combinatorial - Ap p ro xim a tio n Alg o rith m s : C o m...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Appro xim a tio n Alg o rithm s : C o m b ina to ria l  Appro a c he s Lecture 13: March 2
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
The  Ha m ilto nia n Pa th Pro b le m Hamiltonian Path Problem:   G ive n an undire c te d g ra ph, find a  c yc le  vis iting  e ve ry ve rte x e xac tly o nc e . The  Ham ilto nia n Pa th pro b le m  is  NP-c o m ple te . Eulerian Path Problem: G ive n an undire c te d g ra ph, find a  walk vis iting  e ve ry e dg e  e xa c tly o nc e . No tic e  tha t in a  walk s o m e  ve rtic e s  m ay ha ve  be e n vis ite d m o re  than o nc e . The  Eule ria n Pa th pro b le m  is  po lyno m ia l tim e  s o lva ble . A g ra ph ha s  an Eule rian pa th if a nd o nly if e ve ry ve rte x has  a n e ve n de g re e .
Background image of page 2
The  Ha m ilto nia n Pa th Pro b le m Application: Seating assignment problem: G ive n n pe rs o ns  a nd a big  ro und ta ble  o f n s e ats , e a c h pa ir o f pe rs o ns  m a y  like  e ac h o the r o r hate  e a c h o the r.  C an we  find a s e a ting  a s s ig nm e nt s o  that  e ve ryo ne  like s  his /he r ne ig hbo rs  (o nly two  ne ig hbo rs )? C o ns truc t a  g ra ph a s  fo llo ws .  Fo r e ac h pe rs o n, c re a te  a  ve rte x.  Fo r e a c h  pa ir o f ve rtic e s , add an e dg e  if and o nly if the y like  e a c h o the r.  The n the   pro b le m  o f finding  a  s e ating  a s s ig nm e nt is  re duc e d to  the  Ham ilto nian pa th  pro b le m .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
The  Tra ve ling  S a le s m a n Pro b le m Traveling Salesman Problem (TSP): G ive n a c o m ple te  g ra ph with no nne g a tive  e dg e  c o s ts , Find a m inim um  c o s t c yc le  vis iting  e ve ry ve rte x e xa c tly o nc e . G ive n a  num be r o f c itie s  a nd the  c o s ts  o f trave ling  fro m  a ny c ity to   any o the r c ity, wha t is  the  c he ape s t ro und-trip ro ute  tha t vis its   e a c h c ity e xa c tly o nc e  a nd the n re turns  to  the  s ta rting  c ity?   O ne  o f the  m o s t we ll-s tudie d pro b le m  in c o m binato rial o ptim iza tio n.
Background image of page 4
NP-completeness of Traveling Salesman Problem Hamiltonian Path Problem:   G ive n an undire c te d g ra ph, find a  c yc le  vis iting  e ve ry ve rte x e xac tly o nc e .  For each edge, we add an edge of cost 1.  For each non-edge, we add an edge of cost 2. Traveling Salesman Problem (TSP): G ive n a c o m ple te  g ra ph with no nne g a tive  e dg e  c o s ts , Find a m inim um  c o s t c yc le  vis iting  e ve ry ve rte x e xa c tly o nc e . The  Ha m ilto nia n path pro ble m  is  a  s pe c ial c a s e  o f the  Trave ling  S a le s m an Pro b le m .  If there is a Hamiltonian path, then there is a cycle of cost n.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 37

combinatorial - Ap p ro xim a tio n Alg o rith m s : C o m...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online