Reoptimization Algorithms and Persistent Turing Machines

Reoptimization Algorithms and Persistent Machines

Jhoirene B Clemente

December 2, 2014Algorithms and Complexity LabDepartment of Computer Science

University of the Philippines Diliman

33

ReoptimizationAlgorithms and

Persistent Machines

JB Clemente

Introduction2 Optimization Problem

Approximation Algorithm

ReoptimizationModel

SuperTuring Computer

Interaction MachinesPersistent Turing Machine

Dept Computer ScienceUniversity of the Philippines

Diliman

Recall: Combinatorial Optimization Problem

Definition (Combinatorial Optimization Problem[Papadimitriou and Steiglitz, 1998])An optimization problem Π = (DΠ,RΠ, costΠ, goalΠ) consists of1. A set of valid instances DΠ. Let I ∈ DΠ, denote an input

instance.2. Each I ∈ DΠ has a set of feasible solutions, RΠ(I ).3. Objective function, costΠ, that assigns a nonnegative

rational number to each pair (I ,SOL), where I is an instanceand SOL is a feasible solution to I.

4. Either minimization or maximization problem:goalΠ ∈ {min,max}.

33


Persistent Machines

JB Clemente

IntroductionOptimization Problem

3 Approximation Algorithm

ReoptimizationModel




Diliman

Approximation Algorithms

Definition (Recall: Approximation Algorithm[Williamson and Shmoy, 2010] )An ρ-approximation algorithm for an optimization problem is apolynomial-time algorithm that for all instances of the problemproduces a solution whose value is within a factor of ρ of thevalue 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.

I An algorithm for a minimization problem is calledρ-approximative algorithm for some ρ > 1, if the algorithmobtains a maximum cost of ρ · cost(Opt(I )), for any inputinstance I .

I An algorithm for a maximization problem is calledρ-approximative algorithm, for some ρ < 1, if the algorithmobtains a minimum cost of ρ · cost(Opt(I )), for any inputinstance I .

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman





33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman





33


Persistent Machines

JB Clemente



4 ReoptimizationModel




Diliman

Reoptimization

Don’t start from scratch when confronted with aproblem, but try to make good use of prior knowledgeabout similar problem instances whenever they areavailable.

Definition (Reoptimization (Bockenhauer, 2008) )Given a problem instance and an optimal solution for it, we are toefficiently obtain an optimal solution for a locally modifiedinstance of the problem.

INPUT: I , SOL, I ′, where (I , I ′) ∈MOUTPUT: SOL′

33


Persistent Machines

JB Clemente







Diliman

Reoptimization




33


Persistent Machines

JB Clemente







Diliman

Reoptimization




33


Persistent Machines

JB Clemente







Diliman

Reoptimization




33


Persistent Machines

JB Clemente







Diliman

Reoptimization

Definition (Reoptimization [Zych, 2012])Let Π = (DΠ,RΠ, cost, goal) be an optimization problem andM⊆ DΠ ×DΠ be a binary relation (the modification). Thecorresponding reoptimization problem

RM(Π) = (DRM(Π),RRM(Π), costRM(Π), goalRM(Π))

consists of

1. a set of feasible instances defined as

DRM(Π) = {(I , I ′,SOL) : (I , I ′) ∈M and SOL ∈ RΠ(I )};

we refer to I as the original instance and to I ′ as themodified instance

2. a feasibility relation defined as

RRM(Π)((I , I ′,SOL)) = RΠ(I ′)

A solution to a reoptimization variant of the problem isdenoted by SOL′.

