Nonlinearity of davenport—Schinzel sequences and of generalized path compression schemes

Davenport—Schinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. We show that the maximal length of a Davenport—Schinzel sequence composed ofn symbols is Θ (nα(n)), where α(n) is the functional inverse of Ackermann’s function, and is thus very slowly increasing to infinity. This is achieved by establishing an equivalence between such sequences and generalized path compression schemes on rooted trees, and then by analyzing these schemes. Work on this paper by the second author has been supported in part by a grant from the U.S.-Israeli Binational Science Foundation.     

Structural properties of greedoids

Greedoids were introduced by the authors as generalizations of matroids providing a framework for the greedy algorithm. In this paper they are studied from a structural aspect. Definitions of basic matroid-theoretical concepts such as rank and closure can be generalized to greedoids, even though they loose some of their fundamental properties. The rank function of a greedoid is only “locally” submodular. The closure operator is not monotone but possesses a (relaxed) Steinitz—McLane exchange property. We define two classes of subsets, called rank-feasible and closure-feasible, so that the rank and closure behave nicely for them. In particular, restricted to rank-feasible sets the rank function is submodular. Finally we show that Rado’s theorem on independent transversals of subsets of matroids remains valid for feasible transversals of certain sets of greedoids. Dedicated to Paul Erdős on his seventieth birthday Supported by the joint research project “Algorithmic Aspects of Combinatorial Optimization” of the Hungarian Academy of Sciences (Magyar Tudományos Akadémia) and the German Research Association (Deutsche Forschungsgemeinschaft, SFB 21).     

Improved pebbling bounds

Consider a configuration of pebbles distributed on the vertices of a connected graph of order n. A pebbling step consists of removing two pebbles from a given vertex and placing one pebble on an adjacent vertex. A distribution of pebbles on a graph is called solvable if it is possible to place a pebble on any given vertex using a sequence of pebbling steps. The pebbling number of a graph, denoted f(G), is the minimal number of pebbles such that every configuration of f(G) pebbles on G is solvable. We derive several general upper bounds on the pebbling number, improving previous results.     

Decomposition of submodular functions

A decomposition theory for submodular functions is described. Any such function is shown to have a unique decomposition consisting of indecomposable functions and certain highly decomposable functions, and the latter are completely characterized. Applications include decompositions of hypergraphs based on edge and vertex connectivity, the decomposition of matroids based on three-connectivity, the Gomory—Hu decomposition of flow networks, and Fujishige’s decomposition of symmetric submodular functions. Efficient decomposition algorithms are also discussed.     

Lower bounds for the energy of digraphs

The energy of a digraph D is defined as , where z₁,…,z_n are the eigenvalues of D. In this article we find lower bounds for the energy of digraphs in terms of the number of closed walks of length 2, extending in this way the result obtained by Caporossi et al. [G. Caporossi, D. Cvetkovi?, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996]: for all graphs G with m edges. Also, we study digraphs with three eigenvalues.     

Algorithms for minimum length partitions of polygons

We show that it is possible to find a diagonal partition of anyn-vertex simple polygon into smaller polygons, each of at mostm edges, minimizing the total length of the partitioning diagonals, in timeO(n ³ m ²). We derive the same asymptotic upper time-bound for minimum length diagonal partitions of simple polygons into exactlym-gons provided that the input polygon can be partitioned intom-gons. Also, in the latter case, if the input polygon is convex, we can reduce the upper time-bound toO(n ³ logm).     

Covering of graphs by complete bipartite subgraphs; Complexity of 0–1 matrices

We prove that the edge set of an arbitrary simple graphG onn vertices can be covered by at mostn−[log₂ n]+1 complete bipartite subgraphs ofG. If the weight of a subgraph is the number of its vertices, then there always exists a cover with total weightc(n ²/logn) and this bound is sharp apart from a constant factor. Our result answers a problem of T. G. Tarján. Dedicated to Paul Erdős on his seventieth birthday     

Improved processor bounds for combinatorial problems in RNC

Our main result improves the known processor bound by a factor ofn ⁴ (maintaining the expected parallel running time,O(log³ n)) for the following important problem:find a perfect matching in a general or in a bipartite graph with n vertices. A solution to that problem is used in parallel algorithms for several combinatorial problems, in particular for the problems of finding i) a (perfect) matching of maximum weight, ii) a maximum cardinality matching, iii) a matching of maximum vertex weight, iv) a maximums-t flow in a digraph with unit edge capacities. Consequently the known algorithms for those problems are substantially improved.The results of this paper have been presented at the 26-th Annual IEEE Symp. FOCS, Portland, Oregon (October 1985).Partially supported by NSF Grants MCS 8303139 and DCR 8511713.Supporeted by NSF Grants MCS 8203232 and DCR 8507573.     

Expected length of the longest common subsequence for large alphabets

We consider the length L of the longest common subsequence of two randomly uniformly and independently chosen n character words over a k-ary alphabet. Subadditivity arguments yield that E[L]/n converges to a constant γ_k. We prove a conjecture of Sankoff and Mainville from the early 1980s claiming that as k→∞.     

