scholarly journals Polynomial time randomised approximation schemes for the Tutte polynomial of dense graphs

2002 ◽  
Author(s):  
N. Alon ◽  
A. Frieze ◽  
D. Welsh
Keyword(s):  
Polynomial Time ◽  
Tutte Polynomial ◽  
Approximation Schemes ◽  
Dense Graphs

Chromatic, Flow and Reliability Polynomials: The Complexity of their Coefficients

2002 ◽  
Vol 11 (4) ◽  
pp. 403-426 ◽  
Author(s):  
JAMES OXLEY ◽  
DOMINIC WELSH
Keyword(s):  
Finite Fields ◽  
Polynomial Time ◽  
Tutte Polynomial ◽  
Decision Problems ◽  
Approximation Schemes ◽  
Significant Difference ◽  
Classical Polynomials ◽  
Quasi Order

We study the complexity of computing the coefficients of three classical polynomials, namely the chromatic, flow and reliability polynomials of a graph. Each of these is a specialization of the Tutte polynomial Σtijxiyj. It is shown that, unless NP = RP, many of the relevant coefficients do not even have good randomized approximation schemes. We consider the quasi-order induced by approximation reducibility and highlight the pivotal position of the coefficient t10 = t01, otherwise known as the beta invariant.Our nonapproximability results are obtained by showing that various decision problems based on the coefficients are NP-hard. A study of such predicates shows a significant difference between the case of graphs, where, by Robertson–Seymour theory, they are computable in polynomial time, and the case of matrices over finite fields, where they are shown to be NP-hard.

Polynomial-Time Approximation Schemes for Circle Packing Problems

2014 ◽  
pp. 713-724 ◽  
Author(s):  
Flávio K. Miyazawa ◽  
Lehilton L. C. Pedrosa ◽  
Rafael C. S. Schouery ◽  
Maxim Sviridenko ◽  
Yoshiko Wakabayashi
Keyword(s):  
Polynomial Time ◽  
Circle Packing ◽  
Approximation Schemes ◽  
Packing Problems ◽  
Time Approximation ◽  
Polynomial Time Approximation

scholarly journals Scheduling on a Single Machine and Parallel Machines with Batch Deliveries and Potential Disruption

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Hua Gong ◽  
Yuyan Zhang ◽  
Puyu Yuan
Keyword(s):  
Polynomial Time ◽  
Single Machine ◽  
Manufacturing Industry ◽  
Parallel Machines ◽  
Parallel Machine Scheduling ◽  
Parallel Machine ◽  
Approximation Schemes ◽  
Scheduling Problems ◽  
Delivery Scheduling ◽  
Production Part

In this paper, we study several coordinated production-delivery scheduling problems with potential disruption motivated by a supply chain in the manufacturing industry. Both single-machine environment and identical parallel-machine environment are considered in the production part. The jobs finished on the machines are delivered to the same customer in batches. Each delivery batch has a capacity and incurs a delivery cost. There is a situation that a possible disruption in the production part may occur at some particular time and will last for a period of time with a probability. We consider both resumable case and nonresumable case where a job does not need (needs) to restart if it is disrupted for a resumable (nonresumable) case. The objective is to find a coordinated schedule of production and delivery that minimizes the expected total flow times plus the delivery costs. We first present some properties and analyze the NP-hard complexity for four various problems. For the corresponding single-machine and parallel-machine scheduling problems, pseudo-polynomial-time algorithms and fully polynomial-time approximation schemes (FPTASs) are presented in this paper, respectively.

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