Research Article Graphs and Matroids Weighted in a Bounded ...downloads.hindawi.com › journals ›...

Research ArticleGraphs and Matroids Weighted in a Bounded Incline Algebra

Ling-Xia Lu1 and Bei Zhang2

1 School of Mathematics and Science Shijiazhuang University of Economics Shijiazhuang 050031 China2 School of Science Hebei University of Science and Technology Shijiazhuang 050018 China

Correspondence should be addressed to Ling-Xia Lu lu lingxia163com

Received 23 May 2014 Accepted 1 July 2014 Published 14 July 2014

Academic Editor Wieslaw A Dudek

Copyright copy 2014 L-X Lu and B Zhang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Firstly for a graph weighted in a bounded incline algebra (or called a dioid) a longest path problem (LPP for short) is presentedwhich can be considered the uniform approach to the famous shortest path problem the widest path problem and themost reliablepath problem The solutions for LPP and related algorithms are given Secondly for a matroid weighted in a linear matroid themaximum independent set problem is studied

1 Introduction

In graph theory the famous shortest path problem (SPP forshort) is the problem of finding a path between two verticesin a weighted graph such that the sum of the weights of itsconstituent edges is minimized [1] An example is findingthe quickest way to get from one location to another on aroad map In this case the vertices represent locations andthe edges represent segments of road and are weighted by thetime needed to travel that segment

If we assume the weighted function to be nonnegativethen the related algebraic foundation of SPP is the semiring([0 +infin]min +) Therein we use the operation ldquo+rdquo tocompute the length of paths and use the operation ldquominrdquoto find the least one For the widest path problem (WPPfor short) or called the greatest capacity problem (GCPfor short) the related algebraic foundation is the semiring([0 +infin]maxmin) Accordingly we use the operation ldquominrdquoto compute the capacities and use the operation ldquomaxrdquo tofind the greatest one For the most reliable path problem(MRPP for short) the related algebraic foundation is thesemiring ([0 1]max times) Accordingly we use the operationldquotimesrdquo to compute the reliability of paths and use the operationldquomaxrdquo to find the greatest oneThere are many other classicalproblems using various semirings in graph theory [2]

For both ([0 +infin]min +) for SPP and ([0 +infin]maxmin) for WPP as well as the corresponding algorithms

the value ldquo+infinrdquo is used to act as the weight of artificial edgesbetween vertex pairs with no edge For these reasons SPPWPP and MRPP (and other potential problems) can be putinto a more generalized setting the algebraic path problem[2]The first aim of this paper is to unify SPPWPPWPP andother path problems into graphs weighted in an idempotentsemiring (also known as a dioid) [3] We shall give a unifiedapproach to find the shortest path the widest path and themost reliable path as well as their length

In 1935 Whitney introduced matroids as a generalizationof both graphs and linear independence in vector spaces [4]It is well known that matroids play an important role inapplied mathematics especially in optimal theory which areprecisely the structures for the maximum independent setproblem (MISP for short) which the very simple and efficientgreedy algorithm works [5] The second aim of this paper isto study matroids weighted in a linear dioid and the relatedMISP

2 Semirings Incline Algebras Dioids andTheir Properties

Semirings and matrices over semirings are useful tools indiverse areas such as automata theory design of switchingcircuits graph theory medical diagnosis Markov chains

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 912715 4 pageshttpdxdoiorg1011552014912715

2 The Scientific World Journal

informational systems complex systems modeling decision-making theory dynamical programming control theorynervous system probable reasoning psychological measure-ment and clustering [3 6]

Definition 1 (see [3]) A (22)-type algebra (119870 oplus otimes) is calleda semiring if

(K1) both (119870 oplus) and (119870 otimes) are semigroups

(K2) oplus is commutative that is 119886oplus119887 = 119887oplus119886 for all 119886 119887 isin 119870

(K3) otimes is distributive over oplus that is 119886 otimes (119887 oplus 119888) = (119886 otimes 119887) oplus(119886 otimes 119888) (119887 oplus 119888) otimes 119886 = (119887 otimes 119886) oplus (119888 otimes 119886) for all 119886 119887 119888 isin 119870

We call a semiring (119870 oplus otimes) preunital if there is a specialelement 1 isin 119870 such that (119870 otimes 1) is a monoid that is 1otimes119909 =119909 = 119909 otimes 1 for every 119909 isin 119870

Proposition 2 For every preunital semiring (119870 oplus otimes 1) thefollowing conditions are equivalent

(K4) 1 is absorbing with respect to the operation oplus that is1 oplus 119909 = 1 for every 119909 isin 119870

(K41015840) 119909 oplus (119909 otimes 119910) = 119909 and (119909 otimes 119910) oplus 119910 = 119910 for all 119909 119910 isin 119870