33


Persistent Machines

JB Clemente







Diliman

Reoptimization



consists of1. a set of feasible instances defined as




RRM(Π)((I , I ′,SOL)) = RΠ(I ′)


33


Persistent Machines

JB Clemente







Diliman

Reoptimization



consists of1. a set of feasible instances defined as




RRM(Π)((I , I ′,SOL)) = RΠ(I ′)


33


Persistent Machines

JB Clemente







Diliman

Reoptimization

1. Reoptimization can help in providing efficient algorithms toproblems involved in dynamic systems

I Shortest Path Problem [Pallottino and Scutella, 2003][Nardelli et al., 2003]

I Dynamic Minimum Spanning Tree [Thorup, 2000] with edgeweights [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 anefficient 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 [Hromkovič, 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


Persistent Machines

JB Clemente







Diliman

Reoptimization

1. Reoptimization can help in providing efficient algorithms toproblems involved in dynamic systems

I Shortest Path Problem [Pallottino and Scutella, 2003][Nardelli et al., 2003]

I Dynamic Minimum Spanning Tree [Thorup, 2000] with edgeweights [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 anefficient 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 [Hromkovič, 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


Persistent Machines

JB Clemente







Diliman

Metric TSP: with additional information

33


Persistent Machines

JB Clemente







Diliman


33


Persistent Machines

JB Clemente







Diliman


33


Persistent Machines

JB Clemente







Diliman

Metric TSP: Nearest Insert[Ausiello et al., 2009]

33


Persistent Machines

JB Clemente







Diliman


33


Persistent Machines

JB Clemente







Diliman


33


Persistent Machines

JB Clemente







Diliman


10 + 12 + 6 + 4 + 5 + 11 = 48

33


Persistent Machines

JB Clemente







Diliman

Example: Reoptimization for Metric TSP

Definition (Metric TSP)INPUT: In+1OUTPUT: Hn+1

TheoremThere is a 3/2-Approximation algorithm for solving Metric TSP.

Definition (Reoptimization Metric TSP)INPUT: In,H ∗n , In+1OUTPUT: Hn+1

Reoptimization can still improve the approximationratio.

33


Persistent Machines

JB Clemente







Diliman






33


Persistent Machines

JB Clemente







Diliman






33


Persistent Machines

JB Clemente







Diliman






33


Persistent Machines

JB Clemente







Diliman

Reoptimization Solution for Metric TSP

Definition (Metric TSP with additional info)INPUT: In,H ∗n , In+1OUTPUT: Hn+1

OUTPUT Output the best solution between (H1,H2), whereH1 =Nearest Insert(In,H ∗n , In+1)H2 = 3

2 -Approximation Algorithm(In+1)Algorithm 1: 4/3 -Approximation Algorithm for Metric TSP

Proof:c(H1) ≤ c(H ∗n+1) + 2d(v∗,n + 1) (Triangle Inequality)

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


Persistent Machines

JB Clemente







Diliman




2 -Approximation Algorithm(In+1)Algorithm 1: 4/3 -Approximation Algorithm for Metric TSP

Proof:c(H1) ≤ c(H ∗n+1) + 2d(v∗,n + 1) (Triangle Inequality)

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


Persistent Machines

JB Clemente







Diliman




2 -Approximation Algorithm(In+1)Algorithm 1: 4/3 -Approximation Algorithm for Metric TSPProof:

c(H1) ≤ c(H ∗n+1) + 2d(v∗,n + 1) (Triangle Inequality)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


Persistent Machines

JB Clemente



Reoptimization16 Model




Diliman

Another Model for Combinatorial Reoptimiza-tion [Shachnai et al., 2012]

Given an optimization problem Π, let I0 be an input for Π, and let

CI0 = {C 1I0,C 2

I0,C 3

I0, . . . .}

be the set of configurations corresponding to the solution space ofΠ for I0.

In R(Π), we are given C jI0∈ CI0 of an initial instance I0, and a

new instance I obtained from I0

For any i ∈ I and configuration C kI , let δ be the transition cost.

δ(i,C ji0,C k

I )

The goal of reoptimization is to find C ∗I with an optimalcost(C ∗I ) and transition cost

δ(i,C jI0,C ∗I )

33


Persistent Machines

JB Clemente







Diliman



CI0 = {C 1I0,C 2

I0,C 3

I0, . . . .}





δ(i,C ji0,C k

I )


δ(i,C jI0,C ∗I )

33


Persistent Machines

JB Clemente







Diliman



CI0 = {C 1I0,C 2

I0,C 3

I0, . . . .}





δ(i,C ji0,C k

I )


δ(i,C jI0,C ∗I )

