Submodular Knapsack Problem

• Submodular Knapsack Problem is a budget-constrained optimization model that maximizes a diminishing returns function subject to cost constraints.
• It employs algorithms like greedy, local search, and continuous relaxations to achieve near-optimal approximations, often approaching the 1-1/e barrier in monotone cases.
• Extensions include multiple knapsacks, additional combinatorial structures such as matroids and fairness constraints, and even exact solvers for rigorous benchmark settings.

The Submodular Knapsack Problem, also called Budgeted Submodular Maximization, is the problem of maximizing a submodular set function under a budget constraint. In its classical single-knapsack form, one is given a ground set $V$, a non-negative monotone submodular function $f:2^V\to \mathbb{R}$, positive costs $c(v)$, and a budget $B$, and seeks

$$\max_{S\subseteq V,\; c(S)\le B} f(S), \qquad c(S)=\sum_{v\in S} c(v).$$

Broader formulations allow any non-negative submodular function and a fixed constant number $d$ of knapsack constraints, so that feasibility requires $\sum_{i\in S} c_r(i)\le L_r$ for every dimension $r$ (Feldman et al., 2020, Kulik et al., 2011). For monotone objectives, the subject is organized around the classic $1-1/e$ approximation barrier: even for maximum coverage, and therefore for monotone submodular maximization in the oracle model, no algorithm can surpass $1-1/e$ unless $f:2^V\to \mathbb{R}$0 (Sarpatwar et al., 2017).

1. Formal problem and core notions

The standard formulation assumes a finite universe $f:2^V\to \mathbb{R}$1 or $f:2^V\to \mathbb{R}$2, oracle access to a set function $f:2^V\to \mathbb{R}$3, and a knapsack feasibility condition expressed by additive costs. In the single-knapsack setting, the feasible family is

$f:2^V\to \mathbb{R}$4

while the more general SUB formulation under $f:2^V\to \mathbb{R}$5 knapsack constraints is

$f:2^V\to \mathbb{R}$6

This formulation is explicitly studied for non-negative submodular functions, with monotone and non-monotone cases treated separately (Kulik et al., 2011).

Submodularity is used in the standard diminishing-returns sense. In marginal notation, if $f:2^V\to \mathbb{R}$7 and $f:2^V\to \mathbb{R}$8, then

$f:2^V\to \mathbb{R}$9

or equivalently,

$c(v)$0

Monotonicity means $c(v)$1 whenever $c(v)$2. In the monotone regime, many covering, facility-location, and influence-maximization objectives fall into the model; in the non-monotone regime, standard examples include cut functions (Avdiukhin et al., 2019, Kulik et al., 2011).

The terminology is not uniform across the literature. One source states explicitly that Submodular Knapsack Problem and Budgeted Submodular Maximization are used interchangeably for maximizing a non-negative monotone submodular function subject to a single knapsack constraint (Feldman et al., 2020). Other works use SKP for the same task, especially in the context of exact optimization (Hao et al., 15 Jul 2025).

2. Approximation thresholds and representative guarantees

The monotone single-knapsack problem is governed by the $c(v)$3 benchmark. Sviridenko’s classical three-guess greedy method achieves this optimal ratio, and later work shows that two guesses already suffice: 2-Guess $c(v)$4 achieves approximation ratio $c(v)$5, while 1-Guess $c(v)$6 attains

$c(v)$7

with lower running time (Feldman et al., 2020). For fixed constant $c(v)$8, the broader $c(v)$9-knapsack monotone problem admits a randomized $B$0-approximation, and also a deterministic $B$1-approximation assuming the multilinear extension $B$2 can be evaluated deterministically (Kulik et al., 2011).

For non-monotone objectives, the guarantees are weaker but still constant in several settings. Under a fixed constant number $B$3 of knapsack constraints, the continuous-to-discrete reduction of (Kulik et al., 2011) yields a randomized $B$4-approximation, and also a deterministic $B$5-approximation assuming oracle access to $B$6. For a single knapsack and possibly non-monotone objectives, a simple randomized greedy algorithm achieves a $B$7-approximation, i.e. a 5.83-approximation, with a lazy version giving $B$8 using

$B$9

oracle calls (Amanatidis et al., 2020).

Representative guarantees appearing in the sources are summarized below.

