Multi-Submodular Cover Problem

• Multi-Submodular Cover Problem is a framework where multiple monotone submodular functions must meet specified coverage demands, generalizing set cover and common combinatorial problems.
• Algorithmic approaches include aggregate-sum reduction, multi-ratio greedy selection, and bi-criteria approximations that balance cost and coverage, often using randomized rounding and multilinear relaxations.
• Practical applications span sensor placement, facility location, and adaptive testing, with extensions addressing stochastic scenarios and online decision making.

The Multi-Submodular Cover Problem is a fundamental extension of the classical submodular set cover, wherein multiple monotone submodular coverage requirements must be simultaneously satisfied. Given a finite ground set $N$, a nonnegative cost function $c:N \rightarrow \mathbb{R}_{+}$, monotone submodular functions $f_1,\ldots,f_r : 2^N \rightarrow \mathbb{R}_{+}$, and coverage demands $b_1,\ldots,b_r>0$, the objective is to find a minimum-cost subset $S \subseteq N$ such that $f_i(S)\ge b_i$ for every $i$. This generalizes both classical set cover and single-function submodular cover, and unifies a broad array of combinatorial covering problems within a common optimization framework (Iyer et al., 2013, Bajpai et al., 14 Jul 2025).

1. Formal Definitions and Model

Let $N$ be the ground set, $c:N\rightarrow \mathbb{R}_+$ the cost vector, and $f_1,\dots,f_r$ monotone submodular functions. The Multi-Submodular Cover Problem (MSC) is to compute: $$\min_{S\subseteq N} \; c(S) \quad\text{subject to}\quad f_{i}(S)\geq b_{i}\qquad\forall\;i=1,\dots,r$$ When $r=1$, this reduces to the classical (submodular) set cover. Submodularity of $f_i$ means $f_i(A)+f_i(B)\geq f_i(A\cup B)+f_i(A\cap B)$ for all $A,B\subseteq N$, and monotonicity implies $f_i(S)\leq f_i(T)$ whenever $S\subseteq T$.

Multi-Submodular Cover captures numerous problems—sensor placement, data subset selection, multi-objective resource allocation, and more—where coverage requirements represent possibly overlapping, non-modular objectives (Iyer et al., 2013). Applications often demand $r\ll n$, but approaches for large $r$ have also been investigated (Bajpai et al., 14 Jul 2025).

2. Algorithmic Approaches and Approximation Guarantees

For single-constraint submodular cover, the classical greedy algorithm achieves an $H_D$-approximation (where $H_D$ is the $D$th harmonic number, and $D=\max_j f(\{j\})$), and this bound is tight assuming P$\neq$NP (Iyer et al., 2013). In the multi-constraint case with $r>1$, approximation ratios necessarily degrade, as the problem strictly generalizes set cover on $r$ demands.

Multiple approaches have been developed:

• Aggregate-Sum Reduction: Aggregate all $r$ coverage functions into one $G(S)=\sum_{i=1}^r f_i(S)$ and target $L=\sum_i b_i$. Running the standard greedy on $(c,G,L)$ yields $H_{D'}$-approximation, $D'=\max_j G(\{j\})$, but the guarantee degrades in $r$ (Iyer et al., 2013).
• Multi-Ratio Greedy: Selects elements maximizing $\min_{i:f_i(S)<b_i} f_i(j|S)/c(j)$. Still, a logarithmic dependence in $r$ is inevitable when precise satisfaction of all constraints is required (Iyer et al., 2013, Bajpai et al., 14 Jul 2025).
• Bi-criteria Approximation: Accepts approximate coverage $(1-\epsilon)b_i$ for each $i$, allowing improved cost guarantees. For fixed $r$, randomized algorithms achieve for any integer $\alpha\ge1$ and $\epsilon>0$ a solution $S$ with $f_i(S)\ge(1-e^{-\alpha}-\epsilon)b_i$ and $$\mathbb{E}[c(S)]\le(1+\epsilon)\alpha\cdot\mathrm{OPT}$$ (Bajpai et al., 14 Jul 2025). For weighted coverage in deletion-closed set systems, one achieves a $\big(\frac{e}{e-1}\big)(1+\beta)(1+\epsilon)$-approximation, where $\beta$ is the approximation for the underlying set cover LP (Bajpai et al., 14 Jul 2025).

The following table summarizes these core results:

Algorithmic Paradigm	Cost Approximation	Coverage Guarantee	Applicability
Aggregate Sum	$O(\log r)$	All $b_i$ exactly met	Any monotone submodular
Bi-criteria, fixed $r$	$(1+\epsilon)\alpha$	$(1-e^{-\alpha}-\epsilon)b_i$	Constant $r$, all monotone submodular functions
Weighted coverage, LP	$\big(\frac{e}{e-1}\big)(1+\beta)(1+\epsilon)$	All $b_i$	Deletion-closed set systems

These guarantees are best possible up to lower-order factors for constant $r$ and set cover–hard instances (Bajpai et al., 14 Jul 2025).

