On Nontrivial Separators for k-Page Graphs and Simulations by Nondeterministic One-Tape Turing Machines

We show that the following statements are equivalent: 1. Statement 1. 3-pushdown graphs have sublinear separators. 2. Statement 1∗. k-page graphs have sublinear separators. 3. Statement 2. A one-tape nondeterministic Turing machine can simulate a two-tape machine in subquadratic time. None of the statements is known to be true or false at present. However, our proof of equivalence is quantitative-it relates exactly the separator size of the two kinds of graphs to the running time of the simulation in Statement 2. Using this equivalence we derive several graph-theoretic corollaries. There are known examples where upper bounds on graph properties imply upper bounds on computation time or space. There are other examples where lower bounds on graph properties are used to derive lower bounds on computation time in restricted settings. However, our results may constitute the first example where a graph problem is shown to be equivalent to a problem in computational complexity. In a companion paper we construct graphs and prove a lower bound or their separators. Using the equivalence we prove an almost linear lower bound for the size of separators for 3-pushdown graphs and an almost quadratic lower bound for simulating two-tape nondeterministic Turing machines by one-tape machines. Specifically, for an integers s let ls(n), the s-iterated logarithm function, be defined inductively: l°(n)=n, ls+1(n)=log2(ls(n)) for s⩾0. Then: 1. For every fixed s and all n, there is an n-vertex 3-pushdown graph whose smallest separator contains at least ω(n/ls(n)) vertices.2. There is a language L recognizable in real time by a two-tape nondeterministic Turing machine, but every on-line one-tape nondeterministic Turing machine that recognizes L requires ω(n2/ls(n)) time for any positive integer

On 3-Pushdown Graphs With Large Separators

For an integer s let ZS(n), the s-iterated logarithm function, be defined inductively: [O(n) = n, [8+1(n) = log2(l8(n)) for s 2:: o. We show that for every fixed s and all n large enough, there is an n-vertex 3-pushdown graph whose smallest separator contains at least n(n/[8(n)) vertices

Fixed Linear Crossing Minimization by Reduction to the Maximum Cut Problem

Many real-life scheduling, routing and locating problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear layout problem where the order of vertices on the line is fixed, the so-called fixed linear crossing number problem (FLCNP). We show that this NP-hard problem can be reduced to the well-known maximum cut problem. The latter problem was intensively studied in the literature; practically efficient exact algorithms based on the branch-and-cut technique have been developed. By an experimental evaluation on a variety of graphs, we prove that using this reduction for solving FLCNP compares favorably to earlier branch-and-bound algorithms

Why Philosophers Should Care About Computational Complexity

One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue that computational complexity theory---the field that studies the resources (such as time, space, and randomness) needed to solve computational problems---leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume's problem of induction, Goodman's grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis.Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and beyond," MIT Press, 2012. Some minor clarifications and corrections; new references adde

Characterisations and Examples of Graph Classes with Bounded Expansion

Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Ne\v{s}et\v{r}il and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several linear-time algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper we establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with Erd\"os-R\'enyi model of random graphs with constant average degree. In particular, we prove that for every fixed $d>0$, there exists a class with bounded expansion, such that a random graph of order $n$ and edge probability $d/n$ asymptotically almost surely belongs to the class. We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded non-repetitive chromatic number. We also prove that graphs with `linear' crossing number are contained in a topologically-closed class, while graphs with bounded crossing number are contained in a minor-closed class

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..