Submodular Knapsack Problem: Theory & Methods

• Submodular Knapsack Problem (SKP) is a problem of maximizing a submodular function subject to a budget constraint, where item weights and diminishing returns play a key role.
• It combines a discrete approach—using greedy and branch-and-bound methods—with a continuous view via the multilinear extension to transfer approximation guarantees.
• Recent advances address curvature-sensitive formulations, stochastic variants, and exact algorithms, highlighting challenges in scalability and fairness constraints.

The Submodular Knapsack Problem (SKP) is the problem of maximizing a submodular set function under a single knapsack budget. In its standard form, one is given a finite ground set $N$, a non-negative submodular function $f:2^N\to \mathbb{R}_{\ge 0}$, item costs $w_i$, and a budget $B$, and seeks

$$\max\{f(S): S\subseteq N,\ \sum_{i\in S} w_i\le B\}.$$

SKP is the $d=1$ specialization of submodular maximization under $d$ knapsack constraints, and it sits at the intersection of discrete optimization, approximation algorithms, stochastic optimization, and exact combinatorial search. Its modern theory is organized around two complementary views: a discrete view based on greedy, local-search, and branch-and-bound methods, and a continuous view based on the multilinear extension $F(x)=\mathbb{E}[f(R(x))]$, which transfers approximation guarantees from fractional relaxations back to feasible sets (Kulik et al., 2011).

1. Formal model and structural notions

A set function $f$ is submodular if it satisfies diminishing returns. Equivalently, for all $S,T\subseteq N$,

$f:2^N\to \mathbb{R}_{\ge 0}$0

In the monotone case, $f:2^N\to \mathbb{R}_{\ge 0}$1 implies $f:2^N\to \mathbb{R}_{\ge 0}$2; in the non-monotone case, monotonicity is dropped but non-negativity is typically retained (Kulik et al., 2011). In many formulations, especially those centered on classical greedy analysis, $f:2^N\to \mathbb{R}_{\ge 0}$3 is also normalized, $f:2^N\to \mathbb{R}_{\ge 0}$4, and accessed through a value oracle (Klimm et al., 2022).

The single-knapsack constraint assigns each item $f:2^N\to \mathbb{R}_{\ge 0}$5 a nonnegative weight $f:2^N\to \mathbb{R}_{\ge 0}$6 and imposes $f:2^N\to \mathbb{R}_{\ge 0}$7. The marginal gain notation

$f:2^N\to \mathbb{R}_{\ge 0}$8

is central, as are marginal densities $f:2^N\to \mathbb{R}_{\ge 0}$9. These quantities drive both density-greedy algorithms and continuous-relaxation analyses.

Several structural notions recur in the literature. One is the partition of items into “big” and “small” elements relative to an accuracy parameter $w_i$0: for $w_i$1 knapsacks, an item is small if $w_i$2 for all $w_i$3, and big otherwise (Kulik et al., 2011). Another is total curvature, which measures how much an element’s marginal can decay in the presence of others. For normalized, monotone, submodular $w_i$4, the total curvature is

$w_i$5

with $w_i$6 if and only if $w_i$7 is modular and $w_i$8 covering fully curved cases such as matroid rank (Klimm et al., 2022).

2. Multilinear relaxation and continuous-to-discrete transfer

A foundational development for SKP is the extension by expectation, or multilinear extension. For $w_i$9, let $B$0 include each item independently with probability $B$1, and define

$B$2

The continuous relaxation of SKP is

$B$3

Independent Bernoulli sampling is essential here: it gives $B$4 the continuity and diminishing-returns structure needed by continuous optimization methods (Kulik et al., 2011).

For monotone objectives, continuous greedy achieves near-optimal value $B$5, while for non-monotone objectives continuous algorithms achieve $B$6 (Kulik et al., 2011). The central theorem of the 2011 transfer framework states that, for any non-negative submodular function and fixed $B$7, an $B$8-approximation for the continuous relaxation implies a polynomial-time randomized $B$9-approximation for the discrete problem under $$\max\{f(S): S\subseteq N,\ \sum_{i\in S} w_i\le B\}.$$0 knapsack constraints, hence also for SKP (Kulik et al., 2011).

