Approximability of Combinatorial Problems with Multi-agent Submodular Cost Functions

Goel, Gagan; Karande, Chinmay; Tripathi, Pushkar; Wang, Lei

doi:10.1109/focs.2009.81

Cited by 50 publications

(89 citation statements)

References 21 publications

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“…2. Second, recent interest has emerged in the literature regarding the implications of extending classical combinatorial problems (such as Shortest Path, Minimum Spanning Tree, or Set Cover) from a sum-of-weights to submodular cost functions [5,22,23,29,31,36,37,50,70,76]. None of this work, however, has addressed cuts.…”

Section: Relation To the Literaturementioning

confidence: 97%

“…where (22) demands that ϕ must, in addition to satisfying the common flow conservation, reside within the submodular polyhedron P( f ). This more restrictive constraint replaces the edge-wise capacity constraints that occur when f is a sum of weights.…”

Section: Relaxation and The Flow Dualmentioning

confidence: 99%

“…Lower bound References Set cover Ω(ln |U |) [31] Minimum spanning tree Ω(n) [22] Shortest path Ω(n 2/3 ) [22] Perfect matching Ω(n) [22] Minimum cut…”

Section: Problemmentioning

confidence: 99%

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Graph cuts with interacting edge weights: examples, approximations, and algorithms

Jegelka

Bilmes

2016

Math. Program.

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We study an extension of the classical graph cut problem, wherein we replace the modular (sum of edge weights) cost function by a submodular set function defined over graph edges. Special cases of this problem have appeared in different applications in signal processing, machine learning, and computer vision. In this paper, we connect these applications via the generic formulation of "cooperative graph cuts", for which we study complexity, algorithms, and connections to polymatroidal network flows. Finally, we compare the proposed algorithms empirically.

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Section: Relation To the Literaturementioning

confidence: 97%

Section: Relaxation and The Flow Dualmentioning

confidence: 99%

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Graph cuts with interacting edge weights: examples, approximations, and algorithms

Jegelka

Bilmes

2016

Math. Program.

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“…Once i 0 is found, the coefficients can be computed according to (7) in O(n) time. Therefore we can check for a violated inequality of the form (10) in O(n log n) time.…”

Section: Lemmamentioning

confidence: 99%

“…As mentioned earlier, an inequality description of the convex hull of the epigraph of a general submodular function follows from the classical works [15,16]. In the presence of constraints, submodular minimization is in general NP-hard and in most cases very hard to approximate [10,12,17]. Even for the cardinality constraint, the special case of size constrained minimum cut problem is NP-hard since it can be reduced to graph partitioning problem [18].…”

Section: Introductionmentioning

confidence: 99%

Polyhedral results for a class of cardinality constrained submodular minimization problems

Ahmed

2017

Discrete Optimization

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a r t i c l e i n f o Article history: Available online xxxx MSC: 90C11 90C30 a b s t r a c tMotivated by concave cost combinatorial optimization problems, we study the following mixed integer nonlinear set:is a concave function, n and k are positive integers, a ∈ R n is a nonnegative vector, e ∈ R n is a vector of ones, and x ′ y denotes the scalar product of vectors x and y of same dimension. A standard linearization approach for P is to exploit the fact that f (a ′ x) is submodular with respect to the binary vector x. We extend this approach to take the cardinality constraint e ′ x ≤ k into account and provide a full description of the convex hull of P when the vector a has identical components. We also develop a family of facet-defining inequalities when the vector a has nonidentical components. Computational results using the proposed inequalities in a branch-and-cut framework to solve mean-risk knapsack problems show significant decrease in both time and the number of nodes over standard methods.

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Computing a Minimum Color Path in Edge-Colored Graphs

Kumar

2019

Lecture Notes in Computer Science

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Approximability of Combinatorial Problems with Multi-agent Submodular Cost Functions

Cited by 50 publications

References 21 publications

Graph cuts with interacting edge weights: examples, approximations, and algorithms

Graph cuts with interacting edge weights: examples, approximations, and algorithms

Polyhedral results for a class of cardinality constrained submodular minimization problems

Computing a Minimum Color Path in Edge-Colored Graphs

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