channel uses, adaptive greedy strategies optimize only the next batch given the current posterior, without preserving quantum memory across rounds. The paper formulates this via testers and semidefinite programming, and compares adaptive greedy with parallel, sequential, and general higher-order strategies. It finds that greedy can be strictly suboptimal in noiseless SU(2) estimation and in dissipative SU(2), while in the thermometry example for $$j_t \in \arg\max_{j\in V\setminus G_{t-1}} f(j\mid G_{t-1}),$$2 it matches the optimal sequential strategy across all interrogation times considered. A plausible implication is that greedy feedforward is competitive when the task lacks a coherent multi-use advantage, but not when entanglement or quantum memory changes the attainable Bayesian score (André et al., 10 Feb 2026).

Experimental optimization also supplies greedy variants that are strongly domain-specific. In wavefront shaping, the mutate greedy algorithm selects the best current phase mask $$j_t \in \arg\max_{j\in V\setminus G_{t-1}} f(j\mid G_{t-1}),$$3 and generates the next population solely by mutating it with an adaptive mutation rate driven by a Pearson correlation coefficient. With elitism, the best observed intensity is non-decreasing in the absence of measurement noise. In laboratory experiments at $$j_t \in \arg\max_{j\in V\setminus G_{t-1}} f(j\mid G_{t-1}),$$4 measurements, MGA achieved enhancement $$j_t \in \arg\max_{j\in V\setminus G_{t-1}} f(j\mid G_{t-1}),$$5 versus $$j_t \in \arg\max_{j\in V\setminus G_{t-1}} f(j\mid G_{t-1}),$$6 for GA, $$j_t \in \arg\max_{j\in V\setminus G_{t-1}} f(j\mid G_{t-1}),$$7 for CSA, and $$j_t \in \arg\max_{j\in V\setminus G_{t-1}} f(j\mid G_{t-1}),$$8 for SAA (Zhang et al., 2022).

These examples show that greedy strategy in learning and control is often hybridized: it is rarely pure local ascent alone, but instead combined with randomization, posterior updates, mutation, or factorized value learning.

6. Graph cut, clustering, and hierarchical aggregation

Graph-based optimization supplies some of the most direct greedy constructions. For multiway normalized cut, GGC starts from the partition in which each sample is its own cluster and repeatedly merges the two clusters that reduce the global objective the most. For the normalized-cut objective

$$j_t \in \arg\max_{j\in V\setminus G_{t-1}} f(j\mid G_{t-1}),$$9

the paper derives the exact merge gain

$t_j$00

proves $t_j$01, and hence proves monotonic decrease of the objective values. By restricting candidates to neighboring clusters and maintaining them in a red-black tree, GGC attains complexity

$t_j$02

which the paper describes as nearly linear in the number of samples. Because the merge rule is deterministic, the resulting solution is unique once tie-breaking is fixed (Nie et al., 2024).

For $t_j$03-center clustering with outliers, greedy reappears in a modified Gonzalez-style farthest-point scheme. Instead of always taking the farthest point, the algorithm samples from the farthest $t_j$04 points to avoid chasing outliers. In arbitrary metrics, the bi-criteria algorithm yields a $t_j$05 approximation with runtime $t_j$06; in doubling metrics, the same approach yields a $t_j$07 approximation and a coreset of size

$t_j$08

The empirical results reported near-optimal solutions and lower running times than existing methods (Ding et al., 2019).

Hierarchical Aggregation for GNNs presents a more cautionary greedy case. FullGreedy iteratively selects a cover $t_j$09 of size $t_j$10 maximizing the number of right-side receivers that request all elements of $t_j$11. A prior $t_j$12 guarantee is shown to be incorrect, and the paper provides a counterexample showing that one cannot hope for better than a $t_j$13 approximation factor for that greedy algorithm. The proved guarantee is weaker: for the shifted objective $t_j$14,

$t_j$15

in single-layer $t_j$16-HAG. The proof proceeds through a new connection to maximum matching in hypergraphs. The same paper also introduces PartialGreedy, which can outperform FullGreedy but loses any straightforward submodular interpretation (Porter et al., 2021).

7. Limitations, misconceptions, and complexity-theoretic boundaries

A recurring misconception is that greedy strategy is universally justified by the success of a few canonical cases. The supplied literature repeatedly rejects that conclusion. Classical greedy can fail on sparse matrix families by cycling, which motivated selective greedy for Perron optimization (Protasov et al., 2018). Classical greedy performs poorly for non-monotone submodular maximization, which motivated repeated and sampled variants (Feldman et al., 2017). The theorem in the stochastic independent-increments toy model optimizes expected terminal value, not success probability, so extrapolating it to best-arm identification would overstate the result (Yang, 22 Jun 2025). In HAG, the earlier $t_j$17 claim was simply false (Porter et al., 2021).

A second misconception is that greedy is always computationally trivial. Some greedy executions are cheap, but analyzing their behavior can be hard. The Greedy Coin Change paper shows that deciding whether a specific denomination appears in the greedy decomposition is $t_j$18-complete under log-space reductions (Gupta et al., 2024). In contrast, some greedy designs are attractive precisely because they avoid heavier subroutines: GGC avoids eigendecomposition plus $t_j$19-means in normalized cut, and selective greedy for product families usually converges in very few iterations even at large dimension (Nie et al., 2024, Protasov et al., 2018).

The overall picture is therefore sharply conditional. Greedy strategy is exact for some structured problems, admits curvature- or constraint-dependent approximation bounds for others, and can be undermined by non-submodularity, cycling, or combinatorial conflicts elsewhere. The strongest general statement supported by the cited work is not that greedy is globally reliable, but that it becomes technically powerful precisely when the problem exposes the right local-to-global structure.