Submodular Functions Maximization Problems

Buchbinder, Niv; Feldman, Moran

doi:10.1201/9781351236423-42

Cited by 37 publications

(53 citation statements)

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“…Using the monotone (b, 1−e −b b )-balanced CR scheme for the matroid polytope from Chekuri, Vondrák, and Zenklusen [13], one obtains a monotone (b, ( 1−e −b b ) 2 )-balanced CR scheme for the bipartite matching polytope. Moreover, there exists an elegant monotone CR scheme for the general matching polytope that is (b, e −2b )-balanced (see [8] for a description of the scheme reported in [22]). This scheme is very similar to the one we described in Example 1, which we showed to be monotone and (1, 1 8 )-balanced.…”

Section: Prior Results On Contention Resolution Schemes For Matchingsmentioning

confidence: 99%

“…Examples of constraint types for which monotone constant-balanced CR schemes are known include matroid constraints, knapsack constraints, column-restricted packing constraints, interval packing constraints, and matching constraints (see [8,13,19]). These schemes are with respect to the polytope P = P F for matroid, interval packing, and matching constraints, or with respect to a natural linear relaxation for the other constraint types.…”

Section: Theorem 1 ([13]) Let π Be a Monotone (B C)-balanced Cr Schementioning

confidence: 99%

“…What is important to point out is that this CR scheme is not guaranteed to be monotone. 8 However, to obtain the rounding guarantees claimed by Theorem 1, monotonicity is required.…”

Section: Definition 2 Formentioning

confidence: 99%

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An optimal monotone contention resolution scheme for bipartite matchings via a polyhedral viewpoint

Bruggmann

Zenklusen

2020

Math. Program.

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Relaxation and rounding approaches became a standard and extremely versatile tool for constrained submodular function maximization. One of the most common rounding techniques in this context are contention resolution schemes. Such schemes round a fractional point by first rounding each coordinate independently, and then dropping some elements to reach a feasible set. Also the second step, where elements are dropped, is typically randomized. This leads to an additional source of randomization within the procedure, which can complicate the analysis. We suggest a different, polyhedral viewpoint to design contention resolution schemes, which avoids to deal explicitly with the randomization in the second step. This is achieved by focusing on the marginals of a dropping procedure. Apart from avoiding one source of randomization, our viewpoint allows for employing polyhedral techniques. Both can significantly simplify the construction and analysis of contention resolution schemes. We show how, through our framework, one can obtain an optimal monotone contention resolution scheme for bipartite matchings, which has a balancedness of 0.4762. So far, only very few results are known about optimality of monotone contention resolution schemes. Our contention resolution scheme for the bipartite case also improves the lower bound on the correlation gap for bipartite matchings. Furthermore, we derive a monotone contention resolution scheme for matchings that significantly improves over the previously best one. More precisely, we obtain a balancedness of 0.4326, improving on a prior 0.1997-balanced scheme. At the same time, our scheme implies that the currently best lower bound on the correlation gap for matchings is not tight. Our results lead to improved approximation factors for various constrained submodular function maximization problems over a combination of matching constraints with further constraints.

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Section: Prior Results On Contention Resolution Schemes For Matchingsmentioning

confidence: 99%

Section: Theorem 1 ([13]) Let π Be a Monotone (B C)-balanced Cr Schementioning

confidence: 99%

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An optimal monotone contention resolution scheme for bipartite matchings via a polyhedral viewpoint

Bruggmann

Zenklusen

2020

Math. Program.

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“…One of these new ingredients is the multilinear relaxation approach [13] that brought powerful continuous optimization techniques to submodular function maximization. We refer the reader to a recent survey [12] for some pointers to the new developments on greedy and continuous methods, and to [11] on local search methods. Recent applications of submodular function maximization to large data sets, and technological trends, have motivated new directions of research.…”

Section: Introductionmentioning

confidence: 99%

“…To obtain an integer solution we need to round x. Several powerful and general rounding strategies have been developed over the years including pipage rounding, swap rounding, and contention resolution schemes [13,7,17,16,28,26,12]. These establish constant factor integrality gaps for the multilinear relaxation for many constraints of interest.…”

