On the computational complexity of cut-reduction

Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations. Explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way.     

Independence and port oracles for matroids,with an application to computational learning theory

Given a matroidM with distinguished elemente, aport oracie with respect toe reports whether or not a given subset contains a circuit that containse. The first main result of this paper is an algorithm for computing ane-based ear decomposition (that is, an ear decomposition every circuit of which contains elemente) of a matroid using only a polynomial number of elementary operations and port oracle calls. In the case thatM is binary, the incidence vectors of the circuits in the ear decomposition form a matrix representation forM. Thus, this algorithm solves a problem in computational learning theory; it learns the class ofbinary matroid port (BMP) functions with membership queries in polynomial time. In this context, the algorithm generalizes results of Angluin, Hellerstein, and Karpinski [1], and Raghavan and Schach [17], who showed that certain subclasses of the BMP functions are learnable in polynomial time using membership queries. The second main result of this paper is an algorithm for testing independence of a given input set of the matroidM. This algorithm, which uses the ear decomposition algorithm as a subroutine, uses only a polynomial number of elementary operations and port oracle calls. The algorithm proves a constructive version of an early theorem of Lehman [13], which states that the port of a connected matroid uniquely determines the matroid.Research partially funded by NSF PYI Grant No. DDM-91-96083.Research partially funded by NSF Grant No CCR-92-10957.     

Low upper bounds in the LR degrees

We say that A≤_LRB if every B-random real is A-random—in other words, if B has at least as much derandomization power as A. The LR reducibility is a natural weak reducibility in the context of randomness, and generalizes lowness for randomness. We study the existence and properties of upper bounds in the context of the LR degrees. In particular, we show that given two (or even finitely many) low sets, there is a low c.e. set which lies LR above both. This is very different from the situation in the Turing degrees, where Sacks’ splitting theorem shows that two low sets can join to 0^′.     

On the complexity of Boolean matrix ranks

We present a reduction which shows that the fooling set number, tropical and determinantal ranks of a Boolean matrix are NP-hard to compute.     

The complexity of tropical matrix factorization

The tropical arithmetic operations on R are defined by a⊕b=min{a,b}$a\oplus b=\min \{a,b\}$ and a⊗b=a+b$a\otimes b=a+b$. Let A be a tropical matrix and k   a positive integer, the problem of Tropical Matrix Factorization (TMF) asks whether there exist tropical matrices B∈R^m×k$B\in {\mathbf{R}}^{m\times k}$ and C∈R^k×n$C\in {\mathbf{R}}^{k\times n}$ satisfying B⊗C=A$B\otimes C=A$. We show that the TMF problem is NP-hard for every k≥7$k\ge 7$ fixed in advance, thus resolving a problem proposed by Barvinok in 1993.     

The $\mu$-measure as a tool for classifying computational complexity

Two simply typed term systems and are considered, both for representing algorithms computing primitive recursive functions. is based on primitive recursion, on recursion on notation. A purely syntactical method of determining the computational complexity of algorithms in , called $\mu$ -measure, is employed to uniformly integrate traditional results in subrecursion theory with resource-free characterisations of sub-elementary complexity classes. Extending the Schwichtenberg and Müller characterisation of the Grzegorczyk classes for , it is shown $\mathcal{E}_{n+1} = \mathcal{R}^n_1n\ge 1\mathcal{R}^n_i$ denotes the \emph{th modified Heinermann class} based on . The proof does not refer to any machine-based computation model, unlike the Schwichtenberg and Müller proofs. This is due to the notion of modified recursion lying on top of each other provided by . By Ritchie's result, characterises the linear-space computable functions. Using the same method, a short and straightforward proof is presented, showing that characterises the polynomial time computable functions. Furthermore, the classes and coincide at and above level 2. Received: 22 September 1997 / Revised version: 12 May 1999     

Some elementary proofs of lower bounds in complexity theory

We introduce a new and easily applicable criterion called rank immunity for estimating the minimal number of multiplications needed to compute a set of bilinear forms in commuting variables. The result is obtained by an elimination argument after canonically embedding computations in a quotient ring R/I, where I is an appropriately chosen ideal that is left invariant under the eliminations. The criterion combines the well-known arguments based on elimination and on row rank, but in contrast to (for instance) column- and mixed-rank arguments it normally leads to better elementary estimates than were derivable in a uniform manner before.     

