# Algorithmic Graph Problems - From Computer Networks to Graph Embeddings

Abstract: This dissertation is a contribution to the knowledge of the computational complexity of discrete combinatorial problems. 1. The first problem that we consider is to compute the maximum independent set of a box graph, that is, given a set of orthogonal boxes in the plane compute the largest subset such that no boxes in the subset overlap. We provide an \$mathtt{exp}(O(sqrt nlog n))\$-time algorithm for this problem and give an \$mathtt{exp}(Omega(sqrt n))\$ bound unless the {em Exponential Time Hypothesis} (ETH) is false. 2. Next, we consider the problem of computing the Hadwiger number of a graph \$G\$. The Hadwiger number is the largest \$h\$ such that the complete graph on \$h\$ vertices, \$K_h,\$ is a minor of \$G\$. We also study the related problem of computing the maximum homeomorphic clique. That is, determining the largest \$h\$ such that \$K_h\$ is a topological minor of \$G\$. We give upper and lower bounds for the approximability of these problems. For the fixed-vertex subgraph homeomorphism problem we provide an exponential time exact algorithm. 3. Then we study broadcasting in geometric multi-hop radio networks by using analysis techniques from computational complexity. We attempt to minimize the total power consumption of broadcasting a message from a source node to all the other nodes in the network. We also study the number of rounds required to broadcast a message in a known geometric radio network. We also show that an \$h\$-hop broadcasting scheme, in a model that does not account for interference, requiring \$cal E\$ energy can be simulated in \$hlog psi\$ rounds using \$O(cal E)\$ energy in a model that does, where \$psi\$ denotes the ratio between the maximum and the minimum Euclidean distance between nodes in the network. 4. Finally, we establish lower bounds on the computational complexity of counting problems; in particular we study the Tutte polynomial and the permanent under a counting version of the ETH (#ETH). The Tutte polynomial is related to determining the failure probability for computer networks by its relation to the reliability polynomial. We consider the problem of computing the Tutte polynomial in a point \$(x,y)\$, and show that for multigraphs with \$bar m\$ adjacent vertex pairs the problem requires time \$mathtt{exp}(Omega(bar m)),\$ in many points, under the #ETH. We also show that computing the permanent of a \$n imes n\$ matrix with \$bar m\$ nonzero entries requires time \$mathtt{exp}(Omega(bar m))\$,} under the #ETH. This dissertation MIGHT be available in PDF-format. Check this page to see if it is available for download. 