Johannes Köbler

On counting and approximation

We introduce a new class of functions, called span functions which count the different output values that occur at the leaves of the computation tree associated with a nondeterministic polynomial time Turing machine transducer. This function class has natural complete problems; it is placed between Valiants function classes #P and #NP, and contains both Goldberg and Sipsers ranking functions for sets in NP, and Krentels optimization functions. We show that it is unlikely that the span functions coincide with any of the mentioned function classes.

Turning machines with few accepting computations and low sets for PP

The authors study two different ways to restrict the power of NP. They consider languages accepted by nondeterministic polynomial-time machines with a small number of accepting paths in case of acceptance, and they investigate three subclasses of NP that are low for complexity classes not known to be in the polynomial-time hierarchy. The subclasses, UP, FewP, and Few, are all defined in terms of nondeterministic machines with a bounded number of accepting paths for every input string, but for the last two classes this number is not known beforehand and can range over a space of polynomial size. The authors prove lowness properties of the class Few and some other interesting sets that are low for the class PP. The lowness results are used to obtain positive relativizations of complexity classes.<<ETX>>

Interval Graphs: Canonical Representations in Logspace

We present a logspace algorithm for computing a canonical labeling, in fact, a canonical interval representation, for interval graphs. To achieve this, we compute canonical interval representations of interval hypergraphs. This approach also yields a canonical labeling of convex graphs. As a consequence, the isomorphism and automorphism problems for these graph classes are solvable in logspace. For proper interval graphs we also design logspace algorithms computing their canonical representations by proper and by unit interval systems.

The Isomorphism Problem for k-Trees Is Complete for Logspace

We show that k-tree isomorphism can be decided in logarithmic space by giving a logspace canonical labeling algorithm. This improves over the previous StUL upper bound and matches the lower bound. As a consequence, the isomorphism, the automorphism, as well as the canonization problem for k-trees are all complete for deterministic logspace. We also show that even simple structural properties of k-trees are complete for logspace.

Is the Standard Proof System for SAT P-Optimal?

We investigate the question whether there is a (p-)optimal proof system for SAT or for TAUT and its relation to completeness and collapse results for nondeterministic function classes. A p-optimal proof system for SAT is shown to imply (1) that there exists a complete function for the class of all total nondeterministic multi-valued functions and (2) that any set with an optimal proof system has a p-optimal proof system. By replacingthe assumption of the mere existence of a (p-) optimal proof system by the assumption that certain proof systems are (p-)optimal we obtain stronger consequences, namely collapse results for various function classes. Especially we investigate the question whether the standard proof system for SAT is p-optimal. We show that this assumption is equivalent to a variety of complexity theoretical assertions studied before, and to the assumption that every optimal proof system is p-optimal. Finally, we investigate whether there is an optimal proof system for TAUT that admits an effective interpolation, and show some relations between various completeness assumptions.

A general dimension for query learning

We introduce a combinatorial dimension that characterizes the number of queries needed to exactly (or approximately) learn concept classes in various models. Our general dimension provides tight upper and lower bounds on the query complexity for all sorts of queries, not only for example-based queries as in previous works. As an application we show that for learning DNF formulas, unspecified attribute value membership and equivalence queries are not more powerful than standard membership and equivalence queries. Further, in the approximate learning setting, we use the general dimension to characterize the query complexity in the statistical query as well as the learning by distances model. Moreover, we derive close bounds on the number of statistical queries needed to approximately learn DNF formulas.

Graph Isomorphism Is Low for ZPP(NP) and Other Lowness Results

We show the following new lowness results for the probabilistic class ZPPNP. - The class AM ∩ coAM is low for ZPPNP. As a consequence it follows that Graph Isomorphism and several group-theoretic problems known to be in AM ∩ coAM are low for ZPPNP. - The class IP[P=poly], consisting of sets that have interactive proof systems with honest provers in P=poly, is also low for ZPPNP. We consider lowness properties of nonuniform function classes, namely, NPMV/poly, NPSV/poly, NPMVt/poly, and NPSVt/poly. Specifically, we show that - Sets whose characteristic functions are in NPSV/poly and that have program checkers (in the sense of Blum and Kannan [8]) are low for AM and ZPPNP. - Sets whose characteristic functions are in NPMVt/poly are low for Σ2p.

Colored Hypergraph Isomorphism is Fixed Parameter Tractable

We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism, denoted CHI, which has running time (2bN)O(1), where the parameter b is the maximum size of the color classes of the given hypergraphs and N is the input size. We also describe an fpt algorithm for a parameterized coset intersection problem that is used as a subroutine in our algorithm for CHI.

From invariants to canonization in parallel

A function f of a graph is called a complete graph invariant if two given graphs G and H are isomorphic exactly when f(G) = f(H). If additionally, f(G) is a graph isomorphic to G, then f is called a canonical form for graphs. Gurevich [9] proves that any polynomial-time computable complete invariant can be transformed into a polynomial-time computable canonical form. We extend this equivalence to the polylogarithmic-time model of parallel computation for classes of graphs having either bounded rigidity index or small separators. In particular, our results apply to three representative classes of graphs embeddable into a fixed surface, namely, to 3-connected graphs admitting either a polyhedral or a large-edge-width embedding as well as to all embeddable 5-connected graphs. Another application covers graphs with treewidth bounded by a constant k. Since for the latter class of graphs a complete invariant is computable in NC, it follows that graphs of bounded treewidth have a canonical form (and even a canonical labeling) computable in NC.

Approximate graph isomorphism

We study optimization versions of Graph Isomorphism. Given two graphs G1,G2, we are interested in finding a bijection π from V(G1) to V(G2) that maximizes the number of matches (edges mapped to edges or non-edges mapped to non-edges). We give an nO(logn) time approximation scheme that for any constant factor α<1, computes an α-approximation. We prove this by combining the nO(logn) time additive error approximation algorithm of Arora et al. [Math. Program., 92, 2002] with a simple averaging algorithm. We also consider the corresponding minimization problem (of mismatches) and prove that it is NP-hard to α-approximate for any constant factor α. Further, we show that it is also NP-hard to approximate the maximum number of edges mapped to edges beyond a factor of 0.94. We also explore these optimization problems for bounded color class graphs which is a well studied tractable special case of Graph Isomorphism. Surprisingly, the bounded color class case turns out to be harder than the uncolored case in the approximate setting.