Reoptimization Algorithms and Persistent Turing Machines

Reoptimization Algorithms and Persistent Machines
Jhoirene B Clemente
December 2, 2014
Algorithms and Complexity Lab
Department of Computer Science
University of the Philippines Diliman

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
2 Optimization Problem
Approximation Algorithm
Reoptimization
Model
SuperTuring Computer
Interaction Machines
Persistent Turing Machine
Dept Computer Science
University of the Philippines
Diliman
Recall: Combinatorial Optimization Problem
Definition (Combinatorial Optimization Problem
[Papadimitriou and Steiglitz, 1998])
An optimization problem = (D,R, cost, goal) consists of
1. A set of valid instances D. Let I 2 D, denote an input
instance.
2. Each I 2 D has a set of feasible solutions, R(I ).
3. Objective function, cost, that assigns a nonnegative
rational number to each pair (I , SOL), where I is an instance
and SOL is a feasible solution to I.
4. Either minimization or maximization problem:
goal 2 {min, max}.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Optimization Problem
3 Approximation Algorithm
Reoptimization
Model
Diliman
Approximation Algorithms
Definition (Recall: Approximation Algorithm
[Williamson and Shmoy, 2010] )
An -approximation algorithm for an optimization problem is a
polynomial-time algorithm that for all instances of the problem
produces a solution whose value is within a factor of of the
value of an optimal solution.
Given an problem instance I with an optimal solution Opt(I ), i.e.
the cost function cost(Opt(I )) is minimum/maximum.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
I An algorithm for a minimization problem is called
-approximative algorithm for some 1, if the algorithm
obtains a maximum cost of · cost(Opt(I )), for any input
instance I .

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
I An algorithm for a minimization problem is called
-approximative algorithm for some 1, if the algorithm
obtains a maximum cost of · cost(Opt(I )), for any input
instance I .
I An algorithm for a maximization problem is called
-approximative algorithm, for some 1, if the algorithm
obtains a minimum cost of · cost(Opt(I )), for any input
instance I .

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
4 Reoptimization
Model
Diliman
Reoptimization
Don’t start from scratch when confronted with a
problem, but try to make good use of prior knowledge
about similar problem instances whenever they are
available.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
4 Reoptimization
Model
Diliman
Reoptimization
available.
Definition (Reoptimization (Bockenhauer, 2008) )
Given a problem instance and an optimal solution for it, we are to
efficiently obtain an optimal solution for a locally modified
instance of the problem.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
4 Reoptimization
Model
Diliman
Reoptimization
available.
Definition (Reoptimization (Bockenhauer, 2008) )
Given a problem instance and an optimal solution for it, we are to
efficiently obtain an optimal solution for a locally modified
instance of the problem.
INPUT: I , SOL, I 0, where (I , I 0) 2M
OUTPUT: SOL0

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
5 Reoptimization
Model
Diliman
Reoptimization
Definition (Reoptimization [Zych, 2012])
Let = (D,R, cost, goal) be an optimization problem and
M D × D be a binary relation (the modification). The
corresponding reoptimization problem
RM() = (DRM(),RRM(), costRM(), goalRM())
consists of

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
5 Reoptimization
Model
Diliman
Reoptimization
consists of
1. a set of feasible instances defined as
DRM() = {(I , I 0, SOL) : (I , I 0) 2M and SOL 2 R(I )};
we refer to I as the original instance and to I 0 as the
modified instance

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
5 Reoptimization
Model
Diliman
Reoptimization
consists of
1. a set of feasible instances defined as
DRM() = {(I , I 0, SOL) : (I , I 0) 2M and SOL 2 R(I )};
we refer to I as the original instance and to I 0 as the
modified instance
2. a feasibility relation defined as
RRM()((I , I 0, SOL)) = R(I 0)
A solution to a reoptimization variant of the problem is

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
6 Reoptimization
Model
Diliman
Reoptimization
1. Reoptimization can help in providing efficient algorithms to
problems involved in dynamic systems
2. Reoptimization can help in providing a better solution or an
efficient algorithm for computationally hard problems

