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A constant factor approximation algorithm for unsplittable flow on paths
 In Proceedings of the 52th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011
, 2011
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Unsplittable flow in paths and trees and columnrestricted packing integer programs
 IN PROCEEDINGS, INTERNATIONAL WORKSHOP ON APPROXIMATION ALGORITHMS FOR COMBINATORIAL OPTIMIZATION PROBLEMS
, 2009
"... We consider the unsplittable flow problem (UFP) and the closely related columnrestricted packing integer programs (CPIPs). In UFP we are given an edgecapacitated graph G = (V, E) and k request pairs R1,..., Rk, where each Ri consists of a sourcedestination pair (si, ti), a demand di and a weigh ..."
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We consider the unsplittable flow problem (UFP) and the closely related columnrestricted packing integer programs (CPIPs). In UFP we are given an edgecapacitated graph G = (V, E) and k request pairs R1,..., Rk, where each Ri consists of a sourcedestination pair (si, ti), a demand di and a weight wi. The goal is to find a maximum weight subset of requests that can be routed unsplittably in G. Most previous work on UFP has focused on the nobottleneck case in which the maximum demand of the requests is at most the smallest edge capacity. Inspired by the recent work of Bansal et al. [3] on UFP on a path without the above assumption, we consider UFP on paths as well as trees. We give a simple O(log n) approximation for UFP on trees when all weights are identical; this yields an O(log 2 n) approximation for the weighted case. These are the first nontrivial approximations for UFP on trees. We develop an LP relaxation for UFP on paths that has an integrality gap of O(log 2 n); previously there was no relaxation with o(n) gap. We also consider UFP in general graphs and CPIPs without the nobottleneck assumption and obtain new and useful results.
Efficient Submodular Function Maximization under Linear Packing Constraints
"... We study the problem of maximizing a monotone submodular set function subject to linear packing constraints. An instance of this problem consists of a matrix A ∈ [0, 1] m×n, a vector b ∈ [1, ∞) m, and a monotone submodular set function f: 2 [n] → R+. The objective is to find a set S that maximizes ..."
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We study the problem of maximizing a monotone submodular set function subject to linear packing constraints. An instance of this problem consists of a matrix A ∈ [0, 1] m×n, a vector b ∈ [1, ∞) m, and a monotone submodular set function f: 2 [n] → R+. The objective is to find a set S that maximizes f(S) subject to AxS ≤ b. Here, xS stands for the characteristic vector of the set S. A wellstudied special case of this problem is when the objective function f is linear. This special case captures the class of packing integer programs. Our main contribution is an efficient combinatorial algorithm that achieves an approximation ratio of Ω(1/m 1/W), where W = min{bi/Aij: Aij> 0} is the width of the packing constraints. This result matches the best known performance guarantee for the linear case. One immediate corollary of this result is that the algorithm under consideration achieves constant factor approximation when the number of constraints is constant or when the width of the packing constraints is sufficiently large. This motivates us to study the large width setting, trying to determine its exact approximability. We develop an algorithm that has an approximation ratio of (1 − ɛ)(1 − 1/e) when W = Ω(ln m/ɛ 2). This result (almost) matches the theoretical lower bound of 1−1/e, which already holds for maximizing a monotone submodular function subject to a cardinality constraint.
Truthful unsplittable flow for large capacity networks
 In Proceedings 19th Annual ACM Symposium on Parallelism in Algorithms and Architectures
, 2007
"... The unsplittable flow problem is one of the most extensively studied optimization problems in the field of networking. An instance of it consists of an edge capacitated graph and a set of connection requests, each of which is associated with source and target vertices, a demand, and a value. The obj ..."
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The unsplittable flow problem is one of the most extensively studied optimization problems in the field of networking. An instance of it consists of an edge capacitated graph and a set of connection requests, each of which is associated with source and target vertices, a demand, and a value. The objective is to route a maximum value subset of requests subject to the edge capacities. It is a well known fact that as the capacities of the edges are larger with respect to the maximal demand among the requests, the problem can be approximated better. In particular, it is known that for sufficiently large capacities, the integrality gap of the corresponding integer linear program becomes 1 + ɛ, which can be matched by an algorithm that utilizes the randomized rounding technique. In this paper, we focus our attention on the large capacities unsplittable flow problem in a game theoretic setting. In this setting, there are selfish agents, which control some of the requests characteristics, and may be dishonest about them. It is worth noting that in game theoretic settings many standard techniques, such as randomized rounding, violate certain monotonicity properties, which are imperative for truthfulness, and therefore cannot be employed.
A mazing 2+ε approximation for unsplittable flow on a path
 IN: PROCEEDINGS OF SODA 2014
, 2014
"... We study the unsplittable flow on a path problem (UFP), which arises naturally in many applications such as bandwidth allocation, job scheduling, and caching. Here we are given a path with nonnegative edge capacities and a set of tasks, which are characterized by a subpath, a demand, and a profit. ..."