Proof (K4)rArr(K41015840)119909oplus(119909otimes119910) = (119909otimes1)oplus(119909otimes119910) = 119909otimes(1oplus119910) =119909 otimes 1 = 119909 Similarly (119909 otimes 119910) oplus 119910 = 119910

(K41015840)rArr(K4) 1 oplus 119909 = 1 oplus (1 otimes 119909) = 1

A preunital semiringwith condition (K4) is called a unitalsemiring A semiring (119870 oplus otimes) is called idempotent if oplus isidempotent that is 119886 oplus 119886 = 119886 for all 119886 isin 119870 An idempotentsemiring with the condition (K41015840) is called an incline algebra[6]

Proposition 3 Suppose that (119870 oplus otimes) is a unital semiringand 0 is an element in 119870 Then the following conditions areequivalent

(K5) 0 is absorbing with respect to the operation otimes that is0 otimes 119909 = 0 = 119909 otimes 0 for every 119909 isin 119870

(K51015840) (119870 oplus 0) is a monoid that is 0oplus119909 = 119909 for every 119909 isin 119870

Proof (K5)rArr(K51015840) by (K41015840) 0 oplus 119909 = (0 otimes 119909) oplus 119909 = 119909(K51015840)rArr(K5) by (K41015840) 0 otimes 119909 = 0 oplus (0 otimes 119909) = 0

An idempotent and unital semiring (119870 oplus otimes) with condi-tion (K5) is called a dioid that is a dioid is an incline with(K41015840) and (K51015840) which is called a bounded incline algebra[6]

In every dioid (119870 oplus otimes 0 1) we define 119909 ⪯ 119910 iff 119909oplus119910 = 119910Then ⪯ is a partial order on 119870 and (119870 oplus) is a bounded joinsemilattice Clearly 0 is the bottom element and 1 is the topelement so is the name bounded incline

Proposition 4 (see [7]) If 119886 ⪯ 119887 119888 ⪯ 119889 then 119886 oplus 119888 ⪯ 119887 oplus 119889and 119886 otimes 119888 ⪯ 119887 otimes 119889

Proof Since 119886 ⪯ 119887 119888 ⪯ 119889 we have 119886 oplus 119887 = 119887 119888 oplus 119889 = 119889

(1) (119886oplus119888)oplus(119887oplus119888) = (119886oplus119887)oplus(119888oplus119889) = 119887oplus119889 and 119886oplus119888 ⪯ 119887oplus119889(2) (119886otimes119888)oplus(119887otimes119888) = (119886oplus119887)otimes119888 = 119887otimes119888 and then 119886otimes119888 ⪯ 119887otimes119888

Similarly 119887 otimes 119888 ⪯ 119887 otimes 119889 Hence 119886 otimes 119888 ⪯ 119887 otimes 119889

Example 5 (classical examples) (1) ([0 +infin]min + +infin 0)is a dioid which is an algebraic model for SPP The partialorder ⪯ defined above is dual to the usual one le For explicit119886 ⪯ 119887 iff 119887 le 119886 in usual meaning

(2) ([0 +infin]maxmin 0 +infin) is a dioid which is analgebraic model for WPP The partial order ⪯ defined aboveis the same as the usual one le

(3) ([0 1]max times 0 1) is a dioid which is an algebraicmodel for MRPP The partial order ⪯ defined above is thesame as the usual one le

Example 6 (other examples) Consider

(1) ([0 +infin]minmax +infin 0)(2) ([0 1]minmax 1 0)(3) ([0 1]maxmin 0 1)(4) ([1 +infin]min times +infin 1)

Let 119860 be an (119898 times 119899)-matrix and let 119861 be an (119899 times 119897)-matrixover a semiring (119870 oplus otimes) Define 119860 ∘ 119861 = (119901

119894119895)119898times119897

by 119901119894119895=

oplus119899119896=1(119886119894119896otimes 119886119896119895)

Proposition 7 Let 119860119896119897 119861119897119898 119862119898119899

be three matrices Then(119860119896119897∘ 119861119897119898) ∘ 119862119898119899= 119860119896119897∘ (119861119897119898∘ 119862119898119899)

Proof Let 119860119896119897∘ 119861119897119898= (119901119894119895)119896119898 119861119897119898∘ 119862119898119899= (119902119894119895)119897119898

and (119860119896119897∘

119861119897119898) ∘ 119862119898119899= (119906119894119895)119897119899 119860119896119897∘ (119861119897119898∘ 119862119898119899) = (V

119894119895)119897119899 Then

119906119894119895= oplus119898

119904=1(119901119894119904otimes 119888119904119895)

= oplus119898

119904=1((oplus119897

119905=1(119886119894119905otimes 119887119905119904)) otimes 119888119904119895)

= oplus119898

119904=1oplus119897

119905=1[(119886119894119905otimes 119887119905119904) otimes 119888119904119895]