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
6 Reoptimization
Model
Diliman
Reoptimization
1. Reoptimization can help in providing efficient algorithms to
problems involved in dynamic systems
I Shortest Path Problem [Pallottino and Scutella, 2003]
[Nardelli et al., 2003]
I Dynamic Minimum Spanning Tree [Thorup, 2000] with edge
weights [Ribeiro and Toso, 2007] [Cattaneo et al., 2010]
I Vehicle Routing Problem [Secomandi and Margot, 2009]
I Facility Location Problem [Shachnai et al., 2012]
2. Reoptimization can help in providing a better solution or an
efficient algorithm for computationally hard problems
I Traveling salesman problem [Královic and Mömke, 2007]
[Hans-joachim Böckenhauer, 2008] [Ausiello et al., 2011]
I Steiner tree problem [Hromkovic, 2009]
[?][Bilo and Zych, 2012] [Böckenhauer et al., 2012]
I Shortest common superstring [Bilo,2011] [Popov, 2013]
I Hereditary problems on Graphs [Boria et al., 2012]
I Scheduling Problem [Boria et al., 2012]
I Pattern Matching [Yue and Tang, 2008]
[Clemente et al., 2014]

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
7 Reoptimization
Model
Diliman
Metric TSP: with additional information

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
8 Reoptimization
Model
Diliman

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
9 Reoptimization
Model
Diliman

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
10 Reoptimization
Model
Diliman
Metric TSP: Nearest Insert
[Ausiello et al., 2009]

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
11 Reoptimization
Model
Diliman

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
12 Reoptimization
Model
Diliman

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
13 Reoptimization
Model
Diliman
10 + 12 + 6 + 4 + 5 + 11 = 48

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
14 Reoptimization
Model
Diliman
Example: Reoptimization for Metric TSP
Definition (Metric TSP)
INPUT: In+1
OUTPUT: Hn+1

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
14 Reoptimization
Model
Diliman
INPUT: In+1
OUTPUT: Hn+1
Theorem
There is a 3/2-Approximation algorithm for solving Metric TSP.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
14 Reoptimization
Model
Diliman
INPUT: In+1
OUTPUT: Hn+1
Theorem
Definition (Reoptimization Metric TSP)
INPUT: In,H
n , In+1
OUTPUT: Hn+1

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
14 Reoptimization
Model
Diliman
INPUT: In+1
OUTPUT: Hn+1
Theorem
Definition (Reoptimization Metric TSP)
INPUT: In,H
n , In+1
OUTPUT: Hn+1
Reoptimization can still improve the approximation
ratio.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
15 Reoptimization
Model
Diliman
Reoptimization Solution for Metric TSP
Definition (Metric TSP with additional info)
INPUT: In,H
n , In+1
OUTPUT: Hn+1
OUTPUT Output the best solution between (H1,H2), where
H1 =Nearest Insert(In,H
n , In+1)
H2 = 3
2-Approximation Algorithm(In+1)
Algorithm 1: 4/3 -Approximation Algorithm for Metric TSP

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
15 Reoptimization
Model
Diliman
Reoptimization Solution for Metric TSP
Definition (Metric TSP with additional info)
INPUT: In,H
n , In+1
OUTPUT: Hn+1
OUTPUT Output the best solution between (H1,H2), where
H1 =Nearest Insert(In,H
n , In+1)
H2 = 3
2-Approximation Algorithm(In+1)
Algorithm 1: 4/3 -Approximation Algorithm for Metric TSP
Proof:
n+1) + 2d(v, n + 1) (Triangle Inequality)
c(H1) c(H
c(H1) c(H
n+1) + 2max(d(i, n + 1), d(j, n + 1)))
c(H2) 3/2c(H
n+1) − max(d(i, n + 1), d(j, n + 1)))
min(c(H1), c(H2)) (1/3)c(H1) + (2/3)c(H2) 4/3c(H
n+1)

