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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.
Terminal backup, 3d matching, and covering cubic graphs
 In Proceedings of the 39th Annual ACM Symposium on Theory of Computing. ACM
, 2007
"... Abstract. We define a problem called Simplex Matching and show that it is solvable in polynomial time. While Simplex Matching is interesting in its own right as a nontrivial extension of nonbipartite mincost matching, its main value lies in many (seemingly very different) problems that can be solve ..."
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Cited by 7 (4 self)
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Abstract. We define a problem called Simplex Matching and show that it is solvable in polynomial time. While Simplex Matching is interesting in its own right as a nontrivial extension of nonbipartite mincost matching, its main value lies in many (seemingly very different) problems that can be solved using our algorithm. For example, suppose that we are given a graph with terminal nodes, nonterminal nodes, and edge costs. Then, the Terminal Backup problem, which consists of finding the cheapest forest connecting every terminal to at least one other terminal, is reducible to Simplex Matching. Simplex Matching is also useful for various tasks that involve forming groups of at least two members, such as project assignment and variants of facility location. In an instance of Simplex Matching, we are given a hypergraph H with edge costs and edge size at most 3. We show how to find the mincost perfect matching of H efficiently if the edge costs obey a simple and realistic inequality that we call the Simplex Condition. The algorithm we provide is relatively simple to understand and implement but difficult to prove correct. In the process of this proof we show some powerful new results about covering cubic graphs with simple combinatorial objects.
Aspects of Network Design
, 2007
"... In this dissertation we study two problems from the area of network design. The first part discusses the multicommodity buyatbulk network design problem, a problem that occurs naturally in the design of telecommunication and transportation networks. We are given an underlying graph and associated ..."
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In this dissertation we study two problems from the area of network design. The first part discusses the multicommodity buyatbulk network design problem, a problem that occurs naturally in the design of telecommunication and transportation networks. We are given an underlying graph and associated with each edge of the graph, a cost function that represents the price of routing demand along this edge. We are also given a set of demands between pairs of vertices each of which needs to be satisfied by paying for sufficient capacity along a path connecting the vertices of the pair. In the multicommodity network design problem the objective is to minimize the cost of satisfying all demands. There are often situations where there is an initial fixed cost of utilizing an edge, or there is discounting or economies of scale that give rise to concave cost functions. We have an instance of the buyatbulk network design problem when the cost functions along all edges are concave. Unlike the case of linear cost functions, for which polynomial time algorithms exist, the buyatbulk network design problem is NPhard. We give the first nontrivial approx
Capacitated Network Design Problems: Hardness, Approximation Algorithms, and Connections to Group Steiner Tree ⋆
"... Abstract. We design combinatorial approximation algorithms for the Capacitated Steiner Network (CapSN) problem and the Capacitated Multicommodity Flow (CapMCF) problem. These two problems entail satisfying connectivity requirements when edges have costs and hard capacities. In CapSN, the flow has ..."
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Abstract. We design combinatorial approximation algorithms for the Capacitated Steiner Network (CapSN) problem and the Capacitated Multicommodity Flow (CapMCF) problem. These two problems entail satisfying connectivity requirements when edges have costs and hard capacities. In CapSN, the flow has to be supported separately for each commodity while in CapMCF, the flow has to be sent simultaneously for all commodities. We show that the Group Steiner problem on trees ([12]) is a special case of both problems. This implies the first polylogarithmic lower bound for these problems by [17]. We then give various approximations to special cases of the problems. We generalize the well known Source location problem (see for example [19]), to a natural problem called the Connected Rent or Buy Source Location problem. We show that this problem is a a simplification of CapSN and CapMCF and a generalization of Group Steiner on general graphs. We use Group Steiner Tree techniques, and more sophisticated techniques, to derive
NPhard to approximate within an approximation ratio 1.2 Directed Steiner Tree Problem Instance: Directed graph G = (V, E), edge costs w: E → R +, root
"... e∈ET c(e) ..."
Rounding
, 2011
"... In this section we will see an fapproximation algorithm for the Set Cover problem where f is the maximum number of times any element of the universe appears in the sets. Set Cover Input: A universe U = {u1, u2,..., um}, a collection S = {S1, S2,..., Sn} of subsets of U and a cost function c: S → R ..."
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In this section we will see an fapproximation algorithm for the Set Cover problem where f is the maximum number of times any element of the universe appears in the sets. Set Cover Input: A universe U = {u1, u2,..., um}, a collection S = {S1, S2,..., Sn} of subsets of U and a cost function c: S → R + ∪ {0} Goal: To find a minimum cost S ′ ⊆ S such that U = ⋃ S∈S ′ S Consider the following integer program which solves the Set Cover problem. subject to n∑ min c(Sj)xj j=1 ∀ i ∈ [m], xj ≥ 1 ui∈Sj ∀ j ∈ [n], xj ∈ {0, 1}
On the Fixed Cost kFlow Problem and related problems ⋆
"... Abstract. In the Fixed Cost kFlow problem, we are given a graph G = (V, E) with edgecapacities {ue  e ∈ E} and edgecosts {ce  e ∈ E}, sourcesink pair s, t ∈ V, and an integer k. The goal is to find a minimum cost subgraph H of G such that the minimum capacity of an stcut in H is at least k. W ..."
