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The densest subgraph problem in a graph (\dsg), in the simplest form, is the following. Given an undirected graph $G=(V,E)$ find a subset $$S \subseteq V$$ of vertices that maximizes the ratio $|E(S)|/|S|$ where $E(S)$ is the set of edges with both endpoints in $$S$$. \dsg and several of its variants are well-studied in theory and practice and have many applications in data mining and network analysis. In this paper we study fast algorithms and structural aspects of \dsg via the lens of \emph{supermodularity}. For this we consider the densest supermodular subset problem (\dssp): given a non-negative supermodular function $$f: 2^V \rightarrow \mathbb{R}_+$, maximize $f(S)/|S|$$. For \dsg we describe a simple flow-based algorithm that outputs a $$(1-\eps)$-approximation in deterministic $$\tilde{O}(m/\eps)$$ time where $$m$$ is the number of edges. Our algorithm is the first to have a near-linear dependence on $$m$$ and $$1/\eps$$ and improves previous methods based on an LP relaxation. It generalizes to hypergraphs, and also yields a faster algorithm for directed \dsg. Greedy peeling algorithms have been very popular for \dsg and several variants due to their efficiency, empirical performance, and worst-case approximation guarantees. We describe a simple peeling algorithm for \dssp and analyze its approximation guarantee in a fashion that unifies several existing results. Boob et al.\ \cite{bgpstww-20} developed an \emph{iterative} peeling algorithm for \dsg which appears to work very well in practice, and made a conjecture about its convergence to optimality. We affirmatively answer their conjecture, and in fact prove that a natural generalization of their algorithm converges to a $$(1-\eps)$$-approximation for \emph{any} supermodular function $$f$$; the key to our proof is to consider an LP formulation that is derived via the \Lovasz extension of a supermodular function. For \dsg the bound on the number of iterations we prove is $$O(\frac{\Delta \ln |V|}{\lambda^*}\cdot \frac{1}{\eps^2})$ where $$\Delta$$ is the maximum degree and $$\lambda^*$$ is the optimum value. Our work suggests that iterative peeling can be an effective heuristic for several objectives considered in the literature. Finally, we show that the $$2$$-approximation for densest-at-least-$$k$$ subgraph \cite{ks-09} extends to the supermodular setting. We also give a unified analysis of the peeling algorithm for this problem, and via this analysis derive an approximation guarantee for a generalization of \dssp to maximize $$f(S)/g(|S|)$ for a concave function $$g$$.
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Tractable Relaxations of Composite Functions
In this paper, we introduce new relaxations for the hypograph of composite functions assuming that the outer function is supermodular and concave extendable. Relying on a recently introduced relaxation framework, we devise a separation algorithm for the graph of the outer function over P, where P is a special polytope to capture the structure of each inner function using its finitely many bounded estimators. The separation algorithm takes [Formula: see text] time, where d is the number of inner functions and n is the number of estimators for each inner function. Consequently, we derive large classes of inequalities that tighten prevalent factorable programming relaxations. We also generalize a decomposition result and devise techniques to simultaneously separate hypographs of various supermodular, concave-extendable functions using facet-defining inequalities. Assuming that the outer function is convex in each argument, we characterize the limiting relaxation obtained with infinitely many estimators as the solution of an optimal transport problem. When the outer function is also supermodular, we obtain an explicit integral formula for this relaxation.
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- Award ID(s):
- 1727989
- PAR ID:
- 10382117
- Date Published:
- Journal Name:
- Mathematics of Operations Research
- Volume:
- 47
- Issue:
- 2
- ISSN:
- 0364-765X
- Page Range / eLocation ID:
- 1110 to 1140
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1287/moor.2021.1162
