Ingo Schiermeyer

Reverse-Fit: A 2-Optimal Algorithm for Packing Rectangles

We describe and analyze a ”level-oriented” algorithm, called ”Reverse-Fit”, for packing rectangles into a unit-width, infinite-height bin so as to minimize the total height of the packing. For L an arbitrary list of rectangles, all assumed to have width no more than 1, let h OPT denote the minimum possible bin height within the rectangles in L can be packed, and let RF(L) denote the height actually used by Reverse-Fit. We will show that RF(L)≤2·h OPT for an arbitrary list L of rectangles.

Solving 3-Satisfiability in Less Then 1, 579n Steps

In this paper we describe and analyse an improved algorithm for solving the 3-Satisfiability problem. If F is a boolean formula in conjunctive normal form with n variables and r clauses, then we will show that this algorithm solves the Satisfiability problem for formulas with at most three literals per clause in time less than O(1,579n).

Deciding 3-Colourability in Less Than O(1.415^n) Steps

In this paper we describe and analyze an improved algorithm for deciding the 3-Colourability problem. If G is a simple graph on n vertices then we will show that this algorithm tests a graph for 3-Colourability, i.e. an assignment of three colours to the vertices of G such that two adjacent vertices obtain different colours, in less than O(1.415n) steps.

Insertible vertices, neighborhood intersections, and hamiltonicity

Let G be a simple undirected graph of order n. For an independent set S ⊂ V(G) of k vertices, we define the k neighborhood intersections Si = {v ϵ V(G)&n92;S|N(v) ∩ S| = i}, 1 ≦ i ≦ k, with si = |Si|. Using the concept of insertible vertices and the concept of neighborhood intersections, we prove the following theorem.

Computation of the 0-dual closure for hamiltonian graphs

The well-known closure concept of Bondy and Chvatal (1976) is based on degree sums of pairs of nonadjacent vertices. It generalizes six earlier sufficient degree conditions for hamiltonian graphs. Ainouche and Christofides (1987) introduced a more general concept which is called the 0-dual closure. We prove that this concept generalizes five sufficient conditions for hamiltonian graphs which are not generalized by the concept of Bondy and Chvatal (1976).

A closure concept based on neighborhood unions of independent triples

The well-known closure concept of Bondy and Chvatal is based on degree-sums of pairs of nonadjacent (independent) vertices. We show that a more general concept due to Ainouche and Christofides can be restated in terms of degree-sums of independent triples. We introduce a closure concept which is based on neighborhood unions of independent triples and which also generalizes the closure concept of Bondy and Chvatal.

Approximating Maximum Independent Set in k-Clique-Free Graphs

In this paper we study lower bounds and approximation algorithms for the independence number α(G) in k-clique-free graphs G. Ajtai et al. [1] showed that there exists an absolute constant c 1 such that for any k-clique-free graph G on n vertices and with average degree ¯d, α(G) ≥ c 1 log((log ¯d)/k)/d n

A fast sequential and parallel algorithm for the computation of the k -closure of a graph

In 1976 Bondy and Chvatal introduced the k-closure Ck(G) of a graph and described an algorithm which constructs it in O(n4) time. We present an algorithm which requires O(n3) time. However, in the average case, Ck(G) is computed by this algorithm for almost all integers k (in the asymptotic sense) with O ≦ k ≦ 2n − 2 in O(n2) time. We next present a parallel algorithm which requires O(n2 log n) time. In the average case, this algorithm computes Ck(G) for almost all integers k in O(log n) parallel time.

An Approximation Algorithm for 3-Colourability

We present a polynomial time approximation algorithm to colour a 3-colourable graph G with 3f(n) colours, if G has minimum degree δ(G)≥αn/f(n), where Ω(1)≤f(n)<O(n) and α is a positive constant. We also discuss NP—completeness and #P—completeness of restricted k-Colourability problems.

The k -SATISFIABILITY problem remains NP-complete for dense families

Abstract We consider the k - SATISFIABILITY problem ( k -SAT): Given a family F of n clauses c 1 , …, c n in conjunctive normal form, each consisting of k literals corresponding to k different variables of a set of r ⩾ k 1 boolean variables, is F satisfiable? By k -SAT(> n 0 ) we denote the k -SAT problem restricted to families with n > n 0 ( r ) clauses. We prove that for each k ⩾3 and each integer l ⩾4 such that r ⩾ lk 2 , the k- SAT (>( r k ) (2 k −1−4/l)) problem is NP-complete.