Setting	Guarantee	Representative source
Monotone, single knapsack	$$\max_{S\subseteq V,\; c(S)\le B} f(S), \qquad c(S)=\sum_{v\in S} c(v).$$0	(Feldman et al., 2020)
Monotone, fixed $$\max_{S\subseteq V,\; c(S)\le B} f(S), \qquad c(S)=\sum_{v\in S} c(v).$$1 knapsacks	$$\max_{S\subseteq V,\; c(S)\le B} f(S), \qquad c(S)=\sum_{v\in S} c(v).$$2	(Kulik et al., 2011)
Non-monotone, fixed $$\max_{S\subseteq V,\; c(S)\le B} f(S), \qquad c(S)=\sum_{v\in S} c(v).$$3 knapsacks	$$\max_{S\subseteq V,\; c(S)\le B} f(S), \qquad c(S)=\sum_{v\in S} c(v).$$4	(Kulik et al., 2011)
Monotone, knapsack + one matroid	$$\max_{S\subseteq V,\; c(S)\le B} f(S), \qquad c(S)=\sum_{v\in S} c(v).$$5	(Sarpatwar et al., 2017)
Monotone, multiple knapsack	$$\max_{S\subseteq V,\; c(S)\le B} f(S), \qquad c(S)=\sum_{v\in S} c(v).$$6	(Fairstein et al., 2020, Sun et al., 2020)

These guarantees illustrate a recurring structural distinction. The $$\max_{S\subseteq V,\; c(S)\le B} f(S), \qquad c(S)=\sum_{v\in S} c(v).$$7 threshold is essentially optimal for several monotone knapsack families, whereas extensions involving additional combinatorial structure, generalized objective classes, or adaptivity often trade approximation quality for simpler combinatorial methods, stronger robustness properties, or broader modeling power.

3. Greedy, local-search, and combinatorial paradigms

A central algorithmic pattern is marginal density greedy, which selects items according to

$$\max_{S\subseteq V,\; c(S)\le B} f(S), \qquad c(S)=\sum_{v\in S} c(v).$$8

This rule underlies many classical and modern methods, but the literature is explicit that the plain density-based greedy rule is not sufficient by itself. One source states that plain Greedy does not guarantee any approximation for knapsack, and another that the plain density-based greedy algorithm is not guaranteed a constant ratio by itself (Avdiukhin et al., 2019, Feldman et al., 2020). This is a common source of confusion because the same rule is optimal for several simpler constraints.

A simple but influential correction is to augment greedy prefixes with a single additional item. The “Bring Your Own Greedy” + Max framework returns the best set of the form $$\max_{S\subseteq V,\; c(S)\le B} f(S), \qquad c(S)=\sum_{v\in S} c(v).$$9, where $d$0 is a greedy prefix and $d$1 is the best remaining fitting item. In the offline setting, Greedy+Max gives a $d$2-approximation for monotone submodular maximization under a knapsack constraint; streaming and distributed instantiations, Sieve+Max and Distributed+Max, achieve $d$3-approximation with near-optimal resource parameters (Avdiukhin et al., 2019).

Another line keeps the greedy core but adds limited enumeration. In Practical Budgeted Submodular Maximization, the algorithmic templates 2-Guess $d$4 and 1-Guess $d$5 show that fewer guesses than in Sviridenko’s original method suffice to approach or match the optimal monotone ratio. The same source sharpens the analysis of the simpler greedy-plus-singleton rule $d$6, placing its approximation ratio in the interval

$d$7

and thereby distinguishing it sharply from the optimal $d$8 frontier (Feldman et al., 2020).

For knapsack combined with matroid structure, greedy is often embedded in local search. The algorithm of “Constrained Submodular Maximization via Greedy Local Search” begins by guessing two seed elements $d$9, then repeatedly performs feasibility-preserving swaps chosen by maximum profit density

$\sum_{i\in S} c_r(i)\le L_r$0

or a $\sum_{i\in S} c_r(i)\le L_r$1-swap analogue under $\sum_{i\in S} c_r(i)\le L_r$2 matroids. For a single knapsack plus one matroid, this yields the combinatorial ratio

$\sum_{i\in S} c_r(i)\le L_r$3

and for a single knapsack plus the intersection of $\sum_{i\in S} c_r(i)\le L_r$4 matroids, the guarantee becomes

$\sum_{i\in S} c_r(i)\le L_r$5

Polynomial termination is enforced by accepting only multiplicative improvements of factor $\sum_{i\in S} c_r(i)\le L_r$6, leading to running times $\sum_{i\in S} c_r(i)\le L_r$7 for one matroid and $\sum_{i\in S} c_r(i)\le L_r$8 for fixed $\sum_{i\in S} c_r(i)\le L_r$9 (Sarpatwar et al., 2017).