= oplus119897

119905=1oplus119898

119904=1[119886119894119905otimes (119887119905119904otimes 119888119904119895)]

= oplus119897

119905=1[119886119894119905otimes (oplus119898

119904=1(119887119905119904otimes 119888119904119895))]

= oplus119897

119905=1(119886119894119905otimes 119902119905119895)

= V119894119895

(1)

3 Graphs Weighted in a Dioid andthe Longest Path Problem

For the dioid ([0 +infin]min + +infin 0) in SPP since the partialorder ⪯ is dual to the usual one le the SPP in the dioidsituation comes to be a longest path problem (LPP for short)In this section we will study the LPP for graphs weighted in

The Scientific World Journal 3

a dioid which can be considered a unified approach for SPPWPP and MRPP (and so on)

Let 119866 be a graph weighted in a dioid (119870 oplus otimes 0 1) Fortwo vertices 119894 119895 let 119875

119894119895denote the set of paths from 119894 to 119895 For

119901 isin 119875119894119895119908119901= otimes119890isin119901119908(119890) is called the length of the path119901 Since

oplus is idempotent119901119894119895= oplus119901isin119875119894119895

119908119901is the longest path length from

V119894to V119895 For theweighted graph119866 define amatrix119860 = (119886

119894119895)119899times119899

by the following

(1) for 119894 = 119895 if there are some paths from V119894to V119895 then put

119886119894119895as the maximal weight of all parallel edges from V

119894

to V119895 if there is no path from V

119894to V119895 then put 119886

119894119895= 0

(2) for every 119894 put 119886119894119894= 1

For any 1 le 119894 119895 le 119899 define 119886(1)119894119895

= 119886119894119895and 119886(119898+1)

119894119895=

oplus119899119897=1(119886(119898)

119894119897otimes 119886(1)

119897119895) (119898 = 1 2 3 )

Proposition 8 (1) 119886(119898)119894119895

is the longest119898-step path from V119894to V119895

(2) For any119898 = 0 1 2 then 119886(119898)119894119895

⪯ 119886(119898+1)

119894119895

(3) 119886(119898)119894119895

= 119886(119899)

119894119895for all119898 ge 119899

Proof (1)We use the induction to prove this result For 119896 = 1the result holds Suppose that the result holds for 119896 = 119898 For119896 = 119898 + 1 119886(119898+1)

119894119895= oplus119899119896=1(119886(119898)

119894119896otimes 119886(1)

119896119895) Let 119890

1 1198902 119890

119898 119890119898+1

be an (119898 + 1)-step path from V119894to V119895 Then 119890

1 1198902 119890

119898is an

119898-step path from V119894to V119896 where V

119896is the end vertex of 119890

119898

and 119890119898+1

is an edge from V119896to V119895and 119886(1)

119896119895⪰ 119908(119890

119898+1) Then

119886119898119894119896⪰ 119908(119890

1) otimes 119908(119890

2) otimes sdot sdot sdot otimes 119908(119890

119898) and 119886(119898+1)

119894119895⪰ 119886(119898)

119894119896otimes 119886(1)

119896119895⪰

119908(1198901)otimes119908(119890

2)otimessdot sdot sdototimes119908(119890

119898)otimes119908(119890

119898+1)This completes the proof

(2) 119886(119898+1)119894119895

= oplus119899119897=1(119886(119898)

119894119897otimes119886(1)

119897119895) ⪰ 119886(119898)

119894119895otimes119886(1)

119895119895= 119886(119898)

119894119895otimes1 = 119886

(119898)

119894119895

(3) Suppose that119898 ge 119899 By (2) we have 119886(119898)119894119895

⪰ 119886(119899)

119894119895 By (1)

119886(119898)

119894119895is the longest119898-step path from V

119894to V119895 Suppose that the

related path of 119886(119898)119894119895

is 119901 = V1198971 V1198972 V

119897119898 since there is at

most 119899 vertices in119866 and there are some common points in 119901Suppose that V

119897119894and V119897119895(let 119897119894le 119897119895) are the same point In order

to make 119886(119898)119894119895

the longest path it must hold that V119897= V119897119894= V119897119895

for all 119897119894le 119897 le 119897

119895 Then the length 119886(119898)

119894119895is equal to the length

of a path from V119894to V119895with no common point (with at most

119899 vertices) Hence 119886(119898)119894119895

⪯ 119886(119899)

119894119895since 119886(119899)

119894119895is the longest 119899-step

path from V119894to V119895

We now present two algorithms to compute the longestpath length and the corresponding longest path

Algorithm 9 To find the longest path length from a vertex V119894

to another one V119895

input 119860119894119895and 119894 119895

output the longest path length from V119894to V119895

(1) 119886(1)119894119897

= 119886119894119897(119897 = 1 2 119899) 119886(1)

ℎ119895= 119886ℎ119895 (ℎ =

1 2 119899)