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We study the unsplittable flow on a path problem (UFP), which arises naturally in many applications such as bandwidth allocation, job scheduling, and caching. Here we are given a path with nonnegative edge capacities and a set of tasks, which are characterized by a subpath, a demand, and a profit. The goal is to find the most profitable subset of tasks whose total demand does not violate the edge capacities. Not surprisingly, this problem has received a lot of attention in the research community. If the demand of each task is at most a small enough fraction δ of the capacity along its subpath (δsmall tasks), then it has been known for a long time [Chekuri et al., ICALP 2003] how to compute a solution of value arbitrarily close to the optimum via LP rounding. However, much remains unknown for the complementary case, that is, when the demand of each task is at least some fraction δ> 0 of the smallest capacity of its subpath (δlarge tasks). For this setting a constant factor approximation, improving on an earlier logarithmic approximation, was found only recently [Bonsma et al., FOCS 2011]. In this paper we present a PTAS for δlarge tasks, for any constant δ> 0. Key to this result is a complex geometrically inspired dynamic program. Each task is represented as a segment underneath the capacity curve, and we identify a proper mazelike structure so that each corridor of the maze is crossed by only O(1) tasks in the optimal solution. The maze has a tree topology, which guides our dynamic program. Our result implies a 2 + ε approximation for UFP, for any constant ε> 0, improving on the previously best 7 + ε approximation by Bonsma et al. We remark that our improved approximation algorithm matches the best
Polylogarithmic Approximation for Maximum Node Disjoint Paths with Constant Congestion
"... We consider the Maximum Node Disjoint Paths (MNDP) problem in undirected graphs. The input consists of an undirected graph G = (V, E) and a collection {(s1, t1),..., (sk, tk)} of k sourcesink pairs. The goal is to select a maximum cardinality subset of pairs that can be routed/connected via nodedi ..."
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We consider the Maximum Node Disjoint Paths (MNDP) problem in undirected graphs. The input consists of an undirected graph G = (V, E) and a collection {(s1, t1),..., (sk, tk)} of k sourcesink pairs. The goal is to select a maximum cardinality subset of pairs that can be routed/connected via nodedisjoint paths. A relaxed version of MNDP allows up to c paths to use a node, where c is the congestion parameter. We give a polynomial time algorithm that routes Ω(OPT/poly log k) pairs with O(1) congestion, where OPT is the value of an optimum fractional solution to a natural multicommodity flow relaxation. Our result builds on the recent breakthrough of Chuzhoy [17] who gave the first polylogarithmic approximation with constant congestion for the Maximum Edge Disjoint Paths (MEDP) problem.
Maximum Bipartite Flow in Networks with Adaptive Channel Width
, 2010
"... Traditionally, network optimization problems assume that each link in the network has a fixed capacity. Recent research in wireless networking has shown that it is possible to design networks where the capacity of the links can be changed adaptively to suit the needs of specific applications. In par ..."
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Traditionally, network optimization problems assume that each link in the network has a fixed capacity. Recent research in wireless networking has shown that it is possible to design networks where the capacity of the links can be changed adaptively to suit the needs of specific applications. In particular, one gets a choice of having a few high capacity outgoing links or many low capacity ones at any node of the network. This motivates us to have a relook at classical network optimization problems and design algorithms to solve them in this new framework. In particular, we consider the problem of maximum bipartite flow, which has been studied extensively in the fixedcapacity network model. One of the motivations for studying this problem arises from the need to maximize the throughput of an infrastructure wireless network comprising basestations (one set of vertices in the bipartition) and clients (the other set of vertices in the bipartition). We show that this problem has a significantly different
Approximation Algorithms for NETWORK DESIGN AND ORIENTEERING
, 2010
"... This thesis presents approximation algorithms for some N PHard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widelybelieved complexitytheoretic assumption that P ̸ = N P, there are no efficient (i.e., polynomialt ..."
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This thesis presents approximation algorithms for some N PHard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widelybelieved complexitytheoretic assumption that P ̸ = N P, there are no efficient (i.e., polynomialtime) algorithms that solve these problems exactly. Hence, if one desires efficient algorithms for such problems, it is necessary to consider approximate solutions: An approximation algorithm for an N PHard problem is a polynomial time algorithm which, for any instance of the problem, finds a solution whose value is guaranteed to be within a multiplicative factor ρ of the value of an optimal solution to that instance. We attempt to design algorithms for which this factor ρ, referred to as the approximation ratio of the algorithm, is as small as possible. The field of Network Design comprises a large class of problems that deal with constructing networks of low cost and/or high capacity, routing data through existing networks, and many related issues. In this thesis, we focus chiefly on designing faulttolerant networks. Two vertices u, v in a network are said to be kedgeconnected if deleting any set of k − 1 edges leaves u and v connected; similarly, they are kvertex connected if deleting any set of k − 1 other vertices or edges leaves u and v connected. We focus on building networks that are highly connected, meaning
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
On the Admission of Dependent Flows in Powerful Sensor Networks
"... Abstract—In this paper we define and study a new problem, referred to as the Dependent Unsplittable Flow Problem (DUFP). We present and discuss this problem in the context of largescale powerful (radar/camera) sensor networks, but we believe it has important applications on the admission of large ..."
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Abstract—In this paper we define and study a new problem, referred to as the Dependent Unsplittable Flow Problem (DUFP). We present and discuss this problem in the context of largescale powerful (radar/camera) sensor networks, but we believe it has important applications on the admission of large flows in other networks as well. In order to optimize the selection of flows transmitted to the gateway, DUFP takes into account possible dependencies between flows. We show that DUFP is more difficult than NPhard problems for which no good approximation is known. Then, we address two special cases of this problem: the case where all the sensors have a shared channel and the case where the sensors form a mesh and route to the gateway over a spanning tree. I.