Improved approximation for directed cut problems

Agarwal, Amit; Alon, Noga; Charikar, Moses

doi:10.1145/1250790.1250888

Cited by 51 publications

(95 citation statements)

References 16 publications

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“…As we mentioned earlier, Dir-MulC admitsÕ(n 11/23 ) approximation [2] and is hard to approximate within a factor of Ω(2 log 1− n ) assuming N P = ZP P [9]. A k approximation is trivial.…”

Section: Overview Of Resultsmentioning

confidence: 99%

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Approximating Multicut and the Demand Graph

Chekuri

Madan

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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In the minimum Multicut problem, the input is an edgeweighted supply graph G = (V, E) and a demand graph H = (V, F ). Either G and H are directed (Dir-MulC) or both are undirected (Undir-MulC). The goal is to remove a minimum weight set of supply edges E ⊆ E such that in G − E there is no path from s to t for any demand edge (s, t) ∈ F . Undir-MulC admits O(log k)-approximation where k is the number of edges in H while the best known approximation for DirMulC is min{k,Õ(|V | 11/23 )}. These approximations are obtained by proving corresponding results on the multicommodity flow-cut gap. In this paper we consider the role that the structure of the demand graph plays in determining the approximability of Multicut. We obtain several new positive and negative results.In undirected graphs our main result is a 2-approximation in n O(t) time when the demand graph excludes an induced matching of size t. This gives a constant factor approximation for a specific demand graph that motivated this work, and is based on a reduction to uniform metric labeling and not via the flow-cut gap.In contrast to the positive result for undirected graphs, we prove that in directed graphs such approximation algorithms can not exist. We prove that, assuming the Unique Games Conjecture (UGC), that for a large class of fixed demand graphs Dir-MulC cannot be approximated to a factor better than the worstcase flow-cut gap. As a consequence we prove that for any fixed k, assuming UGC, Dir-MulC with k demand pairs is hard to approximate to within a factor better than k. On the positive side, we obtain a k approximation when the demand graph excludes certain graphs as an induced subgraph. This generalizes the known 2 approximation for directed Multiway Cut to a larger class of demand graphs.

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Section: Overview Of Resultsmentioning

confidence: 99%

“…Assuming N P = ZP P it is hard to approximate Dir-MulC to within a factor of Ω(2 log 1− n ) [9]; evidence is also presented in [9] that it could be hard to approximate to within an Ω(n δ ) factor for some fixed δ > 0. The best known approximation is min{k,Õ(n 11/23 )} [2]; here n = |V |. Note that a k-approximation is trivial.…”

Section: Introductionmentioning

confidence: 99%

Approximating Multicut and the Demand Graph

Chekuri

Madan

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…al. [1] to anÕ(n 11/23 ) ratio approximation 2 . In [26], an O(n 2/3 /opt 1/3 ) approximation algorithm for uniform costs directed multicut, is presented.…”

Section: Related Workmentioning

confidence: 98%

“…Here opt is the optimum value. If the optimum is at least n 0.566 , the ratio of [26] is better than the one of [1]. The [26] algorithm is also the only non-trivial combinatorial approximation algorithm for Directed multicut.…”

Section: Related Workmentioning

confidence: 99%

On the Advantage of Overlapping Clusters for Minimizing Conductance

2013

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Graph clustering is an important problem with applications to bioinformatics, community discovery in social networks, distributed computing, and more. While most of the research in this area has focused on clustering using disjoint clusters, many real datasets have inherently overlapping clusters. We compare overlapping and non-overlapping clusterings in graphs in the context of minimizing their conductance. It is known that allowing clusters to overlap gives better results in practice. We prove that overlapping clustering may be significantly better than non-overlapping clustering with respect to conductance, even in a theoretical setting.For minimizing the maximum conductance over the clusters, we give examples demonstrating that allowing overlaps can yield significantly better clusterings, namely, one that has much smaller optimum. In addition for the min-max variant, the overlapping version admits a simple approximation algorithm, while our algorithm for the non-overlapping version is complex and yields a worse approximation ratio due to the presence of the additional constraint. Somewhat surprisingly, for the problem of minimizing the sum of conductances, we found out that allowing overlap does not help. We show how to apply a general technique to transform any overlapping clustering into a non-overlapping one with only a modest increase in the sum of conductances. This uncrossing technique is of independent interest and may find further applications in the future.We consider this work as a step toward rigorous comparison of overlapping and nonoverlapping clusterings and hope that it stimulates further research in this area. * IBM T.J.Watson research center.

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“…Moreover, the best known approximation ratio for IDMC is O(n 11 23 ) (Agarwal et al, 2007), which yields a better (in terms of worst case error) approximation algorithm for MSDAG C . An interesting open problem would compare these two decoding approximation algorithms empirically for semantic parsing decoding and in terms of expected performance (or error) both in general as well as specifically for semantic parsing decoding.…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

Proceedings of the ACL 2014 Workshop on Semantic Parsing

Lewis¹,

Steedman²

2014

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While there has been significant recent work on learning semantic parsers for specific task/ domains, the results don't transfer from one domain to another domains. We describe a project to learn a broad-coverage semantic lexicon for domain independent semantic parsing. The technique involves several bootstrapping steps starting from a semantic parser based on a modest-sized hand-built semantic lexicon. We demonstrate that the approach shows promise in building a semantic lexicon on the scale of WordNet, with more coverage and detail that currently available in widely-used resources such as VerbNet. We view having such a lexicon as a necessary prerequisite for any attempt at attaining broad-coverage semantic parsing in any domain. The approach we described applies to all word classes, but in this paper we focus here on verbs, which are the most critical phenomena facing semantic parsing. Introduction and MotivationRecently we have seen an explosion of work on learning semantic parsers (e.g., Matuszek, et al, 2012;Tellex et al, 2013; Branavan et al, 2010, Chen et al, 2011). While such work shows promise, the results are highly domain dependent and useful only for that domain. One cannot, for instance, reuse a lexical entry learned in one robotic domain in another robotic domain, let alone in a database query domain. Furthermore, the techniques being developed require domains that are simple enough so that the semantic models can be produced, either by hand or induced from the application. Language in general, however, involves much more complex concepts and connections, including discussion of involves abstract concepts, such as plans, theories, political views, and so on. It is not clear how the techniques currently being developed could be generalized to such language.The challenge we are addressing is learning a broad-coverage, domain-independent semantic parser, i.e., a semantic parser that can be used in any domain. At present, there is a tradeoff between the depth of semantic representation produced and the coverage of the techniques. One of the critical gaps in enabling more general, deeper semantic systems is the lack of any broadcoverage deep semantic lexicon. Such a lexicon must contain at least the following information: i. an enumeration of the set of distinct senses for the word (e.g., as in WordNet, PropBank), linked into an ontology that supports reasoning ii. For each sense, we would have• Deep argument structure, i.e., semantic roles with selectional preferences • Constructions that map syntax to the deep argument structure (a.k.a. linking rules) • Lexical entailments that characterize the temporal consequences of the event described by the verb The closest example to such lexical entries can be found in VerbNet (Kipper et al, 2008), a handbuilt resource widely used for a range of general applications. An example entry from VerbNet is seen in Figure 1, which describes a class of verbs called murder-42.1. VerbNet clusters verbs by the constructions they take, not by sense or meaning, alt...

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Improved approximation for directed cut problems

Cited by 51 publications

References 16 publications

Approximating Multicut and the Demand Graph

Approximating Multicut and the Demand Graph

On the Advantage of Overlapping Clusters for Minimizing Conductance

Proceedings of the ACL 2014 Workshop on Semantic Parsing

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