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Abstract. In the Fixed Cost kFlow problem, we are given a graph G = (V, E) with edgecapacities {ue  e ∈ E} and edgecosts {ce  e ∈ E}, sourcesink pair s, t ∈ V, and an integer k. The goal is to find a minimum cost subgraph H of G such that the minimum capacity of an stcut in H is at least k. We show that the Group Steiner on Trees problem is a special case of Fixed Cost kFlow. This implies the first non constant lower bound for Fixed Cost kFlow and the first non constant lower bounds for problems that are more general than Fixed Cost kFlow. In particular, the Capacitated Multicommodity Flow and the Capacitated Steiner Network and the Capacitated Buy at Bulk problem. A special case of both Fixed Cost kFlow and the related NodeWeighted kFlow problem is the NodeMinimum Bibartite kFlow problem: given a bipartite graph G = (A ∪ B, E) with edge capacities and an integer k> 0, find a node subset S ⊆ A ∪ B of minimum size S  such that the minimum capacity of an (S ∩ A, S ∩ B)cut
On Fixed Cost kFlow Problems ⋆
"... Abstract. In the Fixed Cost kFlow problem, we are given a graph G = (V, E) with edgecapacities {ue  e ∈ E} and edgecosts {ce  e ∈ E}, sourcesink pair s, t ∈ V, and an integer k. The goal is to find a minimum cost subgraph H of G such that the minimum capacity of an stcut in H is at least k. W ..."
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Abstract. In the Fixed Cost kFlow problem, we are given a graph G = (V, E) with edgecapacities {ue  e ∈ E} and edgecosts {ce  e ∈ E}, sourcesink pair s, t ∈ V, and an integer k. The goal is to find a minimum cost subgraph H of G such that the minimum capacity of an stcut in H is at least k. We show that Group Steiner is a special case of Fixed Cost kFlow, thus obtaining the first polylogarithmic lower bound for the problem; this also implies the first non constant lower bounds for the Capacitated Steiner Network and Capacitated Multicommodity Flow problems. We then consider two special cases of Fixed Cost kFlow. In the Bipartite FixedCost kFlow problem, we are given a bipartite graph G = (A ∪ B, E) and an integer k> 0. The goal is to find a node subset S ⊆ A ∪ B of minimum size S  such G has k pairwise edgedisjoint paths between S ∩ A and S ∩ B. We give an O ( √ k log k) approximation for this problem. We also show that we can compute a solution of optimum size with Ω(k/polylog(n)) paths, where n = A+B. In the GeneralizedP2P problem we are given an undirected graph G = (V, E) with edgecosts and integer charges {bv: v ∈ V}. The goal is to find a minimumcost spanning subgraph H of G such that every connected component of H has nonnegative charge. This problem originated in a practical project for shift design [11]. Besides that, it generalizes many problems such as Steiner Forest, kSteiner Tree, and Point to Point Connection. We give a logarithmic approximation algorithm for this problem. Finally, we consider a related problem called Connected Rent or Buy Multicommodity Flow and give a log 3+ɛ n approximation scheme for it using Group Steiner techniques. 1
Project Summary
"... One network design problems with hard capacities and kcenters problems with hard capacities Network Design Problems with hard capacities: A network design problem is, given an edge weighted graph G(V, E, d), choose a spanning graph G(V, E ′) that meets some constrains. Many times the constrains are ..."
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One network design problems with hard capacities and kcenters problems with hard capacities Network Design Problems with hard capacities: A network design problem is, given an edge weighted graph G(V, E, d), choose a spanning graph G(V, E ′) that meets some constrains. Many times the constrains are connectivity constrains. A good example is the Steiner network problem. We are given an edge weighted graph G(V, E, d) and connectivity requirements ruv for all pairs uv ∈ V × V some of which may be zero. The goal is to select a min cost subset of the edges E ′ so that in G(V, E ′) there are ruv edge disjoint paths between u and v for every u, v. We define formally the notion of approximation below. As of now we say informally, that a polynomial time algorithm has approximation ratio ρ for a minimization problem, if it always returns a solution for any given instance that has cost no more than ρ times the cost of the minimum solution. For maximization problem it should return a solution at least 1/ρ times the optimum. Steiner network is a seminal problem in computer science and in a seminal paper, Jain [44] gave a ratio 2 approximation for the problem. We study the capacitated version of the Steiner Network problem. This problem is called the Capacitated Network Design (CND) problem. The input is as in the Steiner Network problem except that each edge has