Max-Min-Max Submodular Optimization

• Max-min-max submodular optimization is a framework for selecting a subset that maximizes the worst-case value across multiple monotone submodular functions under a budget constraint.
• The approach employs innovative discrete algorithms that iteratively use greedy selection, linear programming, and Monte Carlo evaluations to achieve scalable, near-optimal results.
• Applications span robust experimental design, fair influence maximization, and novel fair centrality maximization, ensuring balanced performance across diverse objectives.

Max-min-max submodular optimization, often referred to in the literature as multiobjective submodular maximization under a cardinality constraint, considers selecting a subset of elements from a finite ground set so as to simultaneously maximize the worst-case value across multiple monotone submodular objective functions. Formally, given submodular functions $f_c: 2^V \to \mathbb{R}_{\ge 0}$ indexed by $c\in C$ and a budget $B$, the problem is to find $S\subseteq V, |S|\leq B$ such that $\min_{c\in C} f_c(S)$ is maximized. This formulation is central to robust combinatorial optimization, encompassing applications in fair influence maximization, robust experimental design, and (as newly introduced) fair centrality maximization, where ensuring good performance under each objective is essential.

1. Formal Problem Definition and Representative Applications

Let $V$ be a finite ground set of $n$ elements, and let $C$ be an index set of size $k$. For each $c \in C$, $f_c: 2^{V} \to \mathbb{R}_{\ge 0}$ is a monotone submodular function: for all $S \subseteq T \subseteq V$ and $v \in V\setminus T$, $f_c(S) \leq f_c(T)$ and $$f_c(S \cup \{v\}) - f_c(S) \geq f_c(T \cup \{v\}) - f_c(T)$$. The objective is: $$\max_{S \subseteq V,\, |S| \le B} \min_{c \in C} f_c(S)$$ This framework arises in:

• Robust experimental design: Simultaneously maximizing a family $f_\theta(S)$ over uncertain parameters $\theta$.
• Fair influence maximization: Each color $c$ denotes a demographic group, with $f_c(S)$ measuring expected influence spread in group $c$.
• Fair centrality maximization: The new application introduced, optimizing groupwise harmonic centrality in graphs after adding up to $B$ edges.

2. Continuous Relaxation, Multilinear Extension, and Practical Limitations

Theoretical approaches to this problem have explored continuous relaxations via the multilinear extension. For $x \in [0,1]^V$, define $R(x)$ as a random subset of $V$ containing $j$ independently with probability $x_j$; then the multilinear extension is $F_c(x) = \mathbb{E}[f_c(R(x))]$. The relaxed problem is: $$\max_{x \in [0,1]^V,\, \sum_j x_j \le B} \min_{c \in C} F_c(x)$$ However, exactly evaluating $F_c(x)$ involves summing over $2^n$ sets and is thus intractable. Practical approaches rely on Monte Carlo estimation or continuous-greedy methods (e.g., Frank–Wolfe), but these require repeated estimation of $F_c(x)$ and its gradients, leading to significant computational overhead especially as $n$ and $k$ increase.

3. Discrete (Greedy-Style) Asymptotically Optimal Algorithm

A new scalable, discrete algorithm attains a $(1-1/e-\varepsilon)$ approximation with high probability, avoiding the multilinear extension and relying solely on standard submodular oracle calls. The method constructs $S$ iteratively via $B$ rounds, where in each round it solves a linear program (LP) over the simplex to select an element to add:

Algorithm (sketch):

1. Run $r = \lceil \log(2/\delta) \rceil$ independent trials.
2. For each trial: a. Initialize $S \leftarrow \emptyset$. b. For $i = 1$ to $B$, i. Let $S_{prev} = S$. ii. Solve the LP over $x \in \Delta_V$ and $\xi \in \mathbb{R}$: maximize $\xi$ subject to $$\sum_v x_v [B f_c(v|S_{prev}) + f_c(S_{prev})] \ge \xi$$ for every $c \in C$, $\sum_v x_v = 1$, $x_v \ge 0$. iii. Sample $v \sim x$ and add $v$ to $S$. c. If $\min_c f_c(S) > \min_c f_c(S_{best})$, update $S_{best} \leftarrow S$.
3. Return $S_{best}$.

Key performance results are as follows:

• In expectation over the random process, for each $c\in C$, $\mathbb{E}[f_c(S)] \ge (1-1/e)\mathrm{OPT}$.
• With high probability (martingale concentration argument, Theorem 6), $f_c(S)\geq (1-1/e-\varepsilon)\mathrm{OPT}$ for all $c$, provided $\mathrm{OPT} \gg M\log k/\varepsilon^2$, where $M = \max_c \max_v f_c(v|\emptyset)$.

