[Paper Review] Submodular approximation: sampling-based algorithms and lower bounds

Zoya Svitkina, Lisa Fleischer|arXiv (Cornell University)|May 7, 2008

Complexity and Algorithms in GraphsComputer Science37 references71 citations

TL;DR

This paper introduces sampling-based algorithms for submodular optimization problems such as submodular load balancing, sparsest cut, and balanced cut, achieving approximation guarantees of order $\sqrt{n/\ln n}$, and proves matching lower bounds, establishing this as the inherent limit of polynomial-query approximation. It also provides improved lower bounds for approximating submodular functions everywhere and identifies structural constraints under which tighter approximations are possible.

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.

Motivation & Objective

• To generalize classical computer science problems—like load balancing, graph cuts, and knapsack—by replacing simple objective functions with general submodular functions.
• To study the approximability of these generalized problems using only polynomially many queries to a function-value oracle.
• To establish tight upper and lower bounds on the approximation ratio achievable in polynomial time.
• To investigate the limits of approximating submodular functions everywhere, especially under monotonicity and two-partition structure.
• To identify structural conditions under which better approximation guarantees are possible, beyond the general $\sqrt{n/\ln n}$ barrier.

Proposed method

• Designs sampling-based algorithms that query the submodular function oracle to construct approximate solutions with guarantees of order $\sqrt{n/\ln n}$.
• Uses a $K$-biased and $L$-biased walk on a grid to analyze the behavior of submodular functions along paths, leveraging submodularity and concavity.
• Applies a sequence of $K$-steps and $L$-steps to bound the contribution of individual function increments, proving that $K$-steps account for a constant fraction of the total increase.
• Employs a reduction to a two-partition (2P) function structure to derive tighter bounds, showing that stronger lower bounds require more general submodular functions.
• Uses induction and inequalities based on the ratio $K/L$ to prove that balanced pairs in the walk sequences maintain submodular constraints.
• Analyzes the function value along paths from $(0,0)$ to $(k,l)$, using concavity and submodularity to lower-bound $f(k,0)$ and $f(0,l)$ in terms of $F(n)$.

Experimental results

Research questions

• RQ1 What is the best possible approximation ratio achievable for submodular load balancing, sparsest cut, and balanced cut using a polynomial number of queries to a function-value oracle?
• RQ2 Can the $\sqrt{n/\ln n}$ approximation ratio be improved for submodular minimization with cardinality constraints?
• RQ3 Are there structural classes of submodular functions for which better-than-$\sqrt{n/\ln n}$ approximation is possible?
• RQ4 What are the inherent limitations of approximating a submodular function everywhere using only oracle queries?
• RQ5 Can tighter lower bounds be established for approximating submodular functions, especially under monotonicity or two-partition structure?

Key findings

• The paper establishes that $\sqrt{n/\ln n}$ is the tight approximation ratio for submodular load balancing, submodular sparsest cut, and submodular balanced cut, matching both upper and lower bounds.
• It proves that this $\sqrt{n/\ln n}$ bound is inherent for general submodular functions, meaning no polynomial-query algorithm can do better.
• For the problem of approximating a submodular function everywhere, the paper provides a new lower bound, showing that even monotone functions cannot be approximated better than $\Omega(\sqrt{n/\ln n})$ in the worst case.
• The authors show that for two-partition (2P) functions, where $f(S)$ depends only on $|S \cap R|$ and $|S \cap \bar{R}|$, approximation guarantees close to the $\sqrt{n/\ln n}$ lower bound are achievable.
• The results imply that any significantly stronger lower bounds for submodular function approximation must rely on more general submodular functions beyond the 2P class.
• The analysis demonstrates that the $\sqrt{n/\ln n}$ barrier arises from the interplay between submodularity, concavity, and the structure of oracle query sequences.