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.     

Lie algebra computations

In the context of prolongation theory, introduced by Wahlquist and Estabrook, computations of a lot of Jacobi identities in (infinite-dimensional) Lie algebras are necessary. These computations can be done (automatically) using symbolic computations. A package written in REDUCE is demonstrated to give an idea of the chosen approach.     

A lower bound for the recognition of digraph properties

The complexity of a digraph property is the number of entries of the adjacency matrix which must be examined by a decision tree algorithm to recognize the property in the worst case, Aanderaa and Rosenberg conjectured that there is a constant such that every digraph property which is monotone (not destroyed by the deletion of edges) and nontrivial (holds for some but not all digraphs) has complexity at leastv ² wherev is the number of nodes in the digraph. This conjecture was proved by Rivest and Vuillemin and a lower bound ofv ²/4–o(v ²) was subsequently found by Kahn, Saks, and Sturtevant. Here we show a lower bound ofv ²/2–o(v ²). We also prove that a certain class of monotone, nontrivial bipartite digraph properties is evasive (requires that every entry in the adjacency matrix be examined in the worst case).     

A lower bound for the spectral radius of a digraph

We show that the spectral radius ρ(D) of a digraph D with n vertices and c₂ closed walks of length 2 satisfies Moreover, equality occurs if and only if D is the symmetric digraph associated to a -regular graph, plus some arcs that do not belong to cycles. As an application of this result, we construct new sharp upper bounds for the low energy of a digraph, which extends Koolen and Moulton bounds of the energy to digraphs.     

Lower bounds for the game colouring number of partial k-trees and planar graphs

This paper discusses the game colouring number of partial k-trees and planar graphs. Let col_g(PT_k) and col_g(P) denote the maximum game colouring number of partial k trees and the maximum game colouring number of planar graphs, respectively. In this paper, we prove that col_g(PT_k)=3k+2 and col_g(P)?11. We also prove that the game colouring number col_g(G) of a graph is a monotone parameter, i.e., if H is a subgraph of G, then col_g(H)?col_g(G).     

Pfaffian structures and critical problems in finite symplectic spaces

Just as matroids abstract the algebraic properties of determinants in a vector space, Pfaffian structures abstract the algebraic properties of Pfaffians or skew-symmetric determinants in a symplectic space (that is, a vector space with an alternating bilinear form). This is done using an exchange-augmentation axiom which is a combinatorial version of a Laplace expansion or straightening identity for Pfaffians. Using Pfaffian structures, we study a symplectic analogue of the classical critical problem: given a setS of non-zero vectors in a non-singular symplectic spaceV of dimension2m, find its symplectic critical exponent, that is, the minimum of the set {m?dim(U):U∩S=0}, whereU ranges over all the (totally) isotropic subspaces disjoint fromS. In particular, we derive a formula for the number of isotropic subspaces of a given dimension disjoint from the setS by Möbius inversion over the order ideal of isotropic flats in the lattice of flats of the matroid onS given by linear dependence. This formula implies that the symplectic critical exponent ofS depends only on its matroid and Pfaffian structure; however, it may depend on the dimension of the symplectic spaceV.     

Exact Distribution of the Occurrence Number for K-tuples Over an Alphabet of Non-Equal Probability Letters

A nucleotide sequence can be considered as a realization of the non-equal-probability independently and identically distributed (niid) model. In this paper we derive the exact distribution of the occurrence number for each K-tuple with respect to the niid model by means of the Goulden-Jackson cluster method. An application of the probability function to get exact expectation curves [9] is presented, accompanied by comparison between the exact approach and the approximate solution.Received October 31, 2004     

An augmenting path algorithm for linear matroid parity

Linear matroid parity generalizes matroid intersection and graph matching (and hence network flow, degree-constrained subgraphs, etc.). A polynomial algorithm was given by Lovász. This paper presents an algorithm that uses timeO(mn ³), wherem is the number of elements andn is the rank. (The time isO(mn ^2.5) using fast matrix multiplication; both bounds assume the uniform cost model). For graphic matroids the time isO(mn ²). The algorithm is based on the method of augmenting paths used in the algorithms for all subcases of the problem. First author was supported in part by the National Science Foundation under grants MCS 78-18909, MCS-8302648, and DCR-8511991. The research was done while the second author was at the University of Denver and at the University of Colorado at Boulder.     

New upper bounds on the spectral radius of unicyclic graphs

Let G=(V(G),E(G)) be a unicyclic simple undirected graph with largest vertex degree Δ. Let C_r be the unique cycle of G. The graph G-E(C_r) is a forest of r rooted trees T₁,T₂,…,T_r with root vertices v₁,v₂,…,v_r, respectively. Let

     

Cones of alternating and cut submodular set functions

We describe facets of the cones of alternating set functions and cut submodular set functions generated by directed and undirected graphs and by uniform even hypergraphs. This answers a question asked by L. Lovász at the Bonn Mathematical Programming Conference in 1982. We show that there is a network flow algorithm for minimizing a hypergraph cut set function.     

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.