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**1 - 3**of**3**### The All-or-Nothing Flow Problem in Directed Graphs with Symmetric Demand Pairs

, 2014

"... We study the approximability of the All-or-Nothing multicommodity flow problem in directed graphs with symmetric demand pairs (SymANF). The input consists of a directed graph G = (V,E) and a collection of (unordered) pairs of nodesM = {s1t1, s2t2,..., sktk}. A subsetM ′ of the pairs is routable if t ..."

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We study the approximability of the All-or-Nothing multicommodity flow problem in directed graphs with symmetric demand pairs (SymANF). The input consists of a directed graph G = (V,E) and a collection of (unordered) pairs of nodesM = {s1t1, s2t2,..., sktk}. A subsetM ′ of the pairs is routable if there is a feasible multicommodity flow in G such that, for each pair siti ∈ M′, the amount of flow from si to ti is at least one and the amount of flow from ti to si is at least one. The goal is to find a maximum cardinality subset of the given pairs that can be routed. Our main result is a poly-logarithmic approximation with constant congestion for SymANF. We obtain this result by extending the well-linked decomposition framework of [11] to the directed graph setting with symmetric demand pairs. We point out the importance of studying routing problems in this setting and the relevance of our result to future work.

### Excluded Grid Theorem: Improved and Simplified

, 2015

"... We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function f: Z+ → Z+, such that for any integer g> 0, any graph of treewidth at least f(g), contains the (g × g)-grid as a minor. Until recently, the best known ..."

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We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function f: Z+ → Z+, such that for any integer g> 0, any graph of treewidth at least f(g), contains the (g × g)-grid as a minor. Until recently, the best known upper bounds on f were super-exponential in g. A recent work of Chekuri and Chuzhoy provided the first polynomial bound, by showing that treewidth f(g) = O(g98 poly log g) is sufficient to ensure the existence of the (g × g)-grid minor in any graph. In this paper we provide a much simpler proof of the Excluded Grid Theorem, achieving a bound of f(g) = O(g36 poly log g). Our proof is self-contained, except for using prior work to reduce the maxi-mum vertex degree of the input graph to a constant.

### 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 NP-hard and therefore, assuming that P 6 = NP, there do not exist polynomial-time 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 NP-hard and therefore, assuming that P 6 = NP, there do not exist polynomial-time 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 polynomial-time 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 non-metric facility location, multiway cut problems in graphs and hypergraphs, uniform metric labeling and its generalization