Improved approximation algorithms for computing $$k$$ k disjoint paths subject to two constraints

Longkun Guo (), Hong Shen and Kewen Liao
Additional contact information
Longkun Guo: Fuzhou University
Hong Shen: University of Adelaide
Kewen Liao: University of Adelaide

Journal of Combinatorial Optimization, 2015, vol. 29, issue 1, No 9, 153-164

Abstract: Abstract For a given graph $$G$$ G with distinct vertices $$s$$ s and $$t$$ t , nonnegative integral cost and delay on edges, and positive integral bound $$C$$ C and $$D$$ D on cost and delay respectively, the $$k$$ k bi-constraint path ( $$k$$ k BCP) problem is to compute $$k$$ k disjoint $$st$$ s t -paths subject to $$C$$ C and $$D$$ D . This problem is known to be NP-hard, even when $$k=1$$ k = 1 (Garey and Johnson, Computers and Intractability, 1979). This paper first gives a simple approximation algorithm with factor- $$(2,2)$$ ( 2 , 2 ) , i.e. the algorithm computes a solution with delay and cost bounded by $$2*D$$ 2 ∗ D and $$2*C$$ 2 ∗ C respectively. Later, a novel improved approximation algorithm with ratio $$(1+\beta ,\,\max \{2,\,1+\ln (1/\beta )\})$$ ( 1 + β , max { 2 , 1 + ln ( 1 / β ) } ) is developed by constructing interesting auxiliary graphs and employing the cycle cancellation method. As a consequence, we can obtain a factor- $$(1.369,\,2)$$ ( 1.369 , 2 ) approximation algorithm immediately and a factor- $$(1.567,\,1.567)$$ ( 1.567 , 1.567 ) algorithm by slightly modifying the algorithm. Besides, when $$\beta =0$$ β = 0 , the algorithm is shown to be with ratio $$(1,\, O(\ln n))$$ ( 1 , O ( ln n ) ) , i.e. it is an algorithm with only a single factor ratio $$O(\ln n)$$ O ( ln n ) on cost. To the best of our knowledge, this is the first non-trivial approximation algorithm that strictly obeys the delay constraint for the $$k$$ k BCP problem.

Keywords: $$k$$ k -disjoint bi-constraint path; NP-hard; Bifactor approximation algorithm; Auxiliary graph; Cycle cancellation (search for similar items in EconPapers)
Date: 2015
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
http://link.springer.com/10.1007/s10878-013-9693-x Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jcomop:v:29:y:2015:i:1:d:10.1007_s10878-013-9693-x

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10878

DOI: 10.1007/s10878-013-9693-x

Access Statistics for this article

Journal of Combinatorial Optimization is currently edited by Thai, My T.

More articles in Journal of Combinatorial Optimization from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().