33


Persistent Machines

JB Clemente







Diliman



CI0 = {C 1I0,C 2

I0,C 3

I0, . . . .}





δ(i,C ji0,C k

I )


δ(i,C jI0,C ∗I )

33


Persistent Machines

JB Clemente







Diliman

Reoptimization from an Online Environment

I = {I0, I1, I2, I3, . . . , It}

SOL = {SOL0,SOL1,SOL2,SOL3, . . . ,SOLt}

33


Persistent Machines

JB Clemente







Diliman

Online Algorithm

I Many problems such as routing, scheduling, or the pagingproblem work in so called online environments and theiralgorithmic formulation and analysis demand a model inwhich an algorithm deals with such a problem knows only apart of its input at any specific point during runtime.

I These problems are called online problems and the respectivealgorithms are called online algorithms.

I An online algorithm A has to make decisions at any timestep i without knowing what the next chunk of input at timestep i + 1 will be. Algorithm A has to produce part of thefinal output in every step, it cannot revoke decisions it hasalready made.

33


Persistent Machines

JB Clemente







Diliman

Online Algorithm




33


Persistent Machines

JB Clemente







Diliman

Online Algorithm




33


Persistent Machines

JB Clemente







Diliman

Online Algorithm




33


Persistent Machines

JB Clemente







Diliman

Online Algorithm




33


Persistent Machines

JB Clemente



ReoptimizationModel

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, andTM is allowed to use an unbounded but only finiteamount of time or memory resources [(Eberbach, 2003),(Wegner, 2003)].

In Reoptimization,

1. We expect changes in the environment.2. We assume that the initial solution (SOL0) is obtained from

the environment.3. We make use of previous configurations in solving new input

instances.Interaction Machines allow inputs to be generateddynamically and require inputs to be represented by apotentially infinite stream. [Eberbach,Wegner, 2003]

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

Limitations of TM


In Reoptimization,

1. We expect changes in the environment.2. We assume that the initial solution (SOL0) is obtained from



33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

Limitations of TM


In Reoptimization,1. We expect changes in the environment.

2. We assume that the initial solution (SOL0) is obtained fromthe environment.

3. We make use of previous configurations in solving new inputinstances.Interaction Machines allow inputs to be generateddynamically and require inputs to be represented by apotentially infinite stream. [Eberbach,Wegner, 2003]

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

Limitations of TM


In Reoptimization,1. We expect changes in the environment.2. We assume that the initial solution (SOL0) is obtained from

the environment.

3. We make use of previous configurations in solving new inputinstances.Interaction Machines allow inputs to be generateddynamically and require inputs to be represented by apotentially infinite stream. [Eberbach,Wegner, 2003]

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

Limitations of TM




instances.

Interaction Machines allow inputs to be generateddynamically and require inputs to be represented by apotentially infinite stream. [Eberbach,Wegner, 2003]

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

Limitations of TM





33


Persistent Machines

JB Clemente



ReoptimizationModel


Interaction Machines20 Persistent Turing Machine


Diliman

Persistent Turing Machines

The canonical model of interaction machinesI minimal extension of Turing Machines (TMs) that express

interactive behavior.

I reactive systemI multitape machine with a persistent worktape preserved

between interactionsI inputs and outputs are dynamically generated streams of

strings.

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman



interactive behavior.I reactive system

I multitape machine with a persistent worktape preservedbetween interactions

I inputs and outputs are dynamically generated streams ofstrings.

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman



interactive behavior.I reactive systemI multitape machine with a persistent worktape preserved

between interactions

I inputs and outputs are dynamically generated streams ofstrings.

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman



interactive behavior.I reactive systemI multitape machine with a persistent worktape preserved

between interactionsI inputs and outputs are dynamically generated streams of

strings.

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

PTM: Definition

I PTM states are not to be confused with TM states. Unlikefor TMs,the set of PTM states is infinite, represented bystrings of unbounded length.

I Since the worktape (state) at the beginning of a PTMcomputation step is not always the same, the output of aPTM M at the end of the computation step depends both onthe input and on the worktape.

fM : I ×W → O ×W

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

PTM: Definition

I PTM states are not to be confused with TM states. Unlikefor TMs,the set of PTM states is infinite, represented bystrings of unbounded length.