Section: Introductionmentioning

confidence: 99%

Submodular Function Maximization in Parallel via the Multilinear Relaxation

Chekuri

Quanrud

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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Balkanski and Singer [4] recently initiated the study of adaptivity (or parallelism) for constrained submodular function maximization, and studied the setting of a cardinality constraint. Subsequent improvements for this problem by Balkanski, Rubinstein, and Singer [6] and Ene and Nguyen [21] resulted in a near-optimal (1−1/e−ǫ)-approximation in O(log n/ǫ 2 ) rounds of adaptivity. Partly motivated by the goal of extending these results to more general constraints, we describe parallel algorithms for approximately maximizing the multilinear relaxation of a monotone submodular function subject to packing constraints. Formally our problem is to maximize F (x) over x ∈ [0, 1] n subject to Ax ≤ 1 where F is the multilinear relaxation of a monotone submodular function. Our algorithm achieves a near-optimal (1 − 1/e − ǫ)-approximation in O(log 2 m log n/ǫ 4 ) rounds where n is the cardinality of the ground set and m is the number of packing constraints. For many constraints of interest, the resulting fractional solution can be rounded via known randomized rounding schemes that are oblivious to the specific submodular function. We thus derive randomized algorithms with poly-logarithmic adaptivity for a number of constraints including partition and laminar matroids, matchings, knapsack constraints, and their intersections.Our algorithm takes a continuous view point and combines several ideas ranging from the continuous greedy algorithm of [38,13], its adaptation to the MWU framework for packing constraints [19], and parallel algorithms for packing LPs [31,41]. For the basic setting of cardinality constraints, this viewpoint gives rise to an alternative, simple to understand algorithm that matches recent results [6,21]. Our algorithm to solve the multilinear relaxation is deterministic if it is given access to a value oracle for the multilinear extension and its gradient; this is possible in some interesting cases such as the coverage function of an explicitly given set system.

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On Maximizing Sums of Non-monotone Submodular and Linear Functions

2023

Algorithmica

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We study the problem of () as defined by Bodek and Feldman (Maximizing sums of non-monotone submodular and linear functions: understanding the unconstrained case, arXiv:2204.03412, 2022): given query access to a non-negative submodular function $$f:2^{{\mathcal {N}}}\rightarrow {\mathbb {R}}_{\ge 0}$$ f : 2 N → R ≥ 0 and a linear function $$\ell :2^{{\mathcal {N}}}\rightarrow {\mathbb {R}}$$ ℓ : 2 N → R over the same ground set $${\mathcal {N}}$$ N , output a set $$T\subseteq {\mathcal {N}}$$ T ⊆ N approximately maximizing the sum $$f(T)+\ell (T)$$ f ( T ) + ℓ ( T ) . An algorithm is said to provide an $$(\alpha ,\beta )$$ ( α , β ) -approximation for if it outputs a set T such that $${\mathbb {E}}[f(T)+\ell (T)]\ge \max _{S\subseteq {\mathcal {N}}}[\alpha \cdot f(S)+\beta \cdot \ell (S)]$$ E [ f ( T ) + ℓ ( T ) ] ≥ max S ⊆ N [ α · f ( S ) + β · ℓ ( S ) ] . We also consider the setting where S and T are constrained to be independent in a given matroid, which we refer to as Constrained (). The special case of with monotone f has been extensively studied (Sviridenko et al. in Math Oper Res 42(4):1197–1218, 2017; Feldman in Algorithmica 83(3):853–878, 2021; Harshaw et al., in: International conference on machine learning, PMLR, 2634–2643, 2019), whereas we are aware of only one prior work that studies with non-monotone f (Lu et al. in Optimization 1–27, 2023), and that work constrains $$\ell $$ ℓ to be non-positive. In this work, we provide improved $$(\alpha ,\beta )$$ ( α , β ) -approximation algorithms for both and with non-monotone f. Specifically, we are the first to provide nontrivial $$(\alpha ,\beta )$$ ( α , β ) -approximations for where the sign of $$\ell $$ ℓ is unconstrained, and the $$\alpha $$ α we obtain for improves over (Bodek and Feldman in Maximizing sums of non-monotone submodular and linear functions: understanding the unconstrained case, arXiv:2204.03412, 2022) for all $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) . We also prove new inapproximability results for and , as well as 0.478-inapproximability for maximizing a submodular function where S and T are subject to a cardinality constraint, improving a 0.491-inapproximability result due to Oveis Gharan and Vondrak (in: Proceedings of the twenty-second annual ACM-SIAM symposium on discrete algorithms, SIAM, pp 1098–1116, 2011).

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Submodular Functions Maximization Problems

Cited by 37 publications

References 1 publication

An optimal monotone contention resolution scheme for bipartite matchings via a polyhedral viewpoint

An optimal monotone contention resolution scheme for bipartite matchings via a polyhedral viewpoint

Submodular Function Maximization in Parallel via the Multilinear Relaxation

On Maximizing Sums of Non-monotone Submodular and Linear Functions

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