[Paper Review] Multi-Objective Maximization of Monotone Submodular Functions with Cardinality Constraint

Rajan Udwani|arXiv (Cornell University)|Nov 17, 2017

Complexity and Algorithms in Graphs4 citations

TL;DR

This paper presents a fast, asymptotically optimal algorithm for multi-objective maximization of monotone submodular functions under a cardinality constraint, using Multiplicative Weight Updates (MWU) to achieve a $(1-1/e)^2 - \delta$ approximation in $\tilde{O}(n/\delta^3)$ time. It improves upon prior work by enabling super-constant numbers of objectives while maintaining strong theoretical guarantees and outperforming heuristics in experiments.

ABSTRACT

We consider the problem of multi-objective maximization of monotone submodular functions subject to cardinality constraint, often formulated as $\\max_{|A|=k}\\min_{i\\in\\{1,\\dots,m\\}}f_i(A)$. While it is widely known that greedy methods work well for a single objective, the problem becomes much harder with multiple objectives. In fact, Krause et al.\\ (2008) showed that when the number of objectives $m$ grows as the cardinality $k$ i.e., $m=\\Omega(k)$, the problem is inapproximable (unless $P=NP$). On the other hand, when $m$ is constant Chekuri et al.\\ (2010) showed a randomized $(1-1/e)-\\epsilon$ approximation with runtime (number of queries to function oracle) $n^{m/\\epsilon^3}$. %In fact, the result of Chekuri et al.\\ (2010) is for the far more general case of matroid constant. We focus on finding a fast and practical algorithm that has (asymptotic) approximation guarantees even when $m$ is super constant. We first modify the algorithm of Chekuri et al.\\ (2010) to achieve a $(1-1/e)$ approximation for $m=o(\\frac{k}{\\log^3 k})$. This demonstrates a steep transition from constant factor approximability to inapproximability around $m=\\Omega(k)$. Then using Multiplicative-Weight-Updates (MWU), we find a much faster $\ ilde{O}(n/\\delta^3)$ time asymptotic $(1-1/e)^2-\\delta$ approximation. While the above results are all randomized, we also give a simple deterministic $(1-1/e)-\\epsilon$ approximation with runtime $kn^{m/\\epsilon^4}$. Finally, we run synthetic experiments using Kronecker graphs and find that our MWU inspired heuristic outperforms existing heuristics.

Motivation & Objective

• To address the challenge of maximizing multiple monotone submodular objectives simultaneously under a cardinality constraint, especially when the number of objectives $m$ grows with the cardinality $k$.
• To close the gap between known inapproximability results (when $m = \Omega(k)$) and existing approximation algorithms that only work for constant $m$.
• To develop a fast, scalable algorithm with strong theoretical approximation guarantees even when $m$ is super-constant, specifically $m = o(k / \log^3 k)$.
• To design a practical heuristic based on Multiplicative Weight Updates (MWU) that outperforms existing heuristics in synthetic experiments on Kronecker graphs.

Proposed method

• Modifies the randomized algorithm of Chekuri et al. (2010) to achieve a $(1-1/e)$ approximation for $m = o(k / \log^3 k)$, leveraging stage-wise greedy selection with thresholding based on relative gains.
• Introduces a novel MWU-based algorithm that maintains a distribution over elements and updates weights based on marginal gains across all objectives, achieving a $(1-1/e)^2 - \delta$ approximation.
• Uses a randomized sampling process where each element is included independently with probability proportional to its weight, and proves that the expected function value is at least $(1-1/e)^2 - \delta$ times the optimal.
• Employs a two-phase approach: first, a greedy selection phase that builds a core set of size $O(m / \varepsilon^3)$, then applies MWU to refine the solution over multiple rounds.
• Analyzes the MWU algorithm using a potential function argument and submodularity to bound the expected marginal gain, showing convergence to a near-optimal solution.
• Provides a deterministic variant using exhaustive search over candidate sets, achieving a $(1-1/e) - \varepsilon$ approximation with runtime $kn^{m/\varepsilon^4}$.

Experimental results

Research questions

• RQ1 Can we achieve a constant-factor approximation for multi-objective submodular maximization when the number of objectives $m$ grows with the cardinality $k$, beyond the constant $m$ regime?
• RQ2 What is the asymptotic threshold for approximability as $m$ increases relative to $k$, and does a sharp transition from approximability to inapproximability occur?
• RQ3 Can we design a fast algorithm with $\tilde{O}(n/\delta^3)$ runtime that achieves a $(1-1/e)^2 - \delta$ approximation for super-constant $m$?
• RQ4 How does the MWU-based heuristic compare to existing heuristics in practice, especially on large-scale synthetic graphs?

Key findings

• The paper establishes a sharp threshold for approximability: when $m = o(k / \log^3 k)$, a $(1-1/e)$ approximation is achievable, but for $m = \Omega(k)$, the problem becomes inapproximable unless $P = NP$.
• The proposed MWU-based algorithm achieves a $(1-1/e)^2 - \delta$ approximation in $\tilde{O}(n/\delta^3)$ time, significantly improving over the $n^{m/\varepsilon^3}$ runtime of prior randomized algorithms.
• A deterministic algorithm achieves a $(1-1/e) - \varepsilon$ approximation with runtime $kn^{m/\varepsilon^4}$, providing a strong alternative when randomness is undesirable.
• The MWU-inspired heuristic outperforms existing heuristics in synthetic experiments on Kronecker graphs, demonstrating practical superiority despite being a heuristic.
• Theoretical analysis shows that the expected value of the MWU-sampled set is at least $(1-1/e)^2 - \delta$ times the optimal, under submodularity and monotonicity assumptions.
• The analysis proves that each objective function can contribute to the selection of at most $1/\varepsilon^3$ elements in the first phase, ensuring bounded set size and enabling efficient computation.