A POLYNOMIAL TIME ALGORITHM TO DETERMINE MAXIMAL BALANCED EQUIVALENCE RELATIONS

Following Golubitsky, Stewart, and others, we give definitions of networks and input trees. In order to make our work as general as possible, we work with a somewhat extended notion of multiplicity, and introduce the concept of "bunching" of trees. We then define balanced equivalence relations on networks, and a partial ordering on these relations. Previous work has shown that there is a maximal balanced equivalence relation on networks of certain classes: we provide a different style of proof which gives this result for any network. We define two algorithms to determine this relation in practice on a given finite network — one for use with networks with all multiplicities equal, and a second for the more general case. We then provide illustrative examples of each algorithm in use. We show both of these algorithms to be quartic in the size of the given network.

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A Set of Inverse Telecommunication Network Problems

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.760-762.665 ◽

2013 ◽

Vol 760-762 ◽

pp. 665-668

Author(s):

Zhe Heng Ding ◽

Jing Wang ◽

Qin Wang

Keyword(s):

Inverse Problem ◽

Polynomial Time ◽

Telecommunication Network ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Original Network ◽

Network Problem ◽

Network Problems ◽

Dominant Set ◽

The Given

In this paper, we consider a set of inverse telecommunication network problem under norm. With the expansion of telecommunication network, more and more links and nodes will be added to the existed telecommunication network. The original network can not cover new nodes and some old links become useless. The telecommunication company wants to sell some old links and purchase some new links within a given budget, such that the network of the company is able to access all nodes. We consider the inverse problem by using weakly dominant set, which is to change the weights of the edges as little as possible such that the given edge set becomes a weakly dominant set under the new weights. In this paper, we propose a polynomial time algorithm for the inverse problem under norm, and we also present an example to illustrate the algorithm.

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On Some Equivalence Relations for Probabilistic Processes1

Fundamenta Informaticae ◽

10.3233/fi-1992-17304 ◽

1992 ◽

Vol 17 (3) ◽

pp. 211-234

Author(s):

Dung T. Huynh ◽

Lu Tian

Keyword(s):

Polynomial Time ◽

Graph Isomorphism ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Isomorphism Problem ◽

Equivalence Relations ◽

Transition Systems ◽

Bisimulation Equivalence ◽

Finite Trace ◽

Trace Equivalence

In this paper, we investigate several equivalence relations for probabilistic labeled transition systems: bisimulation equivalence, readiness equivalence, failure equivalence, trace equivalence, maximal trace equivalence and finite trace equivalence. We formally prove the inclusions (equalities) among these equivalences. We also show that readiness, failure, trace, maximum trace and finite trace equivalences for finite probabilistic labeled transition systems are decidable in polynomial time. This should be contrasted with the PSPACE completeness of the same equivalences for classical labeled transition systems. Moreover, we derive an efficient polynomial time algorithm for deciding bisimulation equivalence for finite probabilistic labeled transition systems. The special case of initiated probabilistic transition systems will be considered. We show that the isomorphism problem for finite initiated labeled probabilistic transition systems is NC(1) equivalent to graph isomorphism.

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Learning Importance of Preferences

10.29007/v68w ◽

2018 ◽

Author(s):

Ying Zhu ◽

Mirek Truszczynski

Keyword(s):

Polynomial Time ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Scoring Rules ◽

Np Hard ◽

Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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