 NSFPAR ID:
 10362731
 Date Published:
 Journal Name:
 Annual Symposium on Foundations of Computer Science
 ISSN:
 02725428
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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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 wellstudied 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 nonnegative supermodular function $f: 2^V \rightarrow \mathbb{R}_+$, maximize $f(S)/S$. For \dsg we describe a simple flowbased 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 nearlinear 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 worstcase 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{bgpstww20} 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 densestatleast$k$ subgraph \cite{ks09} 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 $g$.more » « less

We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a $11/e\epsilon$ approximation for monotone functions and a $1/e\epsilon$ approximation for nonmonotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $O(\log^2{n}/\epsilon^3)$, which is an exponential speedup over the existing algorithms. We obtain the first parallel algorithm for nonmonotone submodular maximization subject to packing constraints. Our algorithm achieves a $1/e\epsilon$ approximation using $O(\log(n/\epsilon) \log(1/\epsilon) \log(n+m)/ \epsilon^2)$ parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a $11/e\epsilon$ approximation in $O(\log(n/\epsilon)\log(m)/\epsilon^2)$ parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective (Mahoney et al., 2016). Our results apply more generally to the problem of maximizing a diminishing returns submodular (DRsubmodular) function.more » « less

Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground set
N , a weight function ,$$w: N \rightarrow \mathbb {R}_+$$ $w:N\to {R}_{+}$r monotone submodular functions over$$f_1,f_2,\ldots ,f_r$$ ${f}_{1},{f}_{2},\dots ,{f}_{r}$N and requirements the goal is to find a minimum weight subset$$k_1,k_2,\ldots ,k_r$$ ${k}_{1},{k}_{2},\dots ,{k}_{r}$ such that$$S \subseteq N$$ $S\subseteq N$ for$$f_i(S) \ge k_i$$ ${f}_{i}\left(S\right)\ge {k}_{i}$ . We refer to this problem as$$1 \le i \le r$$ $1\le i\le r$MultiSubmodCover and it was recently considered by HarPeled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260 HarPeled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$ $r=1$MultiSubmodCover generalizes the wellknown Submodular Set Cover problem (SubmodSC ), and it can also be easily reduced toSubmodSC . A simple greedy algorithm gives an approximation where$$O(\log (kr))$$ $O(log(kr\left)\right)$ and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm for$$k = \sum _i k_i$$ $k={\sum}_{i}{k}_{i}$MultiSubmodCover that covers each constraint to within a factor of while incurring an approximation of$$(11/e\varepsilon )$$ $(11/e\epsilon )$ in the cost. Second, we consider the special case when each$$O(\frac{1}{\epsilon }\log r)$$ $O(\frac{1}{\u03f5}logr)$ is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover ($$f_i$$ ${f}_{i}$PartialSC ), covering integer programs (CIPs ) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the highlevel model and the lens of submodularity in addressing this class of covering problems. 
Berry, Jonathan ; Shmoys, David ; Cowen, Lenore ; Naumann, Uwe (Ed.)Continuous DRsubmodular functions are a class of functions that satisfy the Diminishing Returns (DR) property, which implies that they are concave along nonnegative directions. Existing works have studied monotone continuous DRsubmodular maximization subject to a convex constraint and have proposed efficient algorithms with approximation guarantees. However, in many applications, e. g., computing the stability number of a graph and meanfield inference for probabilistic logsubmodular models, the DRsubmodular function has the additional property of being strongly concave along nonnegative directions that could be utilized for obtaining faster convergence rates. In this paper, we first introduce and characterize the class of strongly DRsubmodular functions and show how such a property implies strong concavity along nonnegative directions. Then, we study Lsmooth monotone strongly DRsubmodular functions that have bounded curvature, and we show how to exploit such additional structure to obtain algorithms with improved approximation guarantees and faster convergence rates for the maximization problem. In particular, we propose the SDRFW algorithm that matches the provably optimal approximation ratio after only iterations, where c ∈ [0,1] and μ ≥ 0 are the curvature and the strong DRsubmodularity parameter. Furthermore, we study the Projected Gradient Ascent (PGA) method for this problem and provide a refined analysis of the algorithm with an improved approximation ratio (compared to ½ in prior works) and a linear convergence rate. Given that both algorithms require knowledge of the smoothness parameter L, we provide a novel characterization of L for DRsubmodular functions showing that in many cases, computing L could be formulated as a convex optimization problem, i. e., a geometric program, that could be solved efficiently. Experimental results illustrate and validate the efficiency and effectiveness of our algorithms.more » « less

We study the classic Maximum Independent Set problem under the notion of stability introduced by Bilu and Linial (2010): a weighted instance of Independent Set is γstable if it has a unique optimal solution that remains the unique optimal solution under multiplicative perturbations of the weights by a factor of at most γ ≥ 1. The goal then is to efficiently recover this “pronounced” optimal solution exactly. In this work, we solve stable instances of Independent Set on several classes of graphs: we improve upon previous results by solving \tilde{O}(∆/sqrt(log ∆))stable instances on graphs of maximum degree ∆, (k − 1)stable instances on kcolorable graphs and (1 + ε)stable instances on planar graphs (for any fixed ε > 0), using both combinatorial techniques as well as LPs and the SheraliAdams hierarchy. For general graphs, we give an algorithm for (εn)stable instances, for any fixed ε > 0, and lower bounds based on the planted clique conjecture. As a byproduct of our techniques, we give algorithms as well as lower bounds for stable instances of Node Multiway Cut (a generalization of Edge Multiway Cut), by exploiting its connections to Vertex Cover. Furthermore, we prove a general structural result showing that the integrality gap of convex relaxations of several maximization problems reduces dramatically on stable instances. Moreover, we initiate the study of certified algorithms for Independent Set. The notion of a γcertified algorithm was introduced very recently by Makarychev and Makarychev (2018) and it is a class of γapproximation algorithms that satisfy one crucial property: the solution returned is optimal for a perturbation of the original instance, where perturbations are again multiplicative up to a factor of γ ≥ 1 (hence, such algorithms not only solve γstable instances optimally, but also have guarantees even on unstable instances). Here, we obtain ∆certified algorithms for Independent Set on graphs of maximum degree ∆, and (1 + ε)certified algorithms on planar graphs. Finally, we analyze the algorithm of Berman and Fürer (1994) and prove that it is a ((∆+1)/3 + ε)certified algorithm for Independent Set on graphs of maximum degree ∆ where all weights are equal to 1.more » « less