The reduction has three main ingredients. First, profit enumeration guesses a small set $$\max\{f(S): S\subseteq N,\ \sum_{i\in S} w_i\le B\}.$$1 of highly profitable items from the unknown optimum, with enumeration size $$\max\{f(S): S\subseteq N,\ \sum_{i\in S} w_i\le B\}.$$2. For at least one guess, the residual instance preserves at least $$\max\{f(S): S\subseteq N,\ \sum_{i\in S} w_i\le B\}.$$3, and all remaining items are small. Second, one solves the multilinear relaxation on the residual instance to obtain $$\max\{f(S): S\subseteq N,\ \sum_{i\in S} w_i\le B\}.$$4 with $$\max\{f(S): S\subseteq N,\ \sum_{i\in S} w_i\le B\}.$$5. Third, one samples $$\max\{f(S): S\subseteq N,\ \sum_{i\in S} w_i\le B\}.$$6, discards outcomes that violate a $$\max\{f(S): S\subseteq N,\ \sum_{i\in S} w_i\le B\}.$$7-near-feasibility condition, and then applies a fixing procedure that removes a small-value subset to restore exact feasibility while losing only an $$\max\{f(S): S\subseteq N,\ \sum_{i\in S} w_i\le B\}.$$8 fraction of the value. Altogether,

$$\max\{f(S): S\subseteq N,\ \sum_{i\in S} w_i\le B\}.$$9

The same framework admits a deterministic reduction if an oracle for $d=1$0 is available. Pipage rounding shrinks the fractional support to $d=1$1, after which all realizations of the induced product distribution can be enumerated in polynomial time, filtered for near-feasibility, and fixed. This yields deterministic $d=1$2-approximations for the discrete problem from deterministic $d=1$3-approximations to the continuous relaxation (Kulik et al., 2011).

A separate nearly-linear-time contribution showed that the traditional multilinear framework suffers from an $d=1$4 bottleneck because evaluating $d=1$5 and its gradient to high precision is expensive; the proposed remedy was to maintain only $d=1$6 strictly fractional coordinates, so that multilinear values can be evaluated exactly by enumerating the $d=1$7 outcomes on those coordinates (Ene et al., 2017).

3. Approximation algorithms for classical SKP

The modern approximation landscape for classical SKP separates near-optimal but often heavy multilinear methods from lighter combinatorial algorithms with weaker worst-case guarantees.

Setting	Guarantee	Representative method
Monotone SKP	randomized $d=1$8	continuous-relaxation transfer
Non-monotone SKP	randomized $d=1$9	continuous-relaxation transfer
Monotone SKP	$d$0	Greedy+Max
Monotone SKP	deterministic $d$1	threshold/backtracking
Monotone SKP	$d$2	nearly-linear-time multilinear method
Non-monotone SKP	$d$3-approximation	randomized density-greedy

The first two guarantees are immediate specializations of the continuous-to-discrete transfer theorem: monotone SKP admits a randomized $d$4-approximation, and non-monotone SKP admits a randomized $d$5-approximation (Kulik et al., 2011). In settings where $d$6 is available or exactly computable, the same ratios can be obtained deterministically.

On the combinatorial side, “Bring Your Own Greedy” + Max augments every greedy prefix with the best additional item that still fits. In the offline setting it gives a $d$7-approximation for monotone SKP with $d$8 time and oracle calls, where $d$9; it extends to multi-pass streaming and distributed models with $F(x)=\mathbb{E}[f(R(x))]$0-approximation (Avdiukhin et al., 2019). A different deterministic line gives a $F(x)=\mathbb{E}[f(R(x))]$1-approximation for a single knapsack using nearly linear query complexity $F(x)=\mathbb{E}[f(R(x))]$2, and exactly two streaming passes (Li, 2018).

At the high end of the monotone spectrum, a nearly-linear-time algorithm achieves $F(x)=\mathbb{E}[f(R(x))]$3 using $F(x)=\mathbb{E}[f(R(x))]$4 function evaluations and arithmetic operations. Its main theoretical interest is that it breaks the classical multilinear-evaluation bottleneck, although its dependence on $F(x)=\mathbb{E}[f(R(x))]$5 makes it impractical (Ene et al., 2017). A purely combinatorial alternative based on greedy local search also recovers the optimal $F(x)=\mathbb{E}[f(R(x))]$6 ratio when the framework for “one knapsack + intersection of $F(x)=\mathbb{E}[f(R(x))]$7 matroids” is specialized to $F(x)=\mathbb{E}[f(R(x))]$8, though with a higher naive complexity bound (Sarpatwar et al., 2017).

For non-monotone SKP, a simple randomized density-greedy method, SampleGreedy, achieves a $F(x)=\mathbb{E}[f(R(x))]$9-approximation in $f$0 time and can be implemented with lazy evaluations using $f$1 oracle calls for a $f$2-approximation. The same design extends to an adaptive stochastic variant with a $f$3-approximation to the best adaptive policy (Amanatidis et al., 2020).