I Since the worktape (state) at the beginning of a PTMcomputation step is not always the same, the output of aPTM M at the end of the computation step depends both onthe input and on the worktape.

fM : I ×W → O ×W

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

PTM: Example

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

PTM: Example

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

PTM: Computation

I Input streams are generated by the environment.

I The streams have dynamic evaluation semantics, where thenext value is not generated until the previous one isconsumed.

33


Persistent Machines

JB Clemente



ReoptimizationModel




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 isconsumed.

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

PTM: Interaction

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

PTM: Interaction

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

Thank you for listening.

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

References

Ausiello, G., Bonifaci, V., and Escoffier, B. (2011).Complexity and approximation in reoptimization.In Computability in Context: Computation and Logic in the RealWorld, volume 2, pages 101–129.

Ausiello, G., Escoffier, B., Monnot, J., and Paschos, V. (2009).Reoptimization of minimum and maximum traveling salesman’stours.Journal of Discrete Algorithms, 7(4):453–463.

Bilo, D. and Zych, A. (2012).New Advances in Reoptimizing the Minimum Steiner TreeProblem.Lecture Notes in Computer Science, Mathematical Foundationsof Computer Science, 7464:184–197.

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

References (cont.)

Böckenhauer, H.-J., Freiermuth, K., Hromkovič, J., Mömke, T.,Sprock, A., and Steffen, B. (2012).Steiner tree reoptimization in graphs with sharpened triangleinequality.Journal of Discrete Algorithms, 11:73–86.

Boria, N., Monnot, J., and Paschos, V. T. (2012).Reoptimization of the Maximum Weighted Pk-Free SubgraphProblem under Vertex Insertion.pages 76–87.

Cattaneo, G., Faruolo, P., Petrillo, U. F., and Italiano, G.(2010).Maintaining dynamic minimum spanning trees: An experimentalstudy.Discrete Applied Mathematics, 158(5):404–425.

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

References (cont.)

Clemente, J., Aborot, J., and Adorna, H. (2014).Reoptimization of Motif Finding Problem.Proceedings of the International MultiConference of Engineersand 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 Trendsin Theory and Practice of Computer Science (SOFSEM 2008),LNCS, 4910.Hromkovič, J. (2009).Algorithmic adventures: from knowledge to magic.

Královic, R. and Mömke, T. (2007).Approximation Hardness of the Traveling SalesmanReoptimization Problem.MEMICS 2007, 293.

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

References (cont.)

Nardelli, E., Proietti, G., and Widmayer, P. (2003).Swapping a Failing Edge of a Single Source Shortest Paths TreeIs Good and Fast.Algorithmica, pages 56–74.

Pallottino, S. and Scutella, M. (2003).A new algorithm for reoptimizing shortest paths when the arccosts change.Operations Research Letters.

Papadimitriou, C. and Steiglitz, K. (1998).Combinatorial optimization: algorithms and complexity.

Popov, V. (2013).On Reoptimization of the Shortest Common SuperstringProblem.Applied Mathematical Sciences, 7(24):1195–1197.

33


Persistent Machines

JB Clemente



ReoptimizationModel




Diliman

References (cont.)

Ribeiro, C. and Toso, R. (2007).Experimental analysis of algorithms for updating minimumspanning trees on graphs subject to changes on edge weights.Experimental Algorithms.

Secomandi, N. and Margot, F. (2009).Reoptimization approaches for the vehicle-routing problem withstochastic 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 AlgorithmTheory, pages 1–9.

33


Persistent Machines

JB Clemente



ReoptimizationModel




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 localre-optimization.In International Conference on BioMedical Engineering andInformatics, BMEI 2008.Zych, A. (2012).Reoptimization of NP-hard Problems.PhD thesis, ETH Zurich.

Reoptimization Algorithms and Persistent Turing Machines

Engineering

approximative algorithm

maximization problem

minimization problem

problem instance

cost function costopti

apolynomialtime algorithm

maximum cost of costopti

minimum cost of costopti