An Efficient Fixed Parameter Tractable Algorithm for 1-Sided Crossing Minimization

We give an O(^k · n²) fixed parameter tractable algorithm for the 1-Sided Crossing Minimization. The constant in the running time is the golden ratio = (1+5)/2 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings.     

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).     

Exact 3-satisfiability is decidable in time O(20.16254n)

Let F = C ₁ C _m be a Boolean formula in conjunctive normal form over a set V of n propositional variables, s.t. each clause C _i contains at most three literals l over V. Solving the problem exact 3-satisfiability (X3SAT) for F means to decide whether there is a truth assignment setting exactly one literal in each clause of F to true (1). As is well known X3SAT is NP-complete [6]. By exploiting a perfect matching reduction we prove that X3SAT is deterministically decidable in time O(2^0.18674n ). Thereby we improve a result in [2,3] stating X3SAT O(2^0.2072n ) and a bound of O(2^0.200002n ) for the corresponding enumeration problem #X3SAT stated in a preprint [1]. After that by a more involved deterministic case analysis we are able to show that X3SAT O(2^0.16254n ).An extended abstract of this paper was presented at the Fifth International Symposium on the Theory and Applications of Satisfiability Testing (SAT 2002).     

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.     

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.     

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.     

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).     

The optimal control approach to generalized multiprocessor scheduling

In this paper we present several new results in the theory of homogeneous multiprocessor scheduling. We start with some assumptions about the behavior of tasks, with associated precedence constraints, as processor power is applied. We assume that as more processors are applied to a task, the time taken to compute it decreases, yielding some speedup. Because of communication, synchronization, and task scheduling overhead, this speedup increases less than linearly with the number of processors applied. We also assume that the number of processors which can be assigned to a task is a continuous variable, with a view to exploiting continuous mathematics. The optimal scheduling problem is to determine the number of processors assigned to each task, and task sequencing, to minimize the finishing time.These assumptions allow us to recast the optimal scheduling problem in a form which can be addressed by optimal control theory. Various theorems can be proven which characterize the optimal scheduling solution. Most importantly, for the special case where the speedup function of each task isp , wherep is the amount of processing power applied to the task, we can directly solve our equations for the optimal solution. In this case, for task graphs formed from parallel and series connections, the solution can be derived by inspection. The solution can also be shown to be shortest path from the initial to the final state, as measured by anl _1/ distance metric, subject to obstacle constraints imposed by the precedence constraints.This research has been funded in part by the Advanced Research Project Agency monitored by ONR under Grant No. N00014-89-J-1489, in part by Draper Laboratory, in part by DARPA Contract No. N00014-87-K-0825, and in part by NSF Grant No. MIP-9012773. The first author is now with AT&T Bell Laboratories and the second author is with BBN Incorporated.     

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.