(2) for 119898 = 1 to 119899 do Put 119886(119898+1)119894119895

= oplus119899119897=1(119886(119898)

119894119897otimes 119886(1)

119897119895)

If 119886(119898+1)119894119895

= 119886(119898)

119894119895 then print ldquothe longest path length is

119886(119898)

119894119895rdquo

Algorithm 10 Suppose that the longest path length from V119894to

V119895is 119886(119898)119894119895

To find the related longest path

input 119886(1)119894119897 119886(2)

119894119897 119886

(119898)

119894119897for 119897 isin 1 2 119899

output the longest path from V119894to V119895

(1) 119901 = V119895

(2) for ℎ = 119898 to 1 do Find 119897 isin 1 2 119899 such that119886(ℎ)

119894119895= 119886(ℎminus1)

119894119897otimes 119886(1)

119897119895

(3) 119901 larr V119897 cup 119901

(4) print ldquo119901rdquo

4 Matroids Weighted in a Linear Dioid andthe Maximum Independent Set Problem

For a classical graph 119866 = (119864 119881) letI(119866) = 119868 sube 119864 | 119868 has nocircuit Then (119864I(119866)) is a matroid that is

(I1) 0 isin I(I2) 119860 sube 119861 isin I implies 119860 isin I(I3) for 119860 119861 isin I if |119860| lt |119861| then there exists 119890 isin 119861 minus 119860

such that 119860 cup 119890 isin I

Similar to weighted graph matroids also play an impor-tant role in mathematics especially in applied mathematicswhich are precisely the structures for which the very simpleand efficient greedy algorithm works [5]

In this section wewill studymatroids weighted in a lineardioid (notice that all the examples in Examples 5 and 6 arelinear) and the maximum independent set problem

We suppose that (119870 oplus otimes 1 0) is a linear dioid Let 119864 bea finite set and 119872 = (119864I) a matroid weighted in 119870 with119908 119864 rarr 119870 being the weighted function

In the optimization theory the maximum independentset problem (MISP for short) is to find an independent subset119869 isin I(119872) such that 119908(119869) = max119908(119868) | 119868 isin I(119872) We willuse the famous greedy algorithm to deal with this problem

The greedy algorithm

(1) labeling119864 = 1198901 1198902 119890

119898 such that119908(119890

1) ⪰ 119908(119890

2) ⪰

sdot sdot sdot ⪰ (119890119898)

(2) 119869 = 0(3) for 119894 = 1 to 119899 do if 119869 cup 119890

119894isin I(119872) then 119869 larr 119869 cup 119890

119894

Proposition 11 The greedy algorithm has an optimal solution

Proof Suppose that 119869 is a solution of GA Then 119869 isin I(119872)For any 1198691015840 isin I(119872) we need to show that 119908(119869) ⪰ 119908(1198691015840)Suppose that 119869 = 119890

1198941 1198901198942 119890

119894119896 and 1198691015840 = 119890

1198951 1198901198952 119890

119895119897

with 119908(1198901198941) ⪰ 119908(119890

1198942) ⪰ sdot sdot sdot ⪰ 119908(119890

119894119896) and 119908(119890

1198951) ⪰ 119908(119890

1198952) ⪰

sdot sdot sdot ⪰ 119908(119890119895119897) On one hand by the algorithm we have 119894

119896ge 119895119897

that is |119869| ge |1198691015840| On the other hand if 119908(1198691015840) ⪰ 119908(119869) then


there exits a least integral number 119904 gt 0 such that 119908(119890119895119904) ⪰

119908(119908119894119904) Put 119868 = 119890

1198941 1198901198942 119890

119894119904 and 119883 = 119868 cup 119890

1198951 1198901198952 119890

119895119904

We have 119868 isin I(119872 | 119883) By the minimality of 119904 for any119905 isin 1 2 119904 if 119890

119895119905notin 119868 then 119868 cup 119890

119895119905notin I(119872 | 119883)

Hence 119903(119872 | 119883) = 119904 minus 1 which is contradicting with1198901198951 1198901198952 119890

119895119904 isin I(119872 | 119883) For a summary we have

119908(1198691015840) ⪯ 119908(119869) This completes the proof

Proposition 12 Suppose that 119872 = (119864I) is a matroidand 119870 is a linear dioid Let 119875(119872) = 119909 isin 119870119864 | forall119860 sube

119864 119909(119860) ⪯ 119903(119860) Then the set of maximal points is preciselythe characteristic vectors of all independent sets in119872

Proof Suppose that 1199091015840 is a maximal point of 119875(119872) Thenthere exists a vector 119888 isin 119870119864 such that the following optimalproblem has a unique solutionProblem max119908 sdot 119909 | 119909 isin 119875(119872) where 119908 119864 rarr 119870 is aweighted function