3. Rounding Techniques and Multilinear Relaxations

Central to improved bi-criteria guarantees is the multilinear extension framework and randomized rounding (Bajpai et al., 14 Jul 2025):

1. Multilinear Extension: For $x\in[0,1]^N$, $F_i(x)=\mathbb{E}_{R\sim x}[f_i(R)]$ evaluates $f_i$ at a random set $R$ where $j$ is included with probability $x_j$.
2. Continuous Relaxation: Solve the continuous program $\min_{x\in[0,1]^N} c^\top x$ subject to $F_i(x)\ge b_i$ for all $i$.
3. Lipschitz Greedy Prefixes: For each $f_i$, run a short greedy phase to ensure all remaining element marginals are bounded, obtaining Lipschitzness.
4. Randomized Rounding: Independently select elements according to $x_j$, apply a union bound to ensure that for $r=O(1)$ all constraints hold with high probability.
5. Greedy Fix: For unsatisfied $f_i$, run standard submodular maximization with knapsack constraints on the shortfall.

This combination yields the aforementioned $(1+\epsilon)\alpha$-cost, $(1-e^{-\alpha}-\epsilon)$-coverage bicriteria (Bajpai et al., 14 Jul 2025). For weighted coverage and deletion-closed set systems, the rounding is augmented by thresholding "heavy" and "shallow" elements via the natural set cover LP, with the rounding lemma handling the latter (Bajpai et al., 14 Jul 2025).

4. Complexity and Hardness

Hardness results mirror those of set cover and submodular cover (Iyer et al., 2013, Bajpai et al., 14 Jul 2025):

• For fixed $r$, MSC is $\Omega(\log n)$-hard to approximate unless P=NP; this follows via reduction from classical set cover.
• For $r$ growing with $n$, even bicriteria relaxations become hard: if $r = \omega(1)$, no polynomial-time exact or near-exact coverage scheme can avoid an $\Omega(\log r)$ dependency in the approximation factor (Bajpai et al., 14 Jul 2025).
• Submodular cover with multiple constraints is polynomial-time equivalent (via threshold search and duality) to the submodular knapsack problem (Iyer et al., 2013).
• The aggregate-sum algorithm, the main practical baseline when $r$ is large, offers no better than an $O(\log r)$ ratio.

This suggests that for small, fixed $r$, the problem is significantly more tractable: stronger concentration from randomized rounding and controlled enumeration of "failure" events makes near-single-constraint guarantees feasible (Bajpai et al., 14 Jul 2025).

5. Extensions: Stochastic and Adaptive Multi-Submodular Cover

Stochastic and adaptive generalizations arise in online learning, sensor placement, and active testing (Al-Thani et al., 2022):

• In the adaptive setting, the coverage functions become mappings $f_i:2^{N\times\Omega}\rightarrow \mathbb{R}_+$ over partial realizations, and adaptive policies choose elements based on previously observed outcomes.
• Under monotonicity, coverability, and adaptive submodularity, adaptive greedy algorithms attain $O(\log Q)$-approximations for the risk-neutral (expected cost) version, where $Q$ is the target value (Al-Thani et al., 2022).
• For the $k$-function extension, the policy at each decision chooses an item maximizing the sum of expected marginal coverage across all unsatisfied $f_i$, normalized by remaining coverage gaps and cost.
• The approximation factor remains $4(1+\ln(Q/\eta))$, and for higher moments, $(p+1)^{p+1}(1+\ln Q)^p$. All constants are tight, up to lower-order factors, for set cover–hard instances (Al-Thani et al., 2022).

A plausible implication is that, even in adaptive and stochastic models, a single greedy rule suffices to obtain multi-function cover up to logarithmic factors.

6. Representative Applications

Several canonical applications demonstrate the broad modelling power of the multi-submodular cover framework (Bajpai et al., 14 Jul 2025):

• Colorful Vertex Cover: Each color class requires covering a specified number of incident edges. For constant $r$, this MSC instance recovers the best known $(2+\epsilon)$-approximation.
• Facility Location with Multiple Outliers: Each client class must receive minimum coverage, encompassing fair and robust facility assignment.
• Sum-of-Radii with Multiple Outliers: Extends classical geometric k-center and k-median objectives to simultaneously satisfy multiple group-specific coverage goals.
• Information Propagation and Recommendations: For example, in social networks or personalized recommender systems, robust coverage across diverse user or group objectives maps precisely to the multi-submodular model.

In all these cases, the strong guarantees for fixed $r$ and the flexible bicriteria paradigm enable high-quality, scalable algorithms matching or approaching single-objective limits (Bajpai et al., 14 Jul 2025).

7. Connections, Open Questions, and Future Directions

There is a tight connection—via duality and reduction—between the multi-submodular cover and submodular knapsack problems. Any bicriteria or single-criterion approximation for one directly yields guarantees for the other (Iyer et al., 2013). For increasing $r$, the challenge remains to close the logarithmic gap between upper and lower bounds, especially for structured covering functions.

Open directions include:

• Characterizing the landscape for large $r$, especially under additional structure (e.g., matroid or deletion-closed set systems) (Bajpai et al., 14 Jul 2025).
• Improving adaptivity gaps and derandomization for real-world applications with correlated constraints (Al-Thani et al., 2022).
• Exploring combinatorial, polytime algorithms with improved (polylogarithmic or constant) factors for specialized application domains.

A plausible implication is that advances in concentration bounds, LP relaxations, and distributed optimization may further tighten the approximability frontier for multi-submodular cover and its extensions.