1-Guess Greedy in Budgeted k-Submodular Maximization

• 1-Guess Greedy is a combinatorial approximation algorithm that starts by guessing a singleton from an optimal solution and then applies a density-greedy completion rule.
• It achieves a 1/2-approximation for monotone cases and a 1/3-approximation for non-monotone cases, with an efficient nearly O(n²k²) runtime via thresholding.
• Designed for the budgeted k-submodular maximization problem, the algorithm balances exhaustive singleton enumeration with effective greedy selection across disjoint sets.

1-Guess Greedy is a combinatorial approximation algorithm for budgeted $k$-submodular maximization that first guesses one appropriate singleton assignment from an optimal solution and then continues with a density-greedy completion rule. In the formulation studied for the budgeted $k$-submodular maximization problem, the algorithm specializes the more general $q$-Guess Greedy scheme to the case $q=1$: it enumerates feasible support-size-one partial solutions, runs greedy from each guess, and returns the best outcome. Its main known guarantees are a $\frac12$-approximation for the monotone case and a $\frac13$-approximation for the non-monotone case, with a nearly $\tilde O(n^2k^2)$-time implementation obtainable via thresholding (Wang, 17 Jul 2025).

1. Formal problem setting

The algorithm is defined for $k$-submodular functions on $k$ disjoint sets. For a ground set $V=\{e_1,\dots,e_n\}$, the feasible domain is

$k$0

An element may therefore be unassigned or assigned to exactly one of $k$1 labels/components. A function $k$2 is $k$3-submodular if for all $k$4,

$k$5

with the corresponding $k$6-submodular join and meet operations. The paper also uses the standard marginal notation

$k$7

for $k$8, where

$k$9

Each element $q$0 has a nonnegative cost $q$1, and the cost of a solution is

$q$2

The optimization problem is

$q$3

This is the budgeted $q$4-submodular maximization problem, abbreviated $q$5SKM in the source. The monotone case satisfies $q$6, whereas the non-monotone case does not. The knapsack version is harder than the cardinality or matroid versions because costs are heterogeneous, greedy and optimal solutions can have different support sizes, and the algorithm must decide not only which elements to take but also which component each selected element should enter (Wang, 17 Jul 2025).

2. Algorithmic definition

The paper defines a general $q$7-Guess Greedy template and then specializes it to $q$8. The general scheme first enumerates all feasible solutions of support size $q$9, keeps the best of them, and then for every feasible guess $q=1$0 of support size exactly $q=1$1 runs a density-greedy completion: $q=1$2 where $q=1$3 is the set of currently unconsidered elements and $q=1$4 is the current partial solution. If the chosen element fits the remaining budget, it is assigned to component $q=1$5; in either case, it is removed from future consideration. The algorithm returns the best solution obtained over all guesses (Wang, 17 Jul 2025).

For 1-Guess Greedy, $q=1$6. The guessed object is therefore a feasible singleton assignment: one element together with one label/component. The algorithm compares against the best support-size-$q=1$7 feasible solution, which is the empty solution, and then enumerates all feasible support-size-one guesses. For each guess, it starts from that singleton and repeatedly adds the unconsidered pair $q=1$8 with maximum marginal density $q=1$9, subject to the knapsack constraint. In the analysis, the distinguished singleton is taken from the optimum: $\frac12$0 so the “one guess” is not an arbitrary seed but a one-element partial assignment contained in an optimal solution (Wang, 17 Jul 2025).

This name can be misleading if read too literally. The method does not solve the problem after one greedy step. Rather, it performs one guessed initialization and then a full greedy completion. The “1” refers to the size of the guessed partial solution, not to the number of total greedy iterations.

3. Approximation guarantees

The central theorem states that 1-Guess Greedy is a $\frac12$1-approximation for monotone $\frac12$2SKM and a $\frac12$3-approximation for non-monotone $\frac12$4SKM. The same source also gives value-oracle complexity $\frac12$5 for the 1-Guess version, because the general $\frac12$6-Guess Greedy scheme uses $\frac12$7 queries and $\frac12$8 yields the stated bound (Wang, 17 Jul 2025).

The monotone $\frac12$9 factor resolves a question that had remained open for several years for the knapsack-constrained case. The paper further states that this factor is asymptotically tight in the value-oracle model, because a $\frac13$0-approximation requires exponentially many value-oracle queries even without constraints. Since

$\frac13$1

the lower bound approaches $\frac13$2 as $\frac13$3 grows, which is the sense in which the result is asymptotically tight (Wang, 17 Jul 2025).

The algorithm also improves on earlier combinatorial guarantees cited in the same paper. Before this result, the best combinatorial guarantee for monotone $\frac13$4SKM mentioned there was 4-Guess Greedy with

$\frac13$5

and for non-monotone $\frac13$6SKM it was 7-Guess Greedy with

$\frac13$7

The same analysis framework also sharpens existing guarantees for Greedy+Singleton to $\frac13$8 in the monotone case and $\frac13$9 in the non-monotone case (Wang, 17 Jul 2025).