By greedy algorithm the solution of this problem has theform 119909

119869for some independent set 119869 Then 1199091015840 = 119909

119869 On the

other hand if 119869 isin I(119872) then it is easily seen that 119909119869is a

maximal point of 119875(119872)

In [8 9] for a complete lattice 119871 Shi introduced anapproach to fuzzification ofmatroids namely an119871-fuzzifyingmatroid which are successfully characterized by a kindof fuzzy rank functions Consequently the correspondingaxioms of bases and circuits dependent sets and closureoperators are established by which 119871-fuzzifying matroids arealso equivalently characterized [10ndash12]

Of course for a complete lattice 119871 we know that(119871 or and 0 1) is a special dioid So a natural question arisesCan we generalize the truth value table 119871 of Shirsquos 119871-fuzzifyingmatroid to a dioid We here try a first attempt to give apositive answer

Definition 13 Suppose that (119870 oplus otimes 0 1) is a dioid and let 119864be a finite set A map I 2

119864 rarr 119870 is a map satisfying thefollowin

(FI1) I(0) = 1

(FI2) if 119860 sube 119861 thenI(119861) ⪯ I(119860)

(FI3) if |119860| lt |119861| thenI(119860)otimesI(119861) ⪯ oplus119890isin119861minus119860

I(119860cup119890)

The pair (119864I) is called a 119870-fuzzifying matroid

By [12] we know that a graphweighted in the unit interval[0 1] induces a [0 1]-fuzzifying matroid in a natural way Fora dioid119870 we have the similar results

Proposition 14 Suppose that119866 = (119864 119881 119908) is aweighed graphin a dioid (119870 oplus otimes) Define J

119866 2119864 rarr 119870 by I

119866(119860) =

otimes119909isin119860119909 if 119860 is independent and 0 otherwise Then (119864I) is a

119870-fuzzifying matroid

Proof The proof is a routine

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The first author is thankful to the financial support fromNat-ural Science Foundation of Hebei Province (A2014403008)

References

[1] J A Bondy andU S RMurtyGraphTheory vol 244 SpringerNew York NY USA 2008

[2] K K Somasundaram and J S Baras ldquoSolving multi-metricnetwork problems an interplay between idempotent semiringrulesrdquo Linear Algebra and Its Applications vol 435 no 7 pp1494ndash1512 2011

[3] M Gondran and M Minoux Graphs Dioids and SemiringsNew Models and Algorithms Springer 2008

[4] H Whitney ldquoOn the abstract properties of linear dependencerdquoAmerican Journal of Mathematics vol 57 no 3 pp 509ndash5331935

[5] J G Oxley Matroid Theory Oxford Universty Press OxfordUK 1992

[6] Z Q Cao K H Kim and F W Roush Incline Algebra andApplications John Wiley amp Sons New York NY USA 1984

[7] S C HanA study on incline matrices with indices [PhD thesis]Beijing Normal University Beijing China 2005

[8] F Shi ldquoA new approach to the fuzzification of matroidsrdquo FuzzySets and Systems vol 160 no 5 pp 696ndash705 2009

[9] F G Shi ldquo(119871119872)-fuzzy matroidsrdquo Fuzzy Sets and Systems vol160 no 16 pp 2387ndash2400 2009

[10] F-G Shi and L Wang ldquoCharacterizations and applications ofM-fuzzifyingmatroidsrdquo Journal of Intelligent and Fuzzy Systemsvol 25 pp 919ndash930 2013

[11] L Wang and F G Shi ldquoCharacterization of L-fuzzifyingmatroids by L-fuzzifying closure operatorsrdquo Iranian Journal ofFuzzy Systems vol 7 no 1 pp 47ndash58 2010

[12] W Yao and F G Shi ldquoBases axioms and circuits axioms forfuzzifyingmatroidsrdquo Fuzzy Sets and Systems vol 161 no 24 pp3155ndash3165 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


informational systems complex systems modeling decision-making theory dynamical programming control theorynervous system probable reasoning psychological measure-ment and clustering [3 6]

Definition 1 (see [3]) A (22)-type algebra (119870 oplus otimes) is calleda semiring if

(K1) both (119870 oplus) and (119870 otimes) are semigroups

(K2) oplus is commutative that is 119886oplus119887 = 119887oplus119886 for all 119886 119887 isin 119870

(K3) otimes is distributive over oplus that is 119886 otimes (119887 oplus 119888) = (119886 otimes 119887) oplus(119886 otimes 119888) (119887 oplus 119888) otimes 119886 = (119887 otimes 119886) oplus (119888 otimes 119886) for all 119886 119887 119888 isin 119870

We call a semiring (119870 oplus otimes) preunital if there is a specialelement 1 isin 119870 such that (119870 otimes 1) is a monoid that is 1otimes119909 =119909 = 119909 otimes 1 for every 119909 isin 119870

