From Valid Inequalities To Heuristics : A unified view of primaldual approximation algorithms in cov

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From Valid Inequalities To Heuristics A
unified view of primal-dual approximation
algorithms in covering problems

• Dimitris Bertsimas
• Chung-Piaw Teo
• Presented by Alon Alapi Erez Engel

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Lecture Overview

• Introduction
• Primal Dual Algorithm (PD)
• The notion of Strength (?)
• Theorem 1
• Shortest path example
• Conclusion and open problems

3
Introduction

• In the last 20 years, two approaches to discrete
  optimization have emerged Polyhedral
  combinatorics and Approximation Algorithms.
• The success of the first approach critically
  depends on the choice of valid inequalities, but
  it is not clear which class of valid inequalities
  is better at particular instance.
• In the second approach, the quality of solutions
  produced is judged by the worst-case criterion,
  for which the motivation is designing algorithms
  for problems that are robust, i.e. work well for
  all inputs.

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Introduction (cont.)

• In recent years it has been recognized that tight
  LP relaxations can often be used as basis for
  analyzing for heuristics for NP hard problems.
• The Primal-Dual method has been successfully
  applied to analyze a variety of exact and
  approximation algorithms.
• These analyses, how ever, appear to be problem
  specific. Moreover, the analyses do not usually
  offer insight into the design of such algorithms.
  

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Motivation

• This paper is motivated by the authors desire to
  find an algorithmic technique to design
  approximation algorithms when better formulations
  are available.
• It presents a generic frame work to design
  approximation algorithms for covering-type
  problems.

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Lets Recall

• An LP relaxation is a ? approximation if
• Z IZ ?Z
• A proposed heuristic H that produces a solution
  with value is ? approximation algorithm
  if

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??????? ?? ?????

• ???? ????? ????? ?????? ????? ????????? ?????,
  ??????? ??? ????????? ????? ?? ????? ??????
  ????'.
• ???? ?? ???? ?? ????? ????? ????? ??? ??? ?????
  (?) ??? ??-????????? ??? ???? ????? ????????
  ??????????.
• ???? ?? ???? ????? ???' ?????? ? , ???? ?? ???
  ????? ?? ?????????? ?????? (?????????).
• ????? ?????' ???????? ?? ??? ?????? ?????? ????
  ?????? ??? ????' ???????, ?????? ???? ?????.

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Primal-Dual Framework For Covering Problems

• The problem

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Primal-Dual Framework For Covering Problems
(Cont.)

• ?????? ?????? ????? ????? ????????
• Primal-Dual Algorithm , ?????? (PD).
• ??? ??????? ?? ????????? ???? ?? ??????? ???,
  ????? ???? ???? ??????? ???????? ???? ?-1.
• ?????? ???? ?? ????, ????? ????? ?????? ??????
  ????????.
• ???? ????? ????? ?? ????? ????????, ?????? ?????
  ????? ?? ??? ?????? ??????? ??? ??????? .

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PD Algorithm

• Step 1. Initialization
• Step 2. Addition of valid inequalities construct
  a valid inequality Set

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PD Algorithm (Cont.)

• Step 3. Problem modification
• Repeat step 2 and 3 until solution is feasible.
• Else conclude that solution is infeasible.

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PD Algorithm (Cont.)

• Step 4. Reverse deletion
• For r from
  t-1 to 1 do
• Corresponds to a minimal feasible solution to
  problem instance .
• Step 5 Return
  .

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Vertex Cover example

• Step 1. Initialization

9
8
X2
X1
X3
X4
2
7
14
Vertex Cover example (Cont.)

• Step 2. Addition of valid inequalities
• K(1)3, is the chosen variable index. Vertex
  X3 is part of the solution vector.

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Vertex Cover example (Cont.)

• Step 3. Problem modification

0
5
7
1
The new problem
7
8
X2
X1
X3
X4
5
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Vertex Cover example (Cont.)

• Step 2. Addition of valid inequalities

K(2)1, the chosen variable index. Vertex X1 is
part of the solution vector.
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Vertex Cover example (Cont.)

• Step 3. Problem modification

X1
1
X3

• Step 45. Reverse deletion
• Can we do without X1 ? NO!
• Can we do without X3 ? NO!

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??????? ???????? ???????? ?????????

• ???? ??? ??????? ???????? ???????, ???????? ???
  ???????, ?????? ?? ??????? ????????? ?????
  ????'.
• ???????? ?????? ????? ???? ????? ???????, ?? ??
  ???? ????? ???????. ????? ?????? ?????? ???'
  ????? ??? ??????? ??????? ???.
• ????' ???? ?????? ????? ???????? ????? ???????.
  ?????? ??? ????? ??? ??? ?? ????? ???????, ???
  ????? ?? ????? ???? ?? ?? ?? ???????? ??????.
• ?????? ???????? ????? ???? ??? ??????? ??????
  ???? (???? ???? ??????? ????????). ???, ???????
  ??????? ???? ?? ?? (??? ??? ???? yr). ????? ??
  ???? ?????? ??? ???????, ?????? ?? ????? ???? ??
  yr , ??????? ???????? ???????? ????? ????' ??????
  ????? ???????? ????? ???????.