For non-monotone single-knapsack optimization, the combinatorial emphasis shifts to randomization. SampleGreedy samples items with probability $r$0, runs density greedy on the sample, and returns the better of the greedy set and the best singleton. With $r$1, the method is a $r$2-approximation; its robustness further supports an adaptive extension, AdaptiveGreedy, which is a $r$3-approximation to the best adaptive policy in the corresponding stochastic setting (Amanatidis et al., 2020).

4. Continuous relaxations, near-linear algorithms, and curvature

A major conceptual development is the passage from the discrete problem to a continuous relaxation through extension by expectation, also called the multilinear extension in many later works. For $r$4, let $r$5 denote the random subset obtained by including each element independently with probability $r$6, and define

$r$7

The natural relaxation for $r$8 knapsack constraints is

$r$9

The central theorem of (Kulik et al., 2011) states that if there is a polynomial-time $1-1/e$0-approximation algorithm for this continuous relaxation, then for any $1-1/e$1 there is a randomized polynomial-time $1-1/e$2-approximation algorithm for the discrete SUB problem; if $1-1/e$3 can be evaluated deterministically, a deterministic $1-1/e$4-approximation follows as well. This reduction is the main reason that continuous methods dominate the near-optimal approximation landscape.

The difficulty is computational. Standard multilinear-extension frameworks typically encounter an $1-1/e$5 bottleneck because evaluating $1-1/e$6 is expensive. “A Nearly-linear Time Algorithm for Submodular Maximization with a Knapsack Constraint” breaks this bottleneck by maintaining a fractional solution with only

$1-1/e$7

strictly fractional entries. This makes exact evaluation of the multilinear extension combinatorially cheap on the fractional support and leads to a $1-1/e$8 approximation using

$1-1/e$9

function evaluations and

$1-1/e$0

arithmetic operations (Ene et al., 2017). The result is explicitly described as theoretically interesting rather than practical, but it shows that near-linear dependence on $1-1/e$1 is possible without abandoning the continuous paradigm.

A separate structural refinement uses curvature. For a monotone submodular function $1-1/e$2, total curvature is defined as

$1-1/e$3

Under a single knapsack constraint, there exists a polynomial-time algorithm with approximation ratio

$1-1/e$4

and this ratio is tight up to $1-1/e$5 for every $1-1/e$6 (Yoshida, 2016). When $1-1/e$7, the guarantee matches the standard $1-1/e$8 barrier; when $1-1/e$9, it approaches $f:2^V\to \mathbb{R}$00, consistent with the modular case. This suggests a more refined taxonomy than the usual monotone/non-monotone dichotomy: the effective approximability of the knapsack problem depends not only on submodularity, but also on how far the objective is from linearity.

5. Generalizations: multiple knapsacks, matroids, fairness, adaptivity, and generalized objective classes

The classical single-knapsack model has been extended in several orthogonal directions. One axis concerns additional constraints. For a single knapsack combined with matroid independence, greedy local search yields $f:2^V\to \mathbb{R}$01, and for a single knapsack plus the intersection of $f:2^V\to \mathbb{R}$02 matroids it yields $f:2^V\to \mathbb{R}$03 (Sarpatwar et al., 2017). Another axis concerns multiple knapsacks. The monotone submodular multiple knapsack problem admits a randomized $f:2^V\to \mathbb{R}$04-approximation via structuring, leveled bins, continuous greedy, and rounding (Fairstein et al., 2020), and also a deterministic combinatorial $f:2^V\to \mathbb{R}$05-approximation based on greedy density packing, large-element enumeration, reserved bins, and blockwise iterative packing (Sun et al., 2020).

A different family of extensions changes the objective class rather than the feasibility family. For monotone $f:2^V\to \mathbb{R}$06-submodular objectives under a knapsack constraint, a deterministic algorithm based on Sviridenko’s philosophy was originally claimed to achieve

$f:2^V\to \mathbb{R}$07

with $f:2^V\to \mathbb{R}$08 function evaluations; a corrigendum notes that the same algorithm can in fact be analyzed to give a stronger $f:2^V\to \mathbb{R}$09-approximation (Tang et al., 2021). On finite distributive lattices, where feasible solutions are ideals of a poset and knapsack costs are required to be order-consistent, a generalized continuous greedy algorithm on the median complex yields a $f:2^V\to \mathbb{R}$10 approximation for multiple knapsack-constrained monotone DR-submodular maximization (Maehara et al., 2019).