Proposition 2 For every preunital semiring (119870 oplus otimes 1) thefollowing conditions are equivalent

(K4) 1 is absorbing with respect to the operation oplus that is1 oplus 119909 = 1 for every 119909 isin 119870

(K41015840) 119909 oplus (119909 otimes 119910) = 119909 and (119909 otimes 119910) oplus 119910 = 119910 for all 119909 119910 isin 119870

Proof (K4)rArr(K41015840)119909oplus(119909otimes119910) = (119909otimes1)oplus(119909otimes119910) = 119909otimes(1oplus119910) =119909 otimes 1 = 119909 Similarly (119909 otimes 119910) oplus 119910 = 119910

(K41015840)rArr(K4) 1 oplus 119909 = 1 oplus (1 otimes 119909) = 1

A preunital semiringwith condition (K4) is called a unitalsemiring A semiring (119870 oplus otimes) is called idempotent if oplus isidempotent that is 119886 oplus 119886 = 119886 for all 119886 isin 119870 An idempotentsemiring with the condition (K41015840) is called an incline algebra[6]

Proposition 3 Suppose that (119870 oplus otimes) is a unital semiringand 0 is an element in 119870 Then the following conditions areequivalent

(K5) 0 is absorbing with respect to the operation otimes that is0 otimes 119909 = 0 = 119909 otimes 0 for every 119909 isin 119870

(K51015840) (119870 oplus 0) is a monoid that is 0oplus119909 = 119909 for every 119909 isin 119870

Proof (K5)rArr(K51015840) by (K41015840) 0 oplus 119909 = (0 otimes 119909) oplus 119909 = 119909(K51015840)rArr(K5) by (K41015840) 0 otimes 119909 = 0 oplus (0 otimes 119909) = 0

An idempotent and unital semiring (119870 oplus otimes) with condi-tion (K5) is called a dioid that is a dioid is an incline with(K41015840) and (K51015840) which is called a bounded incline algebra[6]

In every dioid (119870 oplus otimes 0 1) we define 119909 ⪯ 119910 iff 119909oplus119910 = 119910Then ⪯ is a partial order on 119870 and (119870 oplus) is a bounded joinsemilattice Clearly 0 is the bottom element and 1 is the topelement so is the name bounded incline

Proposition 4 (see [7]) If 119886 ⪯ 119887 119888 ⪯ 119889 then 119886 oplus 119888 ⪯ 119887 oplus 119889and 119886 otimes 119888 ⪯ 119887 otimes 119889

Proof Since 119886 ⪯ 119887 119888 ⪯ 119889 we have 119886 oplus 119887 = 119887 119888 oplus 119889 = 119889

(1) (119886oplus119888)oplus(119887oplus119888) = (119886oplus119887)oplus(119888oplus119889) = 119887oplus119889 and 119886oplus119888 ⪯ 119887oplus119889(2) (119886otimes119888)oplus(119887otimes119888) = (119886oplus119887)otimes119888 = 119887otimes119888 and then 119886otimes119888 ⪯ 119887otimes119888

Similarly 119887 otimes 119888 ⪯ 119887 otimes 119889 Hence 119886 otimes 119888 ⪯ 119887 otimes 119889

Example 5 (classical examples) (1) ([0 +infin]min + +infin 0)is a dioid which is an algebraic model for SPP The partialorder ⪯ defined above is dual to the usual one le For explicit119886 ⪯ 119887 iff 119887 le 119886 in usual meaning

(2) ([0 +infin]maxmin 0 +infin) is a dioid which is analgebraic model for WPP The partial order ⪯ defined aboveis the same as the usual one le

(3) ([0 1]max times 0 1) is a dioid which is an algebraicmodel for MRPP The partial order ⪯ defined above is thesame as the usual one le

Example 6 (other examples) Consider

(1) ([0 +infin]minmax +infin 0)(2) ([0 1]minmax 1 0)(3) ([0 1]maxmin 0 1)(4) ([1 +infin]min times +infin 1)

Let 119860 be an (119898 times 119899)-matrix and let 119861 be an (119899 times 119897)-matrixover a semiring (119870 oplus otimes) Define 119860 ∘ 119861 = (119901

119894119895)119898times119897

by 119901119894119895=

oplus119899119896=1(119886119894119896otimes 119886119896119895)

Proposition 7 Let 119860119896119897 119861119897119898 119862119898119899

be three matrices Then(119860119896119897∘ 119861119897119898) ∘ 119862119898119899= 119860119896119897∘ (119861119897119898∘ 119862119898119899)

Proof Let 119860119896119897∘ 119861119897119898= (119901119894119895)119896119898 119861119897119898∘ 119862119898119899= (119902119894119895)119897119898

and (119860119896119897∘

119861119897119898) ∘ 119862119898119899= (119906119894119895)119897119899 119860119896119897∘ (119861119897119898∘ 119862119898119899) = (V