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The Notion Of STRENGTH

• The strength ?r of the inequality
• Is defined to be
• wi - ??????? ???????? ???? ?? ?????????
  ???????? ??
• ???????? ?????? ???? ? r .
• ??? ??
  ????????? ????? ?? ??????
  ?????.

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Theorem 1

• In particular
• (a)
• (b) If all inequalities are redundant
• Inequalities for then

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Theorem 1 - Proof

• ???? - ?????? ????? ?????? ?- r ??
  ?????????.
• - ?????? ??????? ???????? ?????
  ?????? ?- r.
• ???? ????? ????? ?????????? ??
• For every rt to 1

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Theorem 1 Proof (cont.)

• ?) ???? ?????????? (rt)
• 
  - ??? ????? ?

?????? ??? ????? ????? ?? ?????? ?- t ?? ????'.
Xt1 ?????? ?- t
?) ???? ?????????? ????? (1) ??????? ???? ?)
????? ???? Kr
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Theorem 1 Proof (cont.)
???? ?????? ?????? ??????? ????? ?????? ?-r

• ?????? ? (?????? ?- r1
  ????? ?? ?????? ????? ?????? ????? ??? ?????).
• ???? ?????
• ????? ??????????
• ??????? ? ???? ??

???? ??????????
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Theorem 1 Proof (cont.)

• ???, ?-(1) ????? ? ??? ????? ????????
  ????? ???????
• ????? ?????? ?????? ????' ???? ?? ??? ? ????
  ?????? ????.
• If, in addition, all the inequalities
• are redundant to then

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????? ????????

• ????? ????? ????? ????? ???????? ?????? ?, ??
  ???? ????? ??? ??? ??-?????? ??????? (strength)
  ????? ??-??? ?.
• ??? ????? ?? ????????? ???? ?? ?? ???? ?????
• ?) ????? ??-?????? ???.
• ?) ????? ?? ??????? ????? ??? ??? ????? ??
  ?????? ????????, ?????? ????? ??????.
• ?) ????? ????? ??????? ??? ??? ?? ?????????.
• ?????? ????? ???????? ????? ????????? ????
  ?????? ?"? ???? ???????, ????????? ???? ????
  ???????????, ?? ????? ?????? ???? ?????? ????
  ???????????.

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The Shortest Path Problem

• The problem of finding the shortest path from s
  to t in an, undirected graph G, can be modeled as
  an edge-covering formulation.

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The Shortest Path Problem (cont.)

• Formulation is reducible.
• Theorem 4. Inequalities

(forward Dijkstra)
(backward Dijkstra)
and
and
(bidirectional Dijkstra)
Have strength 1, i.e.,
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Shortest Path example

• Input Graph G
• Output Shortest path
• Step 1. Initialization

F1
x37
x59
x11
C1
t
s
X1
x612
x23
x44
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Shortest Path example (cont.)

• Step 2. Addition of valid inequalities

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Shortest Path example (cont.)

• Step 3. Problem modification

F2
C2
x37
X2
x37
x59
x59
x11
t
t
s
s
x612
x612
x23
x22
x44
x44
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Shortest Path example (cont.)

• Round 2

x35
x37
x59
x59
F3
t
t
s
s
C3
x612
x22
x612
x44
X3
x44
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Shortest Path example (cont.)
Rolling rolling.
x35
x31
x59
x59
F3
t
t
s
s
C3
x612
x612
x44
X3
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Shortest Path example (cont.)

• And then again

F4
x59
x31
x59
C4
t
t
s
s
x612
X4
x611
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Shortest Path example (cont.)

• ????? ????

X5
Is it over????????????? No!
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Shortest Path example (cont.)

• Step 4. Reverse Deletion

• ??? ???? ??? X5? ??
• ??? ???? ??? X3? ??
• ??? ???? ??? X4? ??
• ??? ???? ??? X2? ??
• ??? ???? ??? X1? ??
• ??? ????? ??????? (???? ??
• ????????) ????? ??????? ???
• 17917
• ????? ?? ??????? ????????
• 1241517

x37
x59
x11
t
s
x23
x44
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?????

• ?????? ??????? ????? ?- strength ?? ??-?????????
  ?????, ?????? ???' ????? ???? ??? ???? ?? ?????
  ?????.
• ?????? ?? ???? ?? ??? ??? ??????? ?? ?????
  ????????? ??????, ???? ??? ?- strength ??
  ??-???????? ???? ????? ???? ?????? ???? ???.
• ????? ?? ?????? ??? ?????? ?? ????? Primal-Dual
  ??????? ??? ?? ???????? ????? ????, ??????? ??
  ????? ??????? ?? ????? ???? ??????? ?????????.

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????? ??????

• ?????? ????? ????? ?????? ?????? ??????, ??? ????
  ????? ?? ??-???????? ????? ?????? ???? ??????
• ????? ??? ?????? ????? ?????? ????? ??????
  ??????? ???? ??-??????????
• ?????? ????? ?????, ???? ????? ?? ???? ??????
  ????? ???????? ????? ?? ??? ????? ????? ??
  ??-??????? ??? ??? ??? ?? ????'. ??? ???? ?????
  ???? ????? ?? ?????-?????? ?????? ????? ??????
  ???? ???? ?? ??-??????? ??? ??? ??? ?? ????'?

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The End