On the Pipage Rounding Algorithm for Submodular Function Maximization — A View From Discrete Convex Analysis

Shioura, Akiyoshi

doi:10.1142/s1793830909000063

Cited by 18 publications

(9 citation statements)

References 27 publications

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“…It is noted that a similar statement is shown in Shioura [42] for a monotone M -concave function defined on 2 N ; we here extend the result to the case of non-monotone M -concave function defined on a subset of 2 N .…”

Section: -Convex Functionssupporting

confidence: 85%

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Polynomial-Time Approximation Schemes for Maximizing Gross Substitutes Utility Under Budget Constraints

Shioura

2015

Mathematics of OR

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We consider the maximization of a gross substitutes utility function under budget constraints. This problem naturally arises in applications such as exchange economies in mathematical economics and combinatorial auctions in (algorithmic) game theory. We show that this problem admits a polynomial-time approximation scheme (PTAS). More generally, we present a PTAS for maximizing a discrete concave function called an M -concave function under budget constraints. Our PTAS is based on rounding an optimal solution of a continuous relaxation problem, which is shown to be solvable in polynomial time by the ellipsoid method. We also consider the maximization of the sum of two M -concave functions under a single budget constraint. This problem is a generalization of the budgeted max-weight matroid intersection problem to the one with certain nonlinear objective functions. We show that this problem also admits a PTAS.2 In the optimal allocation problem discussed in [3,25], we aim at optimally allocating items to consumers, where each item is available by only one unit (see Example 2.5 for details). In contrast, in the problem discussed in [36] and [32, Ch. 11] we consider producers of items in addition to consumers, and allow to have multiple units of items. It is shown in [36] and [32, Ch. 11] that there exists an optimal allocation (equilibrium, more precisely) and such an allocation can be found efficiently under the assumptions such as the gross substitute condition of consumers' utility functions.3 The concept of M -concavity is originally introduced for a function defined on (the set of integral vectors in) an integral generalized polymatroid (see [34]). In this paper a restricted class of M -concave functions is considered; see Section 2.

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Section: -Convex Functionssupporting

confidence: 85%

“…If w(v) = 1 (v ∈ N ), then f is an ordinary rank function of the matroid (N, I). Every weighted rank function is a GS utility function [42]. 2…”

Section: Preliminariesmentioning

confidence: 99%

Polynomial-Time Approximation Schemes for Maximizing Gross Substitutes Utility Under Budget Constraints

Shioura

2015

Mathematics of OR

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“…The maximization problem has also been studied for variants of submodular functions. Shioura [24] investigated the maximization of discrete convex functions. Soma et al [26] provided a (1 − 1/e)-approximation algorithm for maximizing a monotone lattice submodular function under a knapsack constraint.…”

Section: Related Workmentioning

confidence: 99%

“…Any DR-submodular function is lattice submodular; i.e., DR-submodularity is stronger than lattice submodularity. 1 The problem of maximizing DR-submodular functions over Z E naturally appears in the submodular welfare problem [16,24] and the budget allocation problem with decreasing influence probabilities [26]. Nevertheless, only a few studies have considered this problem.…”

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confidence: 99%

Maximizing monotone submodular functions over the integer lattice

Soma

Yoshida

2018

Math. Program.

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The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade. In this paper, we address the problem for functions defined over the integer lattice.Suppose that a non-negative monotone submodular function f : Z n + → R + is given via an evaluation oracle. Assume further that f satisfies the diminishing return property, which is not an immediate consequence of submodularity when the domain is the integer lattice. Given this, we design polynomial-time (1 − 1/e − ǫ)-approximation algorithms for a cardinality constraint, a polymatroid constraint, and a knapsack constraint. For a cardinality constraint, we also provide a (1 − 1/e − ǫ)-approximation algorithm with slightly worse time complexity that does not rely on the diminishing return property.

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“…The applications of discrete convex analysis include economics, system analysis for electrical circuits, phylogenetic analysis, etc. Shioura [22] studied maximization of the sum of M ♮ -concave set functions subject to a matroid constraint. He showed that pipage rounding [2] can be explained from the viewpoint of discrete convexity.…”

Section: Related Workmentioning

confidence: 99%

A New Approximation Guarantee for Monotone Submodular Function Maximization via Discrete Convexity

Soma,

Yoshida

2017

Preprint

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In monotone submodular function maximization, approximation guarantees based on the curvature of the objective function have been extensively studied in the literature. However, the notion of curvature is often pessimistic, and we rarely obtain improved approximation guarantees, even for very simple objective functions.In this paper, we provide a novel approximation guarantee by extracting an M ♮ -concave function h : 2 E → R + , a notion in discrete convex analysis, from the objective function f : 2 E → R + . We introduce the notion of h-curvature, which measures how much f deviates from h, and show that we can obtain a (1 − γ/e − ǫ)-approximation to the problem of maximizing f under a cardinality constraint in polynomial time for any constant ǫ > 0. Then, we show that we can obtain nontrivial approximation guarantees for various problems by applying the proposed algorithm.

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On the Pipage Rounding Algorithm for Submodular Function Maximization — A View From Discrete Convex Analysis

Cited by 18 publications

References 27 publications

Polynomial-Time Approximation Schemes for Maximizing Gross Substitutes Utility Under Budget Constraints

Polynomial-Time Approximation Schemes for Maximizing Gross Substitutes Utility Under Budget Constraints

Maximizing monotone submodular functions over the integer lattice

A New Approximation Guarantee for Monotone Submodular Function Maximization via Discrete Convexity

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