119894119895)119897119899 Then

119906119894119895= oplus119898

119904=1(119901119894119904otimes 119888119904119895)

= oplus119898

119904=1((oplus119897

119905=1(119886119894119905otimes 119887119905119904)) otimes 119888119904119895)

= oplus119898

119904=1oplus119897

119905=1[(119886119894119905otimes 119887119905119904) otimes 119888119904119895]

= oplus119897

119905=1oplus119898

119904=1[119886119894119905otimes (119887119905119904otimes 119888119904119895)]

= oplus119897

119905=1[119886119894119905otimes (oplus119898

119904=1(119887119905119904otimes 119888119904119895))]

= oplus119897

119905=1(119886119894119905otimes 119902119905119895)

= V119894119895

(1)

3 Graphs Weighted in a Dioid andthe Longest Path Problem

For the dioid ([0 +infin]min + +infin 0) in SPP since the partialorder ⪯ is dual to the usual one le the SPP in the dioidsituation comes to be a longest path problem (LPP for short)In this section we will study the LPP for graphs weighted in









119894119895)119899times119899

by the following



119894


119894to V119895 then put 119886

119894119895= 0

(2) for every 119894 put 119886119894119894= 1


= 119886119894119895and 119886(119898+1)

119894119895=

oplus119899119897=1(119886(119898)

119894119897otimes 119886(1)

119897119895) (119898 = 1 2 3 )

Proposition 8 (1) 119886(119898)119894119895


(2) For any119898 = 0 1 2 then 119886(119898)119894119895

⪯ 119886(119898+1)

119894119895

(3) 119886(119898)119894119895

= 119886(119899)

119894119895for all119898 ge 119899


119894119895= oplus119899119896=1(119886(119898)

119894119896otimes 119886(1)

119896119895) Let 119890

1 1198902 119890

119898 119890119898+1


1 1198902 119890

119898is an



119898

and 119890119898+1


119896119895⪰ 119908(119890

119898+1) Then

119886119898119894119896⪰ 119908(119890

1) otimes 119908(119890


119898) and 119886(119898+1)

119894119895⪰ 119886(119898)

119894119896otimes 119886(1)

119896119895⪰

119908(1198901)otimes119908(119890


119898)otimes119908(119890


(2) 119886(119898+1)119894119895

= oplus119899119897=1(119886(119898)

119894119897otimes119886(1)

119897119895) ⪰ 119886(119898)

119894119895otimes119886(1)

119895119895= 119886(119898)

119894119895otimes1 = 119886

(119898)

119894119895


⪰ 119886(119899)

119894119895 By (1)

119886(119898)



related path of 119886(119898)119894119895

is 119901 = V1198971 V1198972 V




to make 119886(119898)119894119895


for all 119897119894le 119897 le 119897

119895 Then the length 119886(119898)



119899 vertices) Hence 119886(119898)119894119895

⪯ 119886(119899)

119894119895since 119886(119899)






input 119860119894119895and 119894 119895


(1) 119886(1)119894119897

= 119886119894119897(119897 = 1 2 119899) 119886(1)

ℎ119895= 119886ℎ119895 (ℎ =

1 2 119899)

(2) for 119898 = 1 to 119899 do Put 119886(119898+1)119894119895

= oplus119899119897=1(119886(119898)

119894119897otimes 119886(1)

119897119895)

If 119886(119898+1)119894119895

= 119886(119898)


119886(119898)

119894119895rdquo


V119895is 119886(119898)119894119895


input 119886(1)119894119897 119886(2)

119894119897 119886

(119898)

119894119897for 119897 isin 1 2 119899


(1) 119901 = V119895


119894119895= 119886(ℎminus1)

119894119897otimes 119886(1)

119897119895

(3) 119901 larr V119897 cup 119901











(1) labeling119864 = 1198901 1198902 119890

119898 such that119908(119890

1) ⪰ 119908(119890

2) ⪰

sdot sdot sdot ⪰ (119890119898)

(2) 119869 = 0(3) for 119894 = 1 to 119899 do if 119869 cup 119890


119894



1198941 1198901198942 119890

119894119896 and 1198691015840 = 119890

1198951 1198901198952 119890

119895119897

with 119908(1198901198941) ⪰ 119908(119890

1198942) ⪰ sdot sdot sdot ⪰ 119908(119890

119894119896) and 119908(119890

1198951) ⪰ 119908(119890

1198952) ⪰


119896ge 119895119897




119908(119908119894119904) Put 119868 = 119890

1198941 1198901198942 119890

119894119904 and 119883 = 119868 cup 119890

1198951 1198901198952 119890

119895119904


119895119905notin 119868 then 119868 cup 119890

119895119905notin I(119872 | 119883)