Improved bounds on the max-flow min-cut ratio for multicommodity flows

In this paper we consider the worst case ratio between the capacity of min-cuts and the value of max-flow for multicommodity flow problems. We improve the best known bounds for the min-cut max-flow ratio for multicommodity flows in undirected graphs, by replacing theO(logD) in the bound byO(logk), whereD denotes the sum of all demands, andk demotes the number of commodities. In essence we prove that up to constant factors the worst min-cut max-flow ratios appear in problems where demands are integral and polynomial in the number of commodities.Klein, Rao, Agrawal, and Ravi have previously proved that if the demands and the capacities are integral, then the min-cut max-flow ratio in general undirected graphs is bounded byO(logClogD), whereC denotes the sum of all the capacities. Tragoudas has improved this bound toO(lognlogD), wheren is the number of nodes in the network. Garg, Vazirani and Yannakakis further improved this toO(logklogD). Klein, Plotkin and Rao have proved that for planar networks, the ratio isO(logD).Our result improves the bound for general networks toO(log² k) and the bound for planar networks toO(logk). In both cases our result implies the first non-trivial bound that is independent of the magnitude of the numbers involved. The method presented in this paper can be used to give polynomial time approximation algorithms to the minimum cuts in the network up to the above factors.Preliminary version appeared in Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, 1993, 691-697.Research supported by U.S. Army Research Office Grant DAAL-03-91-G-0102, and by a grant from Mitsubishi Electric Laboratories.Research supported in part by a Packard Fellowship, an NSF PYI award, a Sloan Fellowship, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

A lower bound for higher topological complexity of real projective space

We obtain an explicit formula for the best lower bound for the higher topological complexity, ${\mathrm{TC}}_k(R{P}^n)$, of real projective space implied by mod 2 cohomology.     

The complexity of detecting crossingfree configurations in the plane

The computational complexity of the following type of problem is studied. Given a geometric graphG=(P, S) whereP is a set of points in the Euclidean plane andS a set of straight (closed) line segments between pairs of points inP, we want to know whetherG possesses a crossingfree subgraph of a special type. We analyze the problem of detecting crossingfree spanning trees, one factors and two factors in the plane. We also consider special restrictions on the slopes and on the lengths of the edges in the subgraphs.Klaus Jansen acknowledges support by the Deutsche Forschungsgemeinschaft. Gerhard J. Woeginger acknowledges support by the Christian Doppler Laboratorium für Diskrete Optimierung.     

The complexity of checking identities for finite matrix rings

We complete the investigations into the word-problem for finite matrix rings. Namely we prove that M₂ (Z₃), the ring of 2 x 2 matrices over Z₃, has a coNP-complete term-equivalence (or identity checking) problem.     

On the complexity of diagram testing

In 1987, Neetil and Rödl [4] claimed to have proved that the problem of finding whether a given graphG can be oriented as the diagram of a partial order is NP-complete. A flaw was discovered in their proof by Thostrup [11]. Neetil and Rödl [5] have since corrected the proof, but the new version is rather complex. We give a simpler and more elementary proof, using a completely different approach.     

On list primitive recursion and the complexity of computinginf

We present a primitive recursive programinf_with_lists computing the minimum of two natural numbersn andp (written in unary notation) and using primitive recursion on lists. This program has at first sight the required property of visiting simultaneously its inputs, so it is a counterexample to a theorem showing that such a program cannot be written in the language of primitive recursion on natural numbers, in the more general framework of primitive recursion on term algebras. However, its complexity is at leastinf(n,p)² so it does not implement the algorithm we have in mind to computeinf(n,p).     

Efficiently implementing two methods of the geometrical theory of differential equations: An experience in algorithm and software design

This paper contains an analysis of the two computational schemes for finding the infinitesimal symmetries and conservation laws of arbitrary systems of differential equations. The design of efficient algorithms implementing these computational schemes is described, together with the design of a portable software system, SCoLAr, supporting these algorithms. A test run is listed in the Appendix and the prospect of using the SCoLAr program on middle class personal computers is discussed.     

On the recognition complexity of some graph properties

By applying a topological approach due to Kahn, Saks and Sturtevant, we prove that all decreasing graph properties consisting of bipartite graphs only are elusive. This is an analogue to a well-known result of Yao.     

The complexity of the fixed point property

It is NP-complete to determine whether a given ordered set has a fixed point free order-preserving self-map. On the way to this result, we establish the NP-completeness of a related problem: Given ordered sets P and Q with t-tuples (p ₁, ... , p _t) and (q ₁, ... , q _t) from P and Q respectively, is there an order-preserving map f: P→Q satisfying f(p _i)≥q _i for each i=1, ... , t?     

The complexity of computing the tutte polynomial on transversal matroids

The complexity of computing the Tutte polynomialT(M,x,y) is determined for transversal matroidM and algebraic numbersx andy. It is shown that for fixedx andy the problem of computingT(M,x,y) forM a transversal matroid is #P-complete unless the numbersx andy satisfy (x−1)(y−1)=1, in which case it is polynomial-time computable. In particular, the problem of counting bases in a transversal matroid, and of counting various types of “matchable” sets of nodes in a bipartite graph, is #P-complete.