scholarly journals A General Framework for Approximating Min Sum Ordering Problems

Author(s):  
Felix Happach ◽  
Lisa Hellerstein ◽  
Thomas Lidbetter

We consider a large family of problems in which an ordering (or, more precisely, a chain of subsets) of a finite set must be chosen to minimize some weighted sum of costs. This family includes variations of min sum set cover, several scheduling and search problems, and problems in Boolean function evaluation. We define a new problem, called the min sum ordering problem (MSOP), which generalizes all these problems using a cost and a weight function defined on subsets of a finite set. Assuming a polynomial time α-approximation algorithm for the problem of finding a subset whose ratio of weight to cost is maximal, we show that under very minimal assumptions, there is a polynomial time [Formula: see text]-approximation algorithm for MSOP. This approximation result generalizes a proof technique used for several distinct problems in the literature. We apply this to obtain a number of new approximation results. Summary of Contribution: This paper provides a general framework for min sum ordering problems. Within the realm of theoretical computer science, these problems include min sum set cover and its generalizations, as well as problems in Boolean function evaluation. On the operations research side, they include problems in search theory and scheduling. We present and analyze a very general algorithm for these problems, unifying several previous results on various min sum ordering problems and resulting in new constant factor guarantees for others.

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Qiufen Ni ◽  
Chuanhe Huang ◽  
Panos M. Pardalos ◽  
Jia Ye ◽  
Bin Fu

We introduce a new two-side approximation method for the channel scheduling problem, which controls the accuracy of approximation in two sides by a pair of parameters f , g . We present a series of simple and practical-for-implementation greedy algorithms which give constant factor approximation in both sides. First, we propose four approximation algorithms for the weighted channel allocation problem: 1. a greedy algorithm for the multichannel with fixed interference radius scheduling problem is proposed and an one side O 1 -IS-approximation is obtained; 2. a greedy O 1 , O 1 -approximation algorithm for single channel with fixed interference radius scheduling problem is presented; 3. we improve the existing algorithm for the multichannel scheduling and show an E O d / ε time 1 − ϵ -approximation algorithm; 4. we speed up the polynomial time approximation scheme for single-channel scheduling through merging two algorithms and show a 1 − ϵ , O 1 -approximation algorithm. Next, we study two polynomial time constant factor greedy approximation algorithms for the unweighted channel allocation with variate interference radius. A greedy O 1 -approximation algorithm for the multichannel scheduling problem and an O 1 , O 1 -approximation algorithm for single-channel scheduling problem are developed. At last, we do some experiments to verify the effectiveness of our proposed methods.


2017 ◽  
Vol 657 ◽  
pp. 111-126 ◽  
Author(s):  
Usha Mohan ◽  
Sivaramakrishnan Ramani ◽  
Sounaka Mishra

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
M. Bouznif ◽  
R. Giroudeau

We investigate complexity and approximation results on a processor networks where the communication delay depends on the distance between the processors performing tasks. We then prove that there is no heuristic with a performance guarantee smaller than 4/3 for makespan minimization for precedence graph on a large class of processor networks like hypercube, grid, torus, and so forth, with a fixed diameter . We extend complexity results when the precedence graph is a bipartite graph. We also design an efficient polynomial-time -approximation algorithm for the makespan minimization on processor networks with diameter .


2021 ◽  
Author(s):  
Ben T. Larson ◽  
Jack Garbus ◽  
Jordan B. Pollack ◽  
Wallace F. Marshall

Cells are complex biochemical systems whose behavior emerges from interactions among myriad molecular components. The idea that cells execute computational processes is often invoked as a general framework for understanding cellular complexity. However, the manner in which cells might embody computational processes in a way that the powerful theories of computation, such as finite state machine models, could be productively applied, remains to be seen. Here we demonstrate finite state machine-like processing embodied in cells, using the walking behavior of Euplotes eurystomus, a ciliate that walks across surfaces using fourteen motile appendages called cirri. We found that cellular walking entails a discrete set of gait states. Transitions between these states are highly regulated, with distinct breaking of detailed balance and only a small subset of possible transitions actually observed. The set of observed transitions decomposes into a small group of high-probability unbalanced transitions forming a cycle and a large group of low-probability balanced transitions, thus revealing stereotypy in sequential patterns of state transitions. Taken together these findings implicate a machine-like process. Cirri are connected by microtubule bundles, and we find an association between the involvement of cirri in different state transitions and the pattern of attachment to the microtubule bundle system, suggesting a mechanical basis for the regularity of state transitions. We propose a model where the actively controlled, unbalanced transitions establish strain in certain cirri, the release of which from the substrate causes the cell to advance forward along a linear trajectory. This demonstration of a finite state machine embodied in a living cell opens up new links between theoretical computer science and cell biology and may provide a general framework for understanding and predicting cell behavior at a super-molecular level.


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