119869 On the







(FI1) I(0) = 1

(FI2) if 119860 sube 119861 thenI(119861) ⪯ I(119860)


I(119860cup119890)




119866 2119864 rarr 119870 by I

119866(119860) =






Acknowledgment


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















119894119895)119899times119899

by the following



119894


119894to V119895 then put 119886

119894119895= 0

(2) for every 119894 put 119886119894119894= 1


= 119886119894119895and 119886(119898+1)

119894119895=

oplus119899119897=1(119886(119898)

119894119897otimes 119886(1)

119897119895) (119898 = 1 2 3 )

Proposition 8 (1) 119886(119898)119894119895


(2) For any119898 = 0 1 2 then 119886(119898)119894119895

⪯ 119886(119898+1)

119894119895

(3) 119886(119898)119894119895

= 119886(119899)

119894119895for all119898 ge 119899


119894119895= oplus119899119896=1(119886(119898)

119894119896otimes 119886(1)

119896119895) Let 119890

1 1198902 119890

119898 119890119898+1


1 1198902 119890

119898is an



119898

and 119890119898+1


119896119895⪰ 119908(119890

119898+1) Then

119886119898119894119896⪰ 119908(119890

1) otimes 119908(119890


119898) and 119886(119898+1)

119894119895⪰ 119886(119898)

119894119896otimes 119886(1)

119896119895⪰

119908(1198901)otimes119908(119890


119898)otimes119908(119890


(2) 119886(119898+1)119894119895

= oplus119899119897=1(119886(119898)

119894119897otimes119886(1)

119897119895) ⪰ 119886(119898)

119894119895otimes119886(1)

119895119895= 119886(119898)

119894119895otimes1 = 119886

(119898)

119894119895


⪰ 119886(119899)

119894119895 By (1)

119886(119898)



related path of 119886(119898)119894119895

is 119901 = V1198971 V1198972 V




to make 119886(119898)119894119895


for all 119897119894le 119897 le 119897

119895 Then the length 119886(119898)



119899 vertices) Hence 119886(119898)119894119895

⪯ 119886(119899)

119894119895since 119886(119899)






input 119860119894119895and 119894 119895


(1) 119886(1)119894119897

= 119886119894119897(119897 = 1 2 119899) 119886(1)

ℎ119895= 119886ℎ119895 (ℎ =

1 2 119899)

(2) for 119898 = 1 to 119899 do Put 119886(119898+1)119894119895

= oplus119899119897=1(119886(119898)

119894119897otimes 119886(1)

119897119895)

If 119886(119898+1)119894119895

= 119886(119898)


119886(119898)

119894119895rdquo


V119895is 119886(119898)119894119895


input 119886(1)119894119897 119886(2)

119894119897 119886

(119898)

119894119897for 119897 isin 1 2 119899


(1) 119901 = V119895


119894119895= 119886(ℎminus1)

119894119897otimes 119886(1)

119897119895

(3) 119901 larr V119897 cup 119901











(1) labeling119864 = 1198901 1198902 119890

119898 such that119908(119890

1) ⪰ 119908(119890

2) ⪰

sdot sdot sdot ⪰ (119890119898)

(2) 119869 = 0(3) for 119894 = 1 to 119899 do if 119869 cup 119890


119894



1198941 1198901198942 119890

119894119896 and 1198691015840 = 119890

1198951 1198901198952 119890

119895119897

with 119908(1198901198941) ⪰ 119908(119890

1198942) ⪰ sdot sdot sdot ⪰ 119908(119890

119894119896) and 119908(119890

1198951) ⪰ 119908(119890

1198952) ⪰


119896ge 119895119897




119908(119908119894119904) Put 119868 = 119890

1198941 1198901198942 119890

119894119904 and 119883 = 119868 cup 119890

1198951 1198901198952 119890

119895119904


119895119905notin 119868 then 119868 cup 119890

119895119905notin I(119872 | 119883)









119869 On the







(FI1) I(0) = 1

(FI2) if 119860 sube 119861 thenI(119861) ⪯ I(119860)


I(119860cup119890)




119866 2119864 rarr 119870 by I

119866(119860) =






Acknowledgment


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119908(119908119894119904) Put 119868 = 119890

1198941 1198901198942 119890

119894119904 and 119883 = 119868 cup 119890

1198951 1198901198952 119890

119895119904


119895119905notin 119868 then 119868 cup 119890

119895119905notin I(119872 | 119883)









119869 On the







(FI1) I(0) = 1

(FI2) if 119860 sube 119861 thenI(119861) ⪯ I(119860)


I(119860cup119890)




119866 2119864 rarr 119870 by I

119866(119860) =






Acknowledgment


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Graphs and Matroids Weighted in a Bounded ...downloads.hindawi.com › journals ›...

Documents

Transcript of Research Article Graphs and Matroids Weighted in a Bounded ...downloads.hindawi.com › journals ›...