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
16 Model
Diliman
Another Model for Combinatorial Reoptimiza-tion
[Shachnai et al., 2012]
Given an optimization problem , let I0 be an input for , and let
I0 ,C2
CI0 = {C1
I0 ,C3
I0 , . . . .}
be the set of configurations corresponding to the solution space of
for I0.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
16 Model
Diliman
I0 ,C2
CI0 = {C1
I0 ,C3
I0 , . . . .}
for I0.
In R(), we are given Cj
I0 2 CI0 of an initial instance I0, and a
new instance I obtained from I0

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
16 Model
Diliman
I0 ,C2
CI0 = {C1
I0 ,C3
I0 , . . . .}
for I0.
For any i 2 I and configuration Ck
I , let be the transition cost.
(i,Cj
i0 ,Ck
I )

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
16 Model
Diliman
I0 ,C2
CI0 = {C1
I0 ,C3
I0 , . . . .}
for I0.
For any i 2 I and configuration Ck
I , let be the transition cost.
(i,Cj
i0 ,Ck
I )
The goal of reoptimization is to find C
I with an optimal
I ) and transition cost
cost(C
(i,Cj
I )
I0 ,C

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
17 Model
Diliman
Reoptimization from an Online Environment
I = {I0, I1, I2, I3, . . . , It}
SOL = {SOL0, SOL1, SOL2, SOL3, . . . , SOLt}

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
18 Model
Diliman
Online Algorithm
I Many problems such as routing, scheduling, or the paging
problem work in so called online environments and their
algorithmic formulation and analysis demand a model in
which an algorithm deals with such a problem knows only a
part of its input at any specific point during runtime.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
18 Model
Diliman
Online Algorithm
I These problems are called online problems and the respective
algorithms are called online algorithms.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
18 Model
Diliman
Online Algorithm
I These problems are called online problems and the respective
algorithms are called online algorithms.
I An online algorithm A has to make decisions at any time
step i without knowing what the next chunk of input at time
step i + 1 will be. Algorithm A has to produce part of the
final output in every step, it cannot revoke decisions it has
already made.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
19 SuperTuring Computer
Diliman
Limitations of TM
The TM is too weak to describe properly the Internet,
evolution or robotics, because it is a closed model,
which requires that all inputs are given in advance, and
TM is allowed to use an unbounded but only finite
amount of time or memory resources [(Eberbach, 2003),
(Wegner, 2003)].

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
Limitations of TM
(Wegner, 2003)].
In Reoptimization,
1. We expect changes in the environment.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
Limitations of TM
(Wegner, 2003)].
In Reoptimization,
2. We assume that the initial solution (SOL0) is obtained from
the environment.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
Limitations of TM
(Wegner, 2003)].
In Reoptimization,
the environment.
3. We make use of previous configurations in solving new input
instances.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
Limitations of TM
(Wegner, 2003)].
In Reoptimization,
the environment.
3. We make use of previous configurations in solving new input
instances.
Interaction Machines allow inputs to be generated
dynamically and require inputs to be represented by a
potentially infinite stream. [Eberbach,Wegner, 2003]

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
20 Persistent Turing Machine
Diliman
Persistent Turing Machines
The canonical model of interaction machines
I minimal extension of Turing Machines (TMs) that express
interactive behavior.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
I reactive system

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
I reactive system
I multitape machine with a persistent worktape preserved
between interactions

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
I reactive system
I multitape machine with a persistent worktape preserved
between interactions
I inputs and outputs are dynamically generated streams of
strings.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
PTM: Definition
I PTM states are not to be confused with TM states. Unlike
for TMs,the set of PTM states is infinite, represented by
strings of unbounded length.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
PTM: Definition
I PTM states are not to be confused with TM states. Unlike
for TMs,the set of PTM states is infinite, represented by
strings of unbounded length.
I Since the worktape (state) at the beginning of a PTM
computation step is not always the same, the output of a
PTM M at the end of the computation step depends both on
the input and on the worktape.
fM : I ×W ! O ×W