4. Curvature-sensitive and uncertain-capacity formulations

Curvature refines the approximation theory of monotone SKP by quantifying deviation from modularity. For any fixed $f$4, there exists a polynomial-time algorithm with approximation ratio

$f$5

where $f$6 is the total curvature. This ratio is tight up to $f$7 for every $f$8. The construction decomposes $f$9 into a monotone submodular part and a linear part, then applies a curvature-aware continuous-greedy scheme and rounding; the improvement is strict over the classical $S,T\subseteq N$0 barrier whenever $S,T\subseteq N$1 (Yoshida, 2016).

A different curvature-sensitive analysis revisits greedy itself. For known capacity, both AGreedy and MGreedy achieve the same approximation guarantee

$S,T\subseteq N$2

where $S,T\subseteq N$3 is the unique solution of

$S,T\subseteq N$4

This yields $S,T\subseteq N$5 in the modular case and $S,T\subseteq N$6 in the fully curved case (Klimm et al., 2022).

The same paper studies the unknown-capacity model, where the capacity is not known in advance and is only revealed through feasibility of attempted items. It constructs a deterministic polynomial-time policy $S,T\subseteq N$7 satisfying

$S,T\subseteq N$8

At $S,T\subseteq N$9, this matches the best possible deterministic robustness factor $f:2^N\to \mathbb{R}_{\ge 0}$00; at $f:2^N\to \mathbb{R}_{\ge 0}$01, it improves the previous best deterministic robustness from $f:2^N\to \mathbb{R}_{\ge 0}$02 to $f:2^N\to \mathbb{R}_{\ge 0}$03 (Klimm et al., 2022).

An earlier uncertain-capacity study distinguishes cancellation-allowed and no-cancellation models. With cancellation allowed, it gives a randomized adaptive policy of robustness ratio $f:2^N\to \mathbb{R}_{\ge 0}$04, a deterministic adaptive policy of robustness ratio $f:2^N\to \mathbb{R}_{\ge 0}$05, and a randomized universal policy of robustness ratio $f:2^N\to \mathbb{R}_{\ge 0}$06. Without cancellation, no randomized adaptive policy achieves a constant robustness ratio in the worst case; with a known distribution over capacities, however, there is a polynomial-time randomized algorithm with approximation ratio $f:2^N\to \mathbb{R}_{\ge 0}$07 (Kawase et al., 2018).

5. Generalizations beyond the classical single-knapsack model

SKP is the base case of a broad family of submodular packing problems. For a fixed number $f:2^N\to \mathbb{R}_{\ge 0}$08 of knapsack constraints, the multilinear-transfer theorem yields a randomized $f:2^N\to \mathbb{R}_{\ge 0}$09-approximation for monotone objectives and $f:2^N\to \mathbb{R}_{\ge 0}$10-approximation for non-monotone ones (Kulik et al., 2011). For monotone submodular multiple knapsack with arbitrary bin capacities, an almost optimal $f:2^N\to \mathbb{R}_{\ge 0}$11-approximation is obtained by structuring bins into leveled blocks, combining partial enumeration, continuous or unified greedy on a block polytope, and randomized rounding with slack (Fairstein et al., 2020). A different deterministic combinatorial treatment reaches the same $f:2^N\to \mathbb{R}_{\ge 0}$12 ratio through a simpler grouping-and-greedy framework (Sun et al., 2020).

Fairness-augmented SKP adds lower and upper bounds on the number of selected elements from each color class. For a constant number of groups $f:2^N\to \mathbb{R}_{\ge 0}$13, there is a polynomial-time algorithm that achieves

$f:2^N\to \mathbb{R}_{\ge 0}$14

with probability at least $f:2^N\to \mathbb{R}_{\ge 0}$15, while strictly satisfying both knapsack and fairness constraints. If either fairness or knapsack is relaxed to hold only in expectation, then a tight $f:2^N\to \mathbb{R}_{\ge 0}$16 expected approximation becomes achievable (Li et al., 17 May 2025).

The $f:2^N\to \mathbb{R}_{\ge 0}$17-submodular generalization replaces binary selection by assignment of each item to one of $f:2^N\to \mathbb{R}_{\ge 0}$18 labels or to label $f:2^N\to \mathbb{R}_{\ge 0}$19. For monotone $f:2^N\to \mathbb{R}_{\ge 0}$20-submodular maximization under a knapsack constraint, one deterministic algorithm based on partial enumeration and density-greedy was originally analyzed as $f:2^N\to \mathbb{R}_{\ge 0}$21, and a corrigendum strengthens the analysis to a $f:2^N\to \mathbb{R}_{\ge 0}$22-approximation with the same $f:2^N\to \mathbb{R}_{\ge 0}$23 query complexity (Tang et al., 2021). Another deterministic greedy-type framework gives $f:2^N\to \mathbb{R}_{\ge 0}$24 for monotone objectives and $f:2^N\to \mathbb{R}_{\ge 0}$25 for non-monotone objectives (Xiao et al., 2023).

