Submodular approximation: sampling-based algorithms and lower bounds (0805.1071v3)

Abstract: We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimum-makespan scheduling, submodular sparsest cut and submodular balanced cut, which generalize their respective graph cut problems, as well as submodular function minimization with a cardinality lower bound. We establish upper and lower bounds for the approximability of these problems with a polynomial number of queries to a function-value oracle. The approximation guarantees for most of our algorithms are of the order of sqrt(n/ln n). We show that this is the inherent difficulty of the problems by proving matching lower bounds. We also give an improved lower bound for the problem of approximately learning a monotone submodular function. In addition, we present an algorithm for approximately learning submodular functions with special structure, whose guarantee is close to the lower bound. Although quite restrictive, the class of functions with this structure includes the ones that are used for lower bounds both by us and in previous work. This demonstrates that if there are significantly stronger lower bounds for this problem, they rely on more general submodular functions.

• The paper introduces sampling-based algorithms that achieve √n/n approximations for generalized submodular optimization problems.
• The paper establishes matching lower bounds, indicating that improving approximations with polynomial queries is unlikely.
• The study leverages submodular minimization and partitioning techniques to generalize classical problems like load balancing and graph cuts.

Submodular Approximation: Sampling-based Algorithms and Lower Bounds

The paper in question explores advancements in approximation algorithms for generalized classical computer science problems incorporating submodular functions, showcasing innovative computational techniques and theoretical insights. Submodular functions, integral due to their discrete convexity-like properties, emerge in various applications such as graph cuts, load balancing, and facility location. This paper extends several classical optimization problems by replacing their objective functions with submodular functions, thereby generalizing problems like load balancing, sparsest cut, and balanced cut, among others.

Problem Generalizations and Definitions

This paper meticulously defines new problems, focusing on a general subset of computer science optimization problems aimed at submodular contexts. These problems include:

• Submodular Load Balancing (SLB): Challenges like partitioning collections into balanced subsets where submodular functions determine "size," expanding minimum-makespan scheduling.
• Submodular Sparsest Cut (SSC) and Submodular Balanced Cut (SBC): Generalizations of graph cut problems simplifying to a search for minimizing cut capacity respective to submodular cost functions.
• Submodular Minimization with Cardinality Lower Bound (SML): Extending knapsack-like problems, where the cost function is submodular instead of linear sums.

Core Contributions

The paper’s principal contributions encompass both algorithmic and lower bound developments across these newly defined problems, yielding substantial insights into their inherent computational difficulty:

1. Upper Bound Approximations: The paper develops polynomial-time sampling-based algorithms offering $\sqrt{n}/{ n}$ approximations, showcasing their complexity relative to their classical counterparts. These algorithms leverage randomized sampling to efficiently explore submodular function spaces.
2. Lower Bound Proofs: Establishes matching lower bounds demonstrating that achieving improved approximation with polynomial queries is unlikely, positioning the approximation orders necessary to efficiently tackle submodular function objectives.
3. Algorithmic Techniques: Innovatively employs submodular function minimization and strategic partitioning for problem-solving, incorporating new avenues for economizing approximation approaches.

Implications and Future Directions

The implications of this work are multi-faceted. Practically, these research insights push forward strategies for problem-solving in complex settings where submodular functions are prevalent, such as network design and resource allocation problems that align with economies of scale scenarios. Theoretically, these results emphasize the complexity inherent within submodular functions despite their typically advantageous properties, suggesting that more refined analytical techniques or focused problem subclasses may necessary for further improvement.

A pivotal frontier for future research suggested by the authors involves developing algorithms for more narrowly defined submodular functions, leveraging specific properties like monotonicity or submodular decomposition structures. Exploring different oracle models or quantifying function characteristics may also yield breakthroughs in approximation efficiency.

The paper adeptly captures submodular functions' intriguing complexity while tracking pathways to harness their benefits, constituting a substantial advancement in theoretical and applied computational sciences.