The algorithm relies only on computing $f_c(S)$ and $f_c(v|S)$ via submodular oracles. The LP can be efficiently approximated via a multiplicative-weights (MWU) subroutine and lazy evaluations in $O(nk)$ oracle calls per outer iteration.

4. Algorithmic Rounding and Ensuring Integral Solutions

Since the main greedy step maintains $S$ as an integral set at all times, explicit rounding is unnecessary. To remove the technical requirement $\mathrm{OPT} \gg M \log k/\varepsilon^2$, a preprocessing phase identifies and includes up to $B'$ elements of highest marginal gain (across colors), forming a set $T$. Modified objectives $\tilde f_c(A) = f_c(A \cup T)$ are constructed, now with all singleton marginals $\leq \mathrm{OPT}/B'$. A continuous relaxation is then run on budget $B-|T|$, yielding a fractional solution that is rounded via "swap rounding" to an integral set $\tilde S$. Lemma 9 and swap rounding analysis ensure a final $(1-1/e-O(\varepsilon))$ approximation guarantee for all $c$.

5. Computational Complexity and Scalability Features

The algorithm achieves practical scalability under the following resource bounds:

• Submodular oracle calls: $O(nBk \log(1/\delta))$.
• Total running time: $O(nB^3(k/\varepsilon^2)\log k\,\log(1/\delta))$.

Crucial speed-ups include:

• The MWU approach to LP solving with $O(B^2 (M^2/\varepsilon^2 \log k))$ rounds and each round using lazy marginal gain bounding.
• Preprocessing to reduce the impact of large-gain elements and further control the $B$ dependence.
• Lazy evaluations of marginal gains (maintaining upper bounds $g_c(v)\geq f_c(v|S)$).

Empirically, the LP can be solved via standard solvers (e.g., Gurobi) or by MWU. The introduction of a "tilt" parameter $\varphi > 1$ in the LP objective biases the allocation toward colors currently yielding the minimum value, which improves practical convergence.

6. Applications: Fair Centrality Maximization

A significant new application is groupwise harmonic centrality in networks. For a node $v$ in a directed graph $G=(V,A)$, classical harmonic centrality is $h_G(v) = \sum_{u \neq v} 1/d_G(u,v)$; adding an edge to $v$ increases this quantity submodularly. The fair variant seeks to maximize

$$h^\mathrm{min}_G(v) = \min_{c\in C} \left[ \frac{1}{|V_c\setminus\{v\}|} \sum_{u\in V_c\setminus\{v\}} \frac{1}{d_G(u,v)} \right]$$

Selecting up to $B$ edges $F$ to add to $G$ defines $f_c(F)$ as the post-edit, groupwise normalized harmonic sum. Each $f_c$ is nonnegative monotone submodular, and the resulting task is $\max_{|F|\leq B} \min_c f_c(F)$. Standard continuous methods fail to scale to graphs with tens of thousands of nodes, whereas the new discrete method retains its theoretical and practical guarantees in this regime.

7. Empirical Performance and Comparative Analysis

Experiments were performed on:

• Max-$k$-cover instances ($k=20$, $n=64$) from stochastic Kronecker, Barabási–Albert, and Erdős–Rényi models.
• 20 Amazon co-purchase networks (up to $n \approx 10^4$ nodes) for fair centrality with $k=2$.
• Simulated Antelope Valley social networks ($n=500$, $k$ up to 13) for fair influence maximization.

Compared algorithms included the LP Greedy method (with both Gurobi and MWU linear program solving, plus lazy updates), round-robin greedy (Udwani-style), Saturate (bi-criteria method), Udwani’s MWU ((1-1/e)$^2$-approximation), and Frank–Wolfe continuous method (for influence). Major findings:

• LP Greedy achieves the highest min-cover on max-$k$-cover, with $\approx 10\times$ fewer oracle calls than MWU.
• On fair centrality, LP Greedy outperforms Saturate and MWU both in objective and running time, solving up to 10,000-node graphs in minutes.
• For fair influence, LP Greedy matches or outperforms Frank–Wolfe for nontrivial budgets and is more broadly applicable.
• Ablation studies reveal that 20 outer repetitions suffice and a tilt factor $\varphi \approx 10$ optimizes practical performance.

Overall, this algorithm bridges the prior theoretical-practical gap, attaining the asymptotically optimal $(1-1/e-\varepsilon)$ approximation via efficient, scalable discrete algorithms deployable on large-scale real-world tasks in fair optimization of submodular objectives.