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
PTM: Example

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
PTM: Computation
I Input streams are generated by the environment.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
PTM: Computation
I Input streams are generated by the environment.
I The streams have dynamic evaluation semantics, where the
next value is not generated until the previous one is
consumed.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
PTM: Interaction

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
Thank you for listening.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
References
Ausiello, G., Bonifaci, V., and Escoffier, B. (2011).
Complexity and approximation in reoptimization.
In Computability in Context: Computation and Logic in the Real
World, volume 2, pages 101–129.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
References (cont.)
Böckenhauer, H.-J., Freiermuth, K., Hromkovic, J., Mömke, T.,
Sprock, A., and Steffen, B. (2012).
Steiner tree reoptimization in graphs with sharpened triangle
inequality.
Journal of Discrete Algorithms, 11:73–86.
Boria, N., Monnot, J., and Paschos, V. T. (2012).
Reoptimization of the Maximum Weighted Pk-Free Subgraph
Problem under Vertex Insertion.
pages 76–87.
Cattaneo, G., Faruolo, P., Petrillo, U. F., and Italiano, G.
(2010).
Maintaining dynamic minimum spanning trees: An experimental
study.
Discrete Applied Mathematics, 158(5):404–425.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
References (cont.)
Clemente, J., Aborot, J., and Adorna, H. (2014).
Reoptimization of Motif Finding Problem.
Proceedings of the International MultiConference of Engineers
and Computer Scientists, I.
Hans-joachim Böckenhauer, J. H. T. M. P. W. (2008).
On the hardness of reoptimization.
In Proc. of the 34th International Conference on Current Trends
in Theory and Practice of Computer Science (SOFSEM 2008),
LNCS, 4910.
Hromkovic, J. (2009).
Algorithmic adventures: from knowledge to magic.
Královic, R. and Mömke, T. (2007).
Approximation Hardness of the Traveling Salesman
Reoptimization Problem.
MEMICS 2007, 293.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
References (cont.)
Nardelli, E., Proietti, G., and Widmayer, P. (2003).
Swapping a Failing Edge of a Single Source Shortest Paths Tree
Is Good and Fast.
Algorithmica, pages 56–74.
Pallottino, S. and Scutella, M. (2003).
A new algorithm for reoptimizing shortest paths when the arc
costs change.
Operations Research Letters.
Papadimitriou, C. and Steiglitz, K. (1998).
Combinatorial optimization: algorithms and complexity.
Popov, V. (2013).
On Reoptimization of the Shortest Common Superstring
Problem.
Applied Mathematical Sciences, 7(24):1195–1197.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
References (cont.)
Ribeiro, C. and Toso, R. (2007).
Experimental analysis of algorithms for updating minimum
spanning trees on graphs subject to changes on edge weights.
Experimental Algorithms.
Secomandi, N. and Margot, F. (2009).
Reoptimization approaches for the vehicle-routing problem with
stochastic demands.
Operations Research, 57:1–11.
Shachnai, H., Tamir, G., and Tamir, T. (2012).
A Theory and Algorithms for Combinatorial Reoptimization.
Lecture Notes in Computer Science, 7256(1574):618–630.
Thorup, M. (2000).
Dynamic Graph Algorithms with Applications.
Proceedings of the 7th Scandinavian Workshop on Algorithm
Theory, pages 1–9.

33
Reoptimization
Algorithms and
Persistent Machines
JB Clemente
Introduction
Reoptimization
Model
Diliman
References (cont.)
Williamson, D. and Shmoy, D. (2010).
The Design of Approximation Algorithms.
Yue, F. and Tang, J. (2008).
A new approach for tree alignment based on local
re-optimization.
In International Conference on BioMedical Engineering and
Informatics, BMEI 2008.
Zych, A. (2012).
Reoptimization of NP-hard Problems.
PhD thesis, ETH Zurich.

Reoptimization Algorithms and Persistent Turing Machines

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