Related online and stochastic models enlarge the scope further. In online vector packing with free disposal, where items arrive online with $f:2^N\to \mathbb{R}_{\ge 0}$26-dimensional weight vectors, there is a deterministic $f:2^N\to \mathbb{R}_{\ge 0}$27-competitive algorithm under an $f:2^N\to \mathbb{R}_{\ge 0}$28-slack assumption, with matching hardness phenomena up to logarithmic factors (Chan et al., 2017). In the correlated stochastic knapsack problem with a lattice-submodular objective, a pseudo-polynomial-time $f:2^N\to \mathbb{R}_{\ge 0}$29 approximation is obtained by combining a time-indexed relaxation, stochastic continuous greedy, and a monotone $f:2^N\to \mathbb{R}_{\ge 0}$30-contention resolution scheme (Yang et al., 2022).

6. Exact algorithms and branch-and-bound methods

Although approximation algorithms dominate the SKP literature, recent work has renewed attention to exact solution methods for monotone SKP. One exact branch-and-bound solver organizes the search tree by feasible partial solutions, uses a fractional knapsack upper bound on local marginal gains,

$f:2^N\to \mathbb{R}_{\ge 0}$31

and augments it with Lazy Evaluations, Early Pruning, and Candidate Reduction. The combined LECR variant gives the best overall performance in experiments, with large reductions in oracle calls and search nodes relative to prior exact baselines (Münch et al., 22 Jul 2025).

A second exact framework introduces a refined subset upper bound

$f:2^N\to \mathbb{R}_{\ge 0}$32

where the minimum ranges over greedy prefixes $f:2^N\to \mathbb{R}_{\ge 0}$33 at node $f:2^N\to \mathbb{R}_{\ge 0}$34. This bound always dominates the plain fractional bound and satisfies a worst-case tightness guarantee: $f:2^N\to \mathbb{R}_{\ge 0}$35 The same work couples the bound with a dual branching strategy that reuses greedy-prefix computations across siblings and yields approximately $f:2^N\to \mathbb{R}_{\ge 0}$36 speedups over standard branching in the reported experiments (Hao et al., 15 Jul 2025).

These exact methods target application domains where approximate solutions may be insufficient. Reported motivations include health-care facility location and risk management, and the benchmarks used in evaluation include weighted coverage, facility location, influence maximization, and partial dominating set (Hao et al., 15 Jul 2025). A parallel exact branch-and-bound paper similarly evaluates on benchmark instances from weighted coverage, facility location, and bipartite influence, and reports improvements over the previously strongest exact solvers of Sakaue and Ishihata (Münch et al., 22 Jul 2025).

7. Applications, neighboring learning formulations, and open problems

SKP subsumes or models maximum coverage with costs, influence maximization with a budget, facility location, and budgeted variants of Max-Cut; these examples motivate both monotone and non-monotone formulations (Kulik et al., 2011). In applied machine learning, knapsack-constrained submodular optimization also appears in query-limited recommendation, summarization, and contextual list prediction. A DAgger-style reduction for contextual sequence prediction under a knapsack constraint learns policies that imitate greedy maximization by normalized marginal benefit

$f:2^N\to \mathbb{R}_{\ge 0}$37

and yields a $f:2^N\to \mathbb{R}_{\ge 0}$38-type guarantee relative to a randomized optimal policy list in extractive multi-document summarization (Zhou et al., 2013).

Several open directions remain explicit in the literature. For fair SKP, when the number of groups is not constant and both knapsack and fairness bounds must hold exactly, the existence of a non-trivial approximation remains open (Li et al., 17 May 2025). For unknown-capacity SKP, tightness is known at $f:2^N\to \mathbb{R}_{\ge 0}$39, but tightness for $f:2^N\to \mathbb{R}_{\ge 0}$40 and intermediate curvature values remains unresolved (Klimm et al., 2022). For curvature-aware SKP, extending the $f:2^N\to \mathbb{R}_{\ge 0}$41 paradigm to multiple knapsacks is identified as open (Yoshida, 2016). For nearly-linear-time near-optimal algorithms, the main unresolved issue is to reduce the prohibitive dependence on $f:2^N\to \mathbb{R}_{\ge 0}$42 and obtain practical implementations (Ene et al., 2017).

Taken together, these results position SKP as a canonical problem whose theory now spans value-oracle approximation, streaming and distributed computation, robustness to unknown budgets, stochastic and online variants, fairness constraints, $f:2^N\to \mathbb{R}_{\ge 0}$43-submodular generalizations, and exact optimization. The single budget constraint remains deceptively rich: it is simple enough to expose fine distinctions among greedy, continuous, and exact paradigms, yet broad enough to serve as a template for much of modern submodular optimization.