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Approximation algorithms for submodular multiway partition
 CoRR
"... Abstract — We study algorithms for the SUBMODULAR MUL ..."
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Cited by 9 (3 self)
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Abstract — We study algorithms for the SUBMODULAR MUL
Submodular cost allocation problem and applications
 PROC. OF ICALP, 354–366
, 2011
"... We study the Minimum SubmodularCost Allocation problem (MSCA). In this problem we are given a finite ground set V and k nonnegative submodular set functions f1,..., fk on V. The objective is to partition V into k (possibly empty) sets A1, · · · , Ak such that the sum ∑k i=1 fi(Ai) is minimi ..."
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Cited by 6 (4 self)
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We study the Minimum SubmodularCost Allocation problem (MSCA). In this problem we are given a finite ground set V and k nonnegative submodular set functions f1,..., fk on V. The objective is to partition V into k (possibly empty) sets A1, · · · , Ak such that the sum ∑k i=1 fi(Ai) is minimized. Several wellstudied problems such as the nonmetric facility location problem, multiwaycut in graphs and hypergraphs, and uniform metric labeling and its generalizations can be shown to be special cases of MSCA. In this paper we consider a convexprogramming relaxation obtained via the Lovászextension for submodular functions. This allows us to understand several previous relaxations and rounding procedures in a unified fashion and also develop new formulations and approximation algorithms for related problems. In particular, we give a (1.5 − 1/k)approximation for the hypergraph multiway partition problem. We also give a min{2(1−1/k), H∆}approximation for the hypergraph multiway cut problem when ∆ is the maximum hyperedge size. Both problems generalize the multiway cut problem in graphs and the hypergraph cut problem is approximation equivalent to the nodeweighted multiway cut problem in graphs.
Convex Relaxations for Learning Bounded Treewidth Decomposable Graphs
 In Proceedings of the 30th Internationl Conference on Machine Learning (ICML
, 2013
"... We consider the problem of learning the structure of undirected graphical models with bounded treewidth, within the maximum likelihood framework. This is an NPhard problem and most approaches consider local search techniques. In this paper, we pose it as a combinatorial optimization problem, which ..."
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Cited by 2 (1 self)
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We consider the problem of learning the structure of undirected graphical models with bounded treewidth, within the maximum likelihood framework. This is an NPhard problem and most approaches consider local search techniques. In this paper, we pose it as a combinatorial optimization problem, which is then relaxed to a convex optimization problem that involves searching over the forest and hyperforest polytopes with special structures, independently. A supergradient method is used to solve the dual problem, with a runtime complexity of O(k3nk+2 log n) for each iteration, where n is the number of variables and k is a bound on the treewidth. We compare our approach to stateoftheart methods on synthetic datasets and classical benchmarks, showing the gains of the novel convex approach. 1
APPROXIMATION ALGORITHMS FOR SUBMODULAR OPTIMIZATION AND GRAPH PROBLEMS
, 2013
"... In this thesis, we consider combinatorial optimization problems involving submodular functions and graphs. The problems we study are NPhard and therefore, assuming that P 6 = NP, there do not exist polynomialtime algorithms that always output an optimal solution. In order to cope with the intracta ..."
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In this thesis, we consider combinatorial optimization problems involving submodular functions and graphs. The problems we study are NPhard and therefore, assuming that P 6 = NP, there do not exist polynomialtime algorithms that always output an optimal solution. In order to cope with the intractability of these problems, we focus on algorithms that construct approximate solutions: An approximation algorithm is a polynomialtime algorithm that, for any instance of the problem, it outputs a solution whose value is within a multiplicative factor ρ of the value of the optimal solution for the instance. The quantity ρ is the approximation ratio of the algorithm and we aim to achieve the smallest ratio possible. Our focus in this thesis is on designing approximation algorithms for several combinatorial optimization problems. In the first part of this thesis, we study a class of constrained submodular minimization problems. We introduce a model that captures allocation problems with submodular costs and we give a generic approach for designing approximation algorithms for problems in this model. Our model captures several problems of interest, such as nonmetric facility location, multiway cut problems in graphs and hypergraphs, uniform metric labeling and its generalization
Complexity of Submodular Partitioning Problems
, 2012
"... KWAY SUBMP is the following optimization problem. Let V be a finite ground set and f: 2V → R+ be a nonnegative submodular set function1 on V that is given by a value oracle. Let k ≥ 2 be a nonnegative integer. Find a partition of V into nonempty sets V1, V2,..., Vk to minimize ∑k i=1 f(Vi). KW ..."
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KWAY SUBMP is the following optimization problem. Let V be a finite ground set and f: 2V → R+ be a nonnegative submodular set function1 on V that is given by a value oracle. Let k ≥ 2 be a nonnegative integer. Find a partition of V into nonempty sets V1, V2,..., Vk to minimize ∑k i=1 f(Vi). KWAY SUBMP is solvable in polynomial time for k = 2, 3 (see [4]. However, its complexity is not known when k is a fixed constant, in particular when k = 4. A special case is when f is symmetric (KWAY SYMSUBMP). The complexity of this special case is also open for fixed k> 4. Let G = (V,E) be a hypergraph. Two problems of interest are the following. One is the KCUT problem: remove as few hyperedges as possible to create at least k connected components. The other is the KPARTITION problem: partition V into nonempty sets V1, V2,..., Vk to minimize