4. Continuous analysis and why one guess suffices

The proof does not use the classical discrete exchange sequence that is standard for cardinality and matroid constraints. Instead, it introduces a continuous transformation from an optimal solution to a greedy solution and evaluates the trajectory via the $\tilde O(n^2k^2)$0-multilinear extension

$\tilde O(n^2k^2)$1

Equivalently,

$\tilde O(n^2k^2)$2

where each element is independently assigned a label or left unassigned according to $\tilde O(n^2k^2)$3. This extension is used only for analysis, not for the algorithm itself (Wang, 17 Jul 2025).

The proof defines a greedy trajectory $\tilde O(n^2k^2)$4, a decaying optimal trajectory $\tilde O(n^2k^2)$5, and the combined path

$\tilde O(n^2k^2)$6

together with an auxiliary path

$\tilde O(n^2k^2)$7

When the greedy partial solution $\tilde O(n^2k^2)$8 and the comparison solution $\tilde O(n^2k^2)$9 have equal total cost, the key monotone lemma proves

$k$0

More generally, if

$k$1

then

$k$2

In the non-monotone case, the differential inequality weakens, and the corresponding equal-cost statement becomes

$k$3

with the unequal-cost version

$k$4

These are the structural estimates that drive the $k$5 and $k$6 final guarantees (Wang, 17 Jul 2025).

The one-guess mechanism is explained through a decomposition of the optimum. Let $k$7 be optimal, $k$8 the best singleton contained in $k$9, $k$0 the most expensive remaining singleton, and

$k$1

Using the residual objective

$k$2

the paper writes

$k$3

The role of the guessed singleton is then precise: once $k$4 is guessed, the continuous lemma shows that greedy captures a constant fraction of the residual part $k$5, while the contribution of the one “missing” expensive singleton $k$6 is covered by the fact that $k$7 was chosen as the best singleton in the optimum. This is why a single guessed element is sufficient in the analysis (Wang, 17 Jul 2025).

5. Complexity, thresholding, and implementation profile

The general $k$8-Guess Greedy scheme uses $k$9 value-oracle queries, so the 1-Guess specialization uses

$V=\{e_1,\dots,e_n\}$0

The paper also states that by using the decreasing-threshold technique of Badanidiyuru and Vondrák, one can reduce the running time by a factor of $V=\{e_1,\dots,e_n\}$1, obtaining a nearly

$V=\{e_1,\dots,e_n\}$2

implementation. In the monotone case this yields a $V=\{e_1,\dots,e_n\}$3-approximation, with polylogarithmic factors suppressed by the $V=\{e_1,\dots,e_n\}$4 notation (Wang, 17 Jul 2025).

The method is described there as simple and parallelizable. The parallelizability claim follows directly from the structure of the algorithm: different singleton guesses can be processed independently, and each guess then runs an ordinary density-greedy completion. The algorithm therefore sits between exhaustive seed enumeration and single-pass greedy: it pays a factor for enumerating all singleton initializations, but avoids the higher-order blowup of larger $V=\{e_1,\dots,e_n\}$5-guess variants (Wang, 17 Jul 2025).

6. Terminology and related uses of “single-decision” greedy

The exact term “1-Guess Greedy” is used in the budgeted $V=\{e_1,\dots,e_n\}$6-submodular maximization setting described above (Wang, 17 Jul 2025). In other areas, related ideas appear, but with different meanings and usually without that exact name.

In the greedy coin change literature, the closest analogue is a decision problem that asks for one bit of information about a greedy output: given $V=\{e_1,\dots,e_n\}$7, a denomination set $V=\{e_1,\dots,e_n\}$8, and a designated coin $V=\{e_1,\dots,e_n\}$9, determine whether $k$00 belongs to the greedy set $k$01. That decision problem is shown to be $k$02-complete under log-space reductions, and it formalizes a “single-aspect prediction” of a greedy computation rather than a guessed initialization for an approximation algorithm (Gupta et al., 2024).

In card-guessing with feedback, the phrase does not appear as a formal term; the closest notion is the greedy strategy in the Yes/No-feedback model, where on each round one makes one guess, namely a card type that is currently most likely to be the next card. That is a posterior-mode rule, not a guessed partial solution followed by a greedy completion (Diaconis et al., 2020).

In generalized Wordle, the paper likewise does not introduce a method called “1-Guess Greedy.” The closest notion there is a one-step minimax greedy heuristic: at each turn it chooses the single next guess minimizing the worst-case size of the remaining candidate set after that one move. This is a one-step lookahead interpretation of greediness, again distinct from the $k$03-Guess Greedy framework used for $k$04-submodular knapsack (Lahiri et al., 2023).

Accordingly, in current arXiv usage the encyclopedia sense of 1-Guess Greedy is narrow and specific: it denotes the $k$05 member of the Guess Greedy family for budgeted $k$06-submodular maximization, where one singleton from the optimum is guessed and then a density-greedy completion is run.