Minimizing Convex Functions with Integral Minimizers

Jiang, Haotian

doi:10.1137/1.9781611976465.61

Cited by 14 publications

(12 citation statements)

References 30 publications

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“…Figuring out what the query complexity of SFM and matroid intersection, regardless of adaptivity, is an intriguing question. Currently, the best known upper bounds on the query complexity for SFM is Õ(N 2 ) [Jia21]. For matroid intersection, the best known upper bounds using rank-oracles is Õ(N 1.5 ) [CLS + 19] and with independence oracles it is Õ(N 9/5 ) [BvdBMN21].…”

Section: Discussionmentioning

confidence: 99%

“…Since then, a lot of work [Cun85, IFF01, Sch00, Orl09, IO09, CJK14, LJJ15, LSW15, CLSW17, DVZ18, ALS20, Jia21] has been done trying to understand the query complexity of SFM. The current best known algorithms are an O(N 3 )-query polynomial-time and an O(N 2 log N )-query exponential time algorithm by Jiang [Jia21] building on the works [LSW15,DVZ18], an Õ(N 2 log M )-query and time algorithm by Lee, Sidford, and Wong [LSW15] where |f (S)| ≤ M for all S ⊆ U , and an Õ(N M 2 ) query and time algorithm by Axelrod, Liu, and Sidford [ALS20] improving upon [CLSW17].…”

Section: Introductionmentioning

confidence: 99%

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A Polynomial Lower Bound on the Number of Rounds for Parallel Submodular Function Minimization and Matroid Intersection

Chakrabarty¹,

Chen²,

Khanna³

2021

Preprint

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Submodular function minimization (SFM) and matroid intersection are fundamental discrete optimization problems with applications in many fields. It is well known that both of these can be solved making poly(N ) queries to a relevant oracle (evaluation oracle for SFM and rank oracle for matroid intersection), where N denotes the universe size. However, all known polynomial query algorithms are highly adaptive, requiring at least N rounds of querying the oracle. A natural question is whether these can be efficiently solved in a highly parallel manner, namely, with poly(N ) queries using only poly-logarithmic rounds of adaptivity.An important step towards understanding the adaptivity needed for efficient parallel SFM was taken recently in the work of Balkanski and Singer who showed that any SFM algorithm making poly(N ) queries necessarily requires Ω(log N/ log log N ) rounds. This left open the possibility of efficient SFM algorithms in poly-logarithmic rounds. For matroid intersection, even the possibility of a constant round, poly(N ) query algorithm was not hitherto ruled out.In this work, we prove that any, possibly randomized, algorithm for submodular function minimization or matroid intersection making poly(N ) queries requires 1 Ω N 1/3 rounds of adaptivity. In fact, we show a polynomial lower bound on the number of rounds of adaptivity even for algorithms that make at most 2 N 1−δ queries, for any constant δ > 0. Therefore, even though SFM and matroid intersection are efficiently solvable, they are not highly parallelizable in the oracle model.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Polynomial Lower Bound on the Number of Rounds for Parallel Submodular Function Minimization and Matroid Intersection

Chakrabarty¹,

Chen²,

Khanna³

2021

Preprint

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“…F , as noted earlier, corresponds to the differential entropy of a Gaussian vector with covariance M , which is a submodular function, plus an additional modular term. The problem of submodular function minimization has a long and rich history, beginning with the seminal works of Grötschel et al [1981Grötschel et al [ , 2012 and continuing to the current day [Iwata et al, 2001, Schrijver, 2000, Lee et al, 2015, Dadush et al, 2018, Jiang, 2021. Thus, we can invoke any of these known polynomial-time algorithms for submodular function minimization to recover the chain components in topological order.…”

Section: Technical Overviewmentioning

confidence: 99%

Identifiability of AMP chain graph models

Wang,

Bhattacharyya

2021

Preprint

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We study identifiability of Andersson-Madigan-Perlman (AMP) chain graph models, which are a common generalization of linear structural equation models and Gaussian graphical models. AMP models are described by DAGs on chain components which themselves are undirected graphs.For a known chain component decomposition, we show that the DAG on the chain components is identifiable if the determinants of the residual covariance matrices of the chain components are monotone non-decreasing in topological order. This condition extends the equal variance identifiability criterion for Bayes nets, and it can be generalized from determinants to any super-additive function on positive semidefinite matrices. When the component decomposition is unknown, we describe conditions that allow recovery of the full structure using a polynomial time algorithm based on submodular function minimization. We also conduct experiments comparing our algorithm's performance against existing baselines * .

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“…Since then a long line of work has developed faster and simpler (combinatorial) algorithms for SFM [Sch00, IFF01, Orl09, LSW15, DVZ21, Jia21]. The work of [Jia21] shows that SFM can be solved by a deterministic algorithm making O(n 2 log n) queries to an evaluation oracle. By the isolating cut lemma of [LP20] this immediately also gives an O(n 2 ) query randomized algorithm for sym-SFM [CQ21,MN21].…”

Section: Introduction and Contributionmentioning

confidence: 99%

Cut query algorithms with star contraction

Apers¹,

Efron²,

Gawrychowski³

et al. 2022

Preprint

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We study the complexity of determining the edge connectivity of a simple graph with cut queries. We show that (i) there is a bounded-error randomized algorithm that computes edge connectivity with O(n) cut queries, and (ii) there is a bounded-error quantum algorithm that computes edge connectivity with Õ( √ n) cut queries. To prove these results we introduce a new technique, called star contraction, to randomly contract edges of a graph while preserving nontrivial minimum cuts. In star contraction vertices randomly contract an edge incident on a small set of randomly chosen "center" vertices. In contrast to the related 2-out contraction technique of Ghaffari, Nowicki, and Thorup [SODA'20], star contraction only contracts vertex-disjoint star subgraphs, which allows it to be efficiently implemented via cut queries.The O(n) bound from item (i) was not known even for the simpler problem of connectivity, and it improves the O(n log 3 n) upper bound by Rubinstein, Schramm, and Weinberg [ITCS'18]. The bound is tight under the reasonable conjecture that the randomized communication complexity of connectivity is Ω(n log n), an open question since the seminal work of Babai, Frankl, and Simon [FOCS'86]. The bound also excludes using edge connectivity on simple graphs to prove a superlinear randomized query lower bound for minimizing a symmetric submodular function. The quantum algorithm from item (ii) gives a nearly-quadratic separation with the randomized complexity, and addresses an open question of Lee, Santha, and Zhang [SODA'21]. The algorithm can alternatively be viewed as computing the edge connectivity of a simple graph with Õ( √ n) matrix-vector multiplication queries to its adjacency matrix. Finally, we demonstrate the use of star contraction outside of the cut query setting by designing a one-pass semi-streaming algorithm for computing edge connectivity in the complete vertex arrival setting. This contrasts with the edge arrival setting where two passes are required.

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Minimizing Convex Functions with Integral Minimizers

Cited by 14 publications

References 30 publications

A Polynomial Lower Bound on the Number of Rounds for Parallel Submodular Function Minimization and Matroid Intersection

A Polynomial Lower Bound on the Number of Rounds for Parallel Submodular Function Minimization and Matroid Intersection

Identifiability of AMP chain graph models

Cut query algorithms with star contraction

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