Zbigniew Lonc

On Complexity of Some Chain and Antichain Partition Problems

In the paper we deal with computational complexity of a problem C k (respectively A k ) of a partition of an ordered set into minimum number of at most k-element chains (resp. antichains). We show that C k , k ≥ 3, is NP-complete even for N-free ordered sets of length at most k, C k and A k are polynomial for series-paralel orders and A k is polynomial for interval orders. We also consider related problems for graphs.

On the number of spanning trees in directed circulant graphs

Let gk(n) [respectively, fk(n)] be the maximum number of spanning trees in directed circulant graphs (respectively, regular directed graphs) with n vertices and out-degrees equal to k > 1. We show that gk(n) = kn(1+o(1)) and fk(n) = kn(1+o(1)). Moreover, we prove that g2(n) = ⌊(2n + 1)/3⌋.

Proof of a conjecture on partitions of a boolean lattice

Let n and c be positive integers. We show that if n is sufficiently large given c then the Boolean lattice consisting of all subsets of an n-element set can be partitioned into chains of size c except for at most c — 1 elements which also form a chain. This settles a conjecture of Griggs.

Sequences of radius k : how to fetch many huge objects into small memory for pairwise computations

Let a1, a2, ..., am be a sequence over [n]={1,...n} We say that a sequence a1, a2, .. am has the k-radius property if every pair of different elements in [n] occurs at least once within distance at most k; the distanced(ai,aj)=|i−j | We demonstrate lower and (asymptotically) matching upper bounds for sequences with the k-radius property Such sequences are applicable, for example, in computations of two-argument functions for all

On the problem of computing the well-founded semantics

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On the Problem of Computing the Well-Founded Semantics

pairs of large objects such as medical images, bitmaps or matrices, when processing occurs in a memory of size capable of storing k + 1 objects, k < n We focus on the model when elements are read into the memory in a FIFO fashion that correspond to streaming the data or a special type of caching We present asymptotically optimal constructions; they are based on the Euler totient theorem and recursion.

Fixed-parameter complexity of semantics for logic programs

The well-founded semantics is one of the most widely studied and used semantics of logic programs with negation. In the case of finite propositional programs, it can be computed in polynomial time, more specifically, in O(∣At(P)∣ × size(P)) steps, where size(P) denotes the total number of occurrences of atoms in a logic program P. This bound is achieved by an algorithm introduced by Van Gelder and known as the alternating-fixpoint algorithm. Improving on the alternating-fixpoint algorithm turned out to be difficult. In this paper we study extensions and modifications of the alternating-fixpoint approach. We then restrict our attention to the class of programs whose rules have no more than one positive occurrence of an atom in their bodies. For programs in that class we propose a new implementation of the alternating-fixpoint method in which false atoms are computed in a top-down fashion. We show that our algorithm is faster than other known algorithms and that for a wide class of programs it is linear and so, asymptotically optimal.

On the asymptotic behavior of the maximum number of spanning trees in circulant graphs

The well-founded semantics is one of the most widely studied and used semantics of logic programs with negation. In the case of finite propositional programs, it can be computed in polynomial time, more specifically, in O(|At(P)| × size(P)) steps, where size(P) denotes the total number of occurrences of atoms in a logic program P. This bound is achieved by an algorithm introduced by Van Gelder and known as the alternating-fixpoint algorithm. Improving on the alternating-fixpoint algorithm turned out to be difficult. In this paper we study extensions and modifications of the alternating-fixpoint approach. We then restrict our attention to the class of programs whose rules have no more than one positive occurrence of an atom in their bodies. For programs in that class we propose a new implementation of the alternating-fixpoint method in which false atoms are computed in a top-down fashion. We show that our algorithm is faster than other known algorithms and that for a wide class of programs it is linear and so, asymptotically optimal.

Edge Decomposition into Isomorphic Copies ofsK1, 2Is Polynomial

A decision problem is called parameterized if its input is a pair of strings. One of these strings is referred to as a parameter. The following problem is an example of a parameterized decision problem with k serving as a parameter: given a propositional logic program P and a nonnegative integer k, decide whether P has a stable model of size no more than k. Parameterized problems that are NP-complete often become solvable in polynomial time if the parameter is fixed. The problem to decide whether a program P has a stable model of size no more than k, where k is fixed and not a part of input, can be solved in time O(mnk), where m is the size of P and n is the number of atoms in P. Thus, this problem is in the class P. However, algorithms with the running time given by a polynomial of order k are not satisfactory even for relatively small values of k.The key question then is whether significantly better algorithms (with the degree of the polynomial not dependent on k) exist. To tackle it, we use the framework of fixed-parameter complexity. We establish the fixed-parameter complexity for several parameterized decision problems involving models, supported models, and stable models of logic programs. We also establish the fixed-parameter complexity for variants of these problems resulting from restricting attention to definite Horn programs and to purely negative programs. Most of the problems considered in the paper have high fixed-parameter complexity. Thus, it is unlikely that fixing bounds on models (supported models, stable models) will lead to fast algorithms to decide the existence of such models.

Covering cycles andk-term degree sums

The following asymptotic estimation of the maximum number of spanning trees fk(n) in 2k-regular circulant graphs (k > 1) on n vertices is the main result of this paper: fk(n) = ((2k)/(rk))n(1+o(1)), where © 1997 John Wiley & Sons, Inc. Networks 30:47–56, 1997