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Submodular Approximation: Samplingbased Algorithms and Lower Bounds
, 2008
"... We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimummakespan scheduling, submodular sparsest cu ..."
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We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimummakespan scheduling, submodular sparsest cut and submodular balanced cut, which generalize their respective graph cut problems, as well as submodular function minimization with a cardinality lower bound. We establish upper and lower bounds for the approximability of these problems with a polynomial number of queries to a functionvalue oracle. The approximation guarantees for most of our algorithms are of the order of √ n/lnn. We show that this is the inherent difficulty of the problems by proving matching lower bounds. We also give an improved lower bound for the problem of approximately learning a monotone submodular function. In addition, we present an algorithm for approximately learning submodular functions with special structure, whose guarantee is close to the lower bound. Although quite restrictive, the class of functions with this structure includes the ones that are used for lower bounds both by us and in previous work. This demonstrates that if there are significantly stronger lower bounds for this problem, they rely on more general submodular functions.
Minmax graph partitioning and small set expansion
, 2011
"... We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are: (i) the k parts need to be of equal s ..."
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Cited by 14 (2 self)
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We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are: (i) the k parts need to be of equal size, and (ii) the parts must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O ( √ log n log k)approximation algorithm. This improves over an O(log 2 n) approximation for the second version due to Svitkina and Tardos [22], and roughly O(k log n) approximation for the first version that follows from other previous work. We also give an improved O(1)approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the SmallSet Expansion problem. In this problem, we are given a graph G and the goal is to find a nonempty set S ⊆ V of size S  ≤ ρn with minimum edgeexpansion. We give an O ( √ log n log (1/ρ)) bicriteria approximation algorithm for the general case of SmallSet Expansion, and O(1) approximation algorithm for graphs that exclude any fixed minor.
Unbalanced graph cuts
 In Proc. 13th European Symp. on Algorithms
, 2005
"... Abstract. We introduce the Minimumsize boundedcapacity cut (MinSBCC) problem, in which we are given a graph with an identified source and seek to find a cut minimizing the number of nodes on the source side, subject to the constraint that its capacity not exceed a prescribed bound B. Besides being ..."
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Abstract. We introduce the Minimumsize boundedcapacity cut (MinSBCC) problem, in which we are given a graph with an identified source and seek to find a cut minimizing the number of nodes on the source side, subject to the constraint that its capacity not exceed a prescribed bound B. Besides being of interest in the study of graph cuts, this problem arises in many practical settings, such as in epidemiology, disaster control, military containment, as well as finding dense subgraphs and communities in graphs. In general, the MinSBCC problem is NPcomplete. We present an efficient ( 1 λ,
ABSTRACT Pricing of Partially Compatible Products
"... In this paper, we examine a duopolistic market where the two firms compete to sell a system of components. Components are digital (firms have unlimited supply at no marginal cost), and customers are homogeneous in their component preferences. Each customer will assemble a utility maximizing system b ..."
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In this paper, we examine a duopolistic market where the two firms compete to sell a system of components. Components are digital (firms have unlimited supply at no marginal cost), and customers are homogeneous in their component preferences. Each customer will assemble a utility maximizing system by purchasing each necessary component from one of the two firms. While components from the same firm are always compatible, pairwise compatibility of components from rival firms may vary; in addition to utility due to the quality of the system purchased, customers have negative utility for purchasing incompatible parts. We investigate algorithms and hardness results for profitmaximizing decisions of the firms with regards to their pricesetting, component valueenhancing and compatibilityenabling strategies. The users ’ behavior can be modeled as a minimum cut computation, and the company’s strategies require addressing novel and interesting questions about graph cuts and flows. We develop a polynomialtime algorithm for finding profitmaximizing prices if the qualities and compatibilities are fixed. On the other hand, we show that finding profitmaximizing quality improvements is equivalent to the Maximum Size Bounded Capacity Cut problem, and thus NPcomplete. Finally, for the problem of improving compatibilities to maximize the price, we give polynomial approximation hardness results even in very restricted cases, but show that if all components have uniform prices, and quality differences are small, then an approximation can be found in polynomial time.
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
MINMAX GRAPH PARTITIONING AND SMALL SET EXPANSION∗
"... Abstract. We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be ..."
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Abstract. We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be of equal size, and where they must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O( logn log k) approximation algorithm. This improves over an O(log2 n) approximation for the second version due to Svitkina and Tardos [Minmax multiway cut, in APPROXRANDOM, 2004, Springer, Berlin, 2004], and roughly O(k logn) approximation for the first version that follows from other previous work. We also give an O(1) approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the smallset expansion problem. In this problem, we are given a graph G and the goal is to find a nonempty set S ⊆ V of size S  ≤ ρn with minimum edge expansion. We give an O( logn log (1/ρ)) bicriteria approximation algorithm for smallset expansion in general graphs, and an improved factor of O(1) for graphs that exclude any fixed minor.
Prophylactic Vaccination . . .
, 2008
"... Motivated by preventative vaccination in a graph against the worstcase outbreak of an infectious disease, we propose new important graph cut problems. In the most basic problem MinMax Component Size, we are given a capacity bound, and the goal is to remove a set of nodes (or edges) whose capacity ..."
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Motivated by preventative vaccination in a graph against the worstcase outbreak of an infectious disease, we propose new important graph cut problems. In the most basic problem MinMax Component Size, we are given a capacity bound, and the goal is to remove a set of nodes (or edges) whose capacity does not exceed the bound, and minimizes the size of the largest resulting connected component. In generalizations of this problem, we consider the objectives of minimizing the size of the k largest components for a fixed k, and the maximum number of special “terminal ” nodes inside any component. Under the assumption that each edge of a network will deterministically transmit a disease from either endpoint to the other, these problems naturally model the goal of targeting limited vaccinations of nodes (or edges) in such a way that the number of infected nodes in a worstcase outbreak is minimized. We present (O(1), O(log n)) bicriteria approximation algorithms for the MinMax Component Size problem and for the generalization to k infected components. Our algorithms are based on LP rounding and region growing techniques. If instead, a bound k on the number of terminals inside any component is given, and the goal is to minimize the total capacity of nodes (or edges) removed, we improve the approximation guarantee to an (O(1), O(log k)) bicriteria result. Finally, if k is a constant, and the edgecut version is considered, we show how to obtain a combinatorial singlecriterion approximation not violating the bound on the component size, and approximating the capacity of edges removed to within O(log k).