Tractability frontiers of the partner units configuration problem

The partner units problem (PUP) is an acknowledged hard benchmark problem for the Logic Programming community with various industrial application fields like surveillance, electrical engineering, computer networks or railway safety systems. Although it is already known for a while that the PUP is NP-complete in its general form, complexity for subproblems reflecting the real problems in industrial fields remained widely unclear so far. In this article we provide all missing complexity results. For the subclass of the PUP that belongs to the complexity class P we present a polynomial-time algorithm and give in-depth algorithmic complexity results.     

Generalized juntas and NP-hard sets

We show that many NP-hard sets have heuristic polynomial-time algorithms with high probability weight of correctness with respect to generalizations of Procaccia and Rosenschein’s junta distributions.     

On finding optimal and near-optimal lineal spanning trees

The search for good lineal, or depth-first, spanning trees is an important aspect in the implementation of a wide assortment of graph algorithms. We consider the complexity of findingoptimal lineal spanning trees under various notions of optimality. In particular, we show that several natural problems, such as constructing a shortest or a tallest lineal tree, are NP-hard. We also address the issue of polynomial-time, near-optimization strategies for these difficult problems, showing that efficient absolute approximation algorithms cannot exist unlessP = NP.This author's research was supported in part by the Sandia University Research Program and by the National Science Foundation under Grant M IP-8603879.This author's research was supported in part by the National Science Foundation under Grants ECS-8403859 and MIP-8603879.     

The Machine Duplication Problem in a Job Shop with Two Jobs

This paper addresses a problem related to the classical job shop scheduling problem with two jobs. The problem consists in concurrently determining the best subset of machines to be duplicated and the optimal scheduling of the operations in order to minimize completion time. Such a problem arises in the tool management for a class of flexible manufacturing cells. The job shop with two jobs is first reviewed, the application of the classical search algorithm A* to this problem is discussed and its performance compared with a previous approach. The complexity of the machine duplication problem is then analysed. The problem is proved to be in general NP-hard in the strong sense, but in a class of special cases, relevant from the applications viewpoint, it can be solved in polynomial time by a dynamic programming algorithm. A heuristic based on such an algorithm and on A* is proposed for the general problem; the results are satisfactory in terms of both efficiency and quality of the solution.     

Efficient combinator code

Some combinatory logics are examined as object code for functional programs. The worst-case performances of certain algorithms for abstracting variables from combinatory expressions are analysed. A lower bound on the performance of any abstraction algorithm for a finite set of combinators is given. Using the combinators S, K, I, B, C, S′, B′, C′ and Y, the problem of finding an optimal abstraction algorithm is shown to be NP-complete. Some methods of improving abstraction algorithms for those combinators are examined, including “balancing” (for asymptotic performance) and “peephole” optimisations (for smaller cases).     

On variable-weighted exact satisfiability problems

We show that the NP-hard optimization problems minimum and maximum weight exact satisfiability (XSAT) for a CNF formula C over n propositional variables equipped with arbitrary real-valued weights can be solved in O(||C||2^0.2441n ) time. To the best of our knowledge, the algorithms presented here are the first handling weighted XSAT optimization versions in non-trivial worst case time. We also investigate the corresponding weighted counting problems, namely we show that the number of all minimum, resp. maximum, weight exact satisfiability solutions of an arbitrarily weighted formula can be determined in O(n ²·||C||?+?2^0.40567n ) time. In recent years only the unweighted counterparts of these problems have been studied (Dahllöf and Jonsson, An algorithm for counting maximum weighted independent sets and its applications. In: Proceedings of the 13th ACM-SIAM Symposium on Discrete Algorithms, pp. 292–298, 2002; Dahllöf et al., Theor Comp Sci 320: 373–394, 2004; Porschen, On some weighted satisfiability and graph problems. In: Proceedings of the 31st Conference on Current Trends in Theory and Practice of Informatics (SOFSEM 2005). Lecture Notes in Comp. Science, vol. 3381, pp. 278–287. Springer, 2005).     

The Tractability of Segmentation and Scene Analysis

One of the fundamental problems in computer vision is the segmentation of an image into semantically meaningful regions, based only on image characteristics. A single segmentation can be determined using a linear number of evaluations of a uniformity predicate. However, minimising the number of regions is shown to be an NP-complete problem. We also show that the variational approach to segmentation, based on minimising a criterion combining the overall variance of regions and the number of regions, also gives rise to an NP-complete problem.When a library of object models is available, segmenting the image becomes a problem of scene analysis. A sufficient condition for the reconstruction of a 3D scene from a 2D image to be solvable in polynomial time is that the scene contains no cycles of mutually occluding objects and that no range information can be deduced from the image. It is known that relaxing the no cycles condition renders the problem NP-complete. We show that relaxing the no range information condition also produces an NP-complete problem.     

超协调限制逻辑的计算复杂性分析

超协调限制逻辑LPc是一种同时具有非单调性和超协调性的非经典逻辑，它可作为在不完全与不协调知识下常识推理的形式化.给出了命题LPc的计算复杂性结果和算法实现，指出LPc是NP完全问题，并给出了将LPc转化为等价的优先限制逻辑的线性时间算法，由于限制逻辑具有实用的实现算法且可用归结方法实现，因而该算法为LPc的实现提供了新的途径.     

The Difficulty of Reduced Error Pruning of Leveled Branching Programs

Induction of decision trees is one of the most successful approaches to supervised machine learning. Branching programs are a generalization of decision trees and, by the boosting analysis, exponentially more efficiently learnable than decision trees. However, this advantage has not been seen to materialize in experiments. Decision trees are easy to simplify using pruning. Reduced error pruning is one of the simplest decision tree pruning algorithms. For branching programs no pruning algorithms are known. In this paper we prove that reduced error pruning of branching programs is infeasible. Finding the optimal pruning of a branching program with respect to a set of pruning examples that is separate from the set of training examples is NP-complete. Because of this intractability result, we have to consider approximating reduced error pruning. Unfortunately, it turns out that even finding an approximate solution of arbitrary accuracy is computationally infeasible. In particular, reduced error pruning of branching programs is APX-hard. Our experiments show that, despite the negative theoretical results, heuristic pruning of branching programs can reduce their size without significantly altering the accuracy.