Recent work also adds fairness constraints. In Fair Knapsack-constrained Submodular Maximization, each element has a weight and a color, and the selected set must satisfy both a knapsack constraint and per-group lower and upper bounds. When the number of colors is constant, there is a polynomial-time algorithm achieving

$f:2^V\to \mathbb{R}$11

with probability at least

$f:2^V\to \mathbb{R}$12

while relaxing either the knapsack or fairness constraint to expected satisfaction yields the tight $f:2^V\to \mathbb{R}$13 ratio in expectation (Li et al., 17 May 2025). The same paper notes that the general case with many groups remains open.

Adaptive, uncertain-capacity, and stochastic formulations move beyond static feasibility. For non-monotone adaptive submodular maximization under a knapsack constraint, a sampling-based randomized policy achieves a $f:2^V\to \mathbb{R}$14 approximation without assuming pointwise submodularity; the paper presents this as the first constant-factor approximation for that knapsack setting under adaptive submodularity alone (Tang, 2021). When the knapsack capacity is unknown and only feasibility feedback is available, a randomized adaptive policy with cancellation allowed achieves robustness ratio $f:2^V\to \mathbb{R}$15, a deterministic adaptive policy achieves $f:2^V\to \mathbb{R}$16, and a randomized universal policy achieves $f:2^V\to \mathbb{R}$17; without cancellation, no randomized adaptive policy achieves a constant robustness ratio (Kawase et al., 2018). A curvature-sensitive refinement for the unknown-capacity setting gives robustness factor

$f:2^V\to \mathbb{R}$18

where $f:2^V\to \mathbb{R}$19 is the unique root of

$f:2^V\to \mathbb{R}$20

at the extremes, this yields $f:2^V\to \mathbb{R}$21 for $f:2^V\to \mathbb{R}$22 and approximately $f:2^V\to \mathbb{R}$23 for $f:2^V\to \mathbb{R}$24 (Klimm et al., 2022).

Stochasticity can also enter through item realizations. For the correlated stochastic knapsack problem with a lattice-submodular objective and optional partition-matroid side constraints, a pseudo-polynomial-time algorithm based on stochastic continuous greedy and a new contention-resolution-compatible rounding scheme achieves

$f:2^V\to \mathbb{R}$25

improving on the $f:2^V\to \mathbb{R}$26 approximation reported for the earlier model (Yang et al., 2022). These variants show that the phrase “Submodular Knapsack Problem” now denotes not a single isolated model, but a family of optimization problems sharing a budget structure and a diminishing-returns objective.

6. Exact optimization and contemporary solver design

Although the dominant literature is approximation-oriented, recent work has revived exact optimization for monotone submodular knapsack. Two 2025 papers develop branch-and-bound solvers tailored specifically to SKP. The first proposes an exact depth-first branch-and-bound method built around a greedy primal heuristic, a novel refined subset upper bound, and a dual branching strategy. At a node $f:2^V\to \mathbb{R}$27, the refined bound minimizes a fractional-knapsack-style upper bound over intermediate greedy subsets $f:2^V\to \mathbb{R}$28, producing a bound $f:2^V\to \mathbb{R}$29 that is at least as tight as the usual fractional knapsack bound and that can be computed in

$f:2^V\to \mathbb{R}$30

time. The same work proves a worst-case tightness guarantee using Wolsey’s lemma and reports that the variant dual-rs is the best overall performer, with dual branching providing about a 2× speedup on average (Hao et al., 15 Jul 2025).

A second exact line studies a depth-first branch-and-bound framework with three acceleration techniques: Lazy Evaluations, Early Pruning, and Candidate Reduction. The core upper bound is the fractional knapsack relaxation

$f:2^V\to \mathbb{R}$31

and the strongest empirical variant combines lazy evaluations with candidate reduction in the solver LECR. The same source also examines a tighter bound $f:2^V\to \mathbb{R}$32 obtained by solving the inner knapsack subproblem exactly, but concludes empirically that the stronger bound is too expensive in practice; the structural accelerations are more effective than replacing the relaxation by an exact integer knapsack computation at every node (Münch et al., 22 Jul 2025).

These exact methods do not alter the NP-hardness of the problem or the approximation barrier for polynomial-time algorithms, but they shift the practical discussion. Approximation guarantees remain central when scale dominates, yet the exact-solvers literature emphasizes settings in which approximate solutions are not sufficient in practice, including facility location and risk-sensitive decision-making (Hao et al., 15 Jul 2025, Münch et al., 22 Jul 2025). A plausible implication is that the contemporary theory of the Submodular Knapsack Problem now has two mature fronts: approximation algorithms that optimize worst-case ratios under rich structural assumptions, and exact solvers that exploit submodularity aggressively enough to make optimal search competitive on benchmark families.