MohammadTaghi Hajiaghayi

TL;DR: A new framework for designing fixed-parameter algorithms with subexponential running time---2O(&kradic;) nO(1) is introduced, which applies to a broad family of graph problems, called bidimensional problems, which includes many domination and problems such as vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominate set, disk dimension, and many others restricted to bounded-genus graphs.

TL;DR: The algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces is developed, and an O(√log n) pseudo-approximation for finding balanced vertices in general graphs is exhibited.

TL;DR: The foundation of this work is the topological theory of drawings of graphs on surfaces and the results regarding the relation of the size of the largest grid minor in terms of treewidth in bounded-genus graphs and more generally in graphs excluding a fixed graph H as a minor.

TL;DR: It is shown that embeddings into $L_1$ are insufficient but that the additional structure provided by many embedding theorems does suffice for the authors' purposes, and an optimal $O(\log k)$-approximate max-flow/min-vertex-cut theorem for arbitrary vertex-capacitated multicommodity flow instances on $k$ terminals is proved.

TL;DR: A polynomial-time algorithm is developed using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a clique-sum of pieces almost-embeddable into bounded-genus surfaces.