Exponential Time Complexity of the Permanent and the Tutte Polynomial

We show conditional lower bounds for well-studied #P-hard problems: (a) The number of satisfying assignments of a 2-CNF formula with n variables cannot be counted in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph. (b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed in time exp(o(n)). (c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs. Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting

On the complexity of approximating the Hadwiger number

Unless P = NP there is no polynomial time approximation scheme (PTAS) for the problem of finding, for a graph G, the largest h such that the complete graph K-h is a minor of G. (C) 2008 Elsevier B.V. All rights reserved

Algorithmic Graph Problems - From Computer Networks to Graph Embeddings

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

Subexponential-time algorithms for maximum independent set and related problems on box graphs

A box graph is the intersection graph of orthogonal rectangles in the plane. We consider such basic combinatorial problems on box graphs as maximum independent set, minimum vertex cover and maximum induced subgraph with polynomial-time testable hereditary property Pi. We show that they can be exactly solved in subexponential time, more precisely, in time 2(O(rootnlog n)), by applying Miller's simple cycle planar separator theorem [6] (in spite of the fact that the input box graph might be strongly non-planar). Furthermore we extend our idea to include the intersection graphs of orthogonal d-cubes of bounded aspect ratio and dimension. We present an algorithm that solves maximum independent set and the other aforementioned problems in time 2(O(d2dbn1-1/dlogn)) on, such box graphs in d-dimensions. We do this by applying a separator theorem by Smith and Wormald [7]. Finally, we show that in general graph case substantially subexponential algorithms for maximum independent set and the maximum induced subgraph with polynomial-time testable hereditary property Pi problems can yield non-trivial upper bounds on approximation factors achievable in polynomial time

An exact algorithm for subgraph homeomorphism

The subgraph homeomorphism problem is to decide if there is an injective mapping of the vertices of a pattern graph into vertices of a host graph so that the edges of the pattern graph can be mapped into (internally) vertex-disjoint paths in the host graph. The restriction of subgraph homeomorphism where an injective mapping of the vertices of the pattern graph into vertices of the host graph is already given in the input instance is termed fixed-vertex subgraph homeomorphism. We show that fixed-vertex subgraph homeomorphism for a pattern graph on p vertices and a host graph on n vertices can be solved in time 2n−pnO(1) or in time 3n−pnO(1) and polynomial space. In effect, we obtain new non-trivial upper bounds on the time complexity of the problem of finding k vertex-disjoint paths and general subgraph homeomorphism

On exact complexity of subgraph homeomorphism

The subgraph homeomorphism problem is to decide whether there is an injective mapping of the vertices of a pattern graph into vertices of a host graph so that the edges of the pattern graph can be mapped into (internally) vertex-disjoint paths in the host graph. The restriction of subgraph homeomorphism where an injective mapping of the vertices of the pattern graph into vertices of the host graph is already given is termed fixed-vertex subgraph homeomorphism. We show that fixed-vertex subgraph homeomorphism for a pattern graph on p vertices and a host graph on n vertices can be solved in time O(2n − pnO(1)) or in time O(3n − pn6) and polynomial space. In effect, we obtain new non-trivial upper time-bounds on the exact complexity of the problem of finding k vertex-disjoint paths and general subgraph homeomorphism

A note on maximum independent set and related problems on box graphs

A box graph is the intersection graph of orthogonal rectangles in the plane. We show that maximum independent set and minimum vertex cover on box graphs can be solved in subexponential time, more precisely, in time 2(O(rootn log n)), by applying Miller's simple cycle planar separator theorem [J. Comput. System Sci. 32 (1986) 265-279] (in spite of the fact that the input box graph might be strongly non-planar)

Minimum-energy broadcasting in wireless networks in the d-dimensional Euclidean space (the alpha

We consider the problem of minimizing the total energy assigned to nodes of wireless network so that broadcasting from the source node to all other nodes is possible. This problem has been extensively studied especially under the assumption that the nodes correspond to points in the Euclidean two- or three-dimensional space and the broadcast range of a node is proportional to at most the α root of the energy assigned to the node where α is not less than the dimension d of the space. In this paper, we study the case α≤d, providing several tight upper and lower bounds on approximation factors of known heuristics for minimum energy broadcasting in the d-dimensional Euclidean space

Approximating the maximum clique minor and some subgraph homeomorphism problems.

We consider the “minor” and “homeomorphic” analogues of the maximum clique problem, i.e., the problems of determining the largest h such that the input graph (on n vertices) has a minor isomorphic to Kh or a subgraph homeomorphic to Kh, respectively, as well as the problem of finding the corresponding subgraphs. We term them as the maximum clique minor problem and the maximum homeomorphic clique problem, respectively. We observe that a known result of Kostochka and Thomason supplies an O(sqrt n) source bound on the approximation factor for the maximum clique minor problem achievable in polynomial time. We also provide an independent proof of nearly the same approximation factor with explicit polynomial-time estimation, by exploiting the minor separator theorem of Plotkin et al. Next, we show that another known result of Bollobás and Thomason and of Komlós and Szemerédi provides an O(sqrt n) source bound on the approximation factor for the maximum homeomorphic clique achievable in polynomial time. On the other hand, we show an Ω(n1/2−O(1/(logn)γ)) lower bound (for some constant γ, unless NP subset ZPTIME(2^(logn)^O(1)) on the best approximation factor achievable efficiently for the maximum homeomorphic clique problem, nearly matching our upper bound. Finally, we derive an interesting trade-off between approximability and subexponential time for the problem of subgraph homeomorphism where the guest graph has maximum degree not exceeding three and low treewidth