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 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.     

Sensor selection for parameterized random field estimation in wireless sensor networks

We consider the random field estimation problem with parametric trend in wireless sensor networks where the field can be described by unknown parameters to be estimated. Due to the limited resources, the network selects only a subset of the sensors to perform the estimation task with a desired performance under the D-optimal criterion. We propose a greedy sampling scheme to select the sensor nodes according to the information gain of the sensors. A distributed algorithm is also developed by consensus-based ...     

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).     

Efficient algorithms for single- and two-layer linear placement of parallel graphs

This paper outlines an algorithm for optimum linear ordering (OLO) of a weighted parallel graph with O(n log k) worst-case time complexity, and O(n + k log(n/k) log k) expected-case time complexity, where n is the total number of nodes and k is the number of chains in the parallel graph. Next, the two-layer OLO problem is considered, where the goal is to place the nodes linearly in two routing layers minimizing the total wire length. The two-layer problem is shown to subsume the maxcut problem and a befitting heuristic algorithm is proposed. Experimental results on randomly generated samples show that the heuristic algorithm runs very fast and outputs optimum solutions in more than 90% instances.     

Maximal and minimal representations of gapped and non-gapped motifs of a string

The problems of finding maximal and minimal equivalent representations for gapped and non-gapped motifs as well as finding motifs that characterize a fixed set of occurrence locations for a given string are studied. We apply two equivalence relations on representations. The first one is the well-known occurrence-equivalence of motifs. The second equivalence is introduced for patterns of occurrence locations, to characterize such patterns by motifs. For both equivalences, quadratic-time algorithms are given for finding a maximal representative of an equivalence class. Finding a minimal representative is shown to be NP-complete in both cases. For non-gapped motifs suffix-tree-based linear-time algorithms are given for finding maximal and minimal representatives. Maximal (minimal) gapped motifs are composed of blocks that are maximal (minimal) non-gapped motifs, maximal and minimal non-gapped motifs thus making up a small basis for all motifs. The implied bound on the number of gapped motifs that have a fixed number of non-gapped blocks is also given.     

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.     

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.