2020 [ nach oben ]

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  Doerr, Benjamin; Kötzing, Timo; Lagodzinski, J. A. Gregor; Lengler, JohannesThe impact of lexicographic parsimony pressure for ORDER/MAJORITY on the run time. Theoretical Computer Science 2020: 144--168
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  While many optimization problems work with a fixed number of decision variables and thus a fixed-length representation of possible solutions, genetic programming (GP) works on variable-length representations. A naturally occurring problem is that of bloat, that is, the unnecessary growth of solution lengths, which may slow down the optimization process. So far, the mathematical runtime analysis could not deal well with bloat and required explicit assumptions limiting bloat. In this paper, we provide the first mathematical runtime analysis of a GP algorithm that does not require any assumptions on the bloat. Previous performance guarantees were only proven conditionally for runs in which no strong bloat occurs. Together with improved analyses for the case with bloat restrictions our results show that such assumptions on the bloat are not necessary and that the algorithm is efficient without explicit bloat control mechanism. More specifically, we analyzed the performance of the \($(1+1)~$\)GP on the two benchmark functions Order and Majority. When using lexicographic parsimony pressure as bloat control, we show a tight runtime estimate of \($\mathrm{O(T_init+n \log n)$\) iterations both for Order and Majority. For the case without bloat control, the bounds \($\mathrm{O(T_init log T_init + n (\log n)^3)$\) and \($\Omega(T_init + n \log n)$\) (and \($Omega (T_init + \log T_{init})$\) for \($n=1$\)) hold for Majority.
  @article{doerr2020impact, abstract = {While many optimization problems work with a fixed number of decision variables and thus a fixed-length representation of possible solutions, genetic programming (GP) works on variable-length representations. A naturally occurring problem is that of bloat, that is, the unnecessary growth of solution lengths, which may slow down the optimization process. So far, the mathematical runtime analysis could not deal well with bloat and required explicit assumptions limiting bloat. In this paper, we provide the first mathematical runtime analysis of a GP algorithm that does not require any assumptions on the bloat. Previous performance guarantees were only proven conditionally for runs in which no strong bloat occurs. Together with improved analyses for the case with bloat restrictions our results show that such assumptions on the bloat are not necessary and that the algorithm is efficient without explicit bloat control mechanism. More specifically, we analyzed the performance of the $(1+1)~$GP on the two benchmark functions Order and Majority. When using lexicographic parsimony pressure as bloat control, we show a tight runtime estimate of $\mathrm{O}(T_{init}+n \log n)$ iterations both for Order and Majority. For the case without bloat control, the bounds $\mathrm{O}(T_{init} \log T_{init} + n (\log n)^3)$ and $\Omega(T_{init} + n \log n)$ (and $\Omega (T_{init} + \log T_{init})$ for $n=1$) hold for Majority.}, author = {Doerr, Benjamin and Kötzing, Timo and Lagodzinski, J. A. Gregor and Lengler, Johannes}, journal = {Theoretical Computer Science}, keywords = {benjamindoerr gregorlagodzinski johanneslengler tcs timokoetzing year2020}, pages = {144--168}, title = {The impact of lexicographic parsimony pressure for ORDER/MAJORITY on the run time}, volume = 816, year = 2020 }
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  Kötzing, Timo; Lagodzinski, J. A. Gregor; Lengler, Johannes; Melnichenko, AnnaDestructiveness of lexicographic parsimony pressure and alleviation by a concatenation crossover in genetic programming. Theoretical Computer Science 2020: 96--113
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  For theoretical analyses there are two specifics distinguishing GP from many other areas of evolutionary computation: the variable size representations, in particular yielding a possible bloat (i.e. the growth of individuals with redundant parts); and also the role and the realization of crossover, which is particularly central in GP due to the tree-based representation. Whereas some theoretical work on GP has studied the effects of bloat, crossover had surprisingly little share in this work. We analyze a simple crossover operator in combination with randomized local search, where a preference for small solutions minimizes bloat (lexicographic parsimony pressure); we denote the resulting algorithm Concatenation Crossover GP. We consider three variants of the well-studied Majority test function, adding large plateaus in different ways to the fitness landscape and thus giving a test bed for analyzing the interplay of variation operators and bloat control mechanisms in a setting with local optima. We show that the Concatenation Crossover GP can efficiently optimize these test functions, while local search cannot be efficient for all three variants independent of employing bloat control.
  @article{ktzing2020destructiveness, abstract = {For theoretical analyses there are two specifics distinguishing GP from many other areas of evolutionary computation: the variable size representations, in particular yielding a possible bloat (i.e. the growth of individuals with redundant parts); and also the role and the realization of crossover, which is particularly central in GP due to the tree-based representation. Whereas some theoretical work on GP has studied the effects of bloat, crossover had surprisingly little share in this work. We analyze a simple crossover operator in combination with randomized local search, where a preference for small solutions minimizes bloat (lexicographic parsimony pressure); we denote the resulting algorithm Concatenation Crossover GP. We consider three variants of the well-studied Majority test function, adding large plateaus in different ways to the fitness landscape and thus giving a test bed for analyzing the interplay of variation operators and bloat control mechanisms in a setting with local optima. We show that the Concatenation Crossover GP can efficiently optimize these test functions, while local search cannot be efficient for all three variants independent of employing bloat control.}, author = {Kötzing, Timo and Lagodzinski, J. A. Gregor and Lengler, Johannes and Melnichenko, Anna}, journal = {Theoretical Computer Science}, keywords = {annamelnichenko gregorlagodzinski johanneslengler tcs timokoetzing year2020}, pages = {96--113}, title = {Destructiveness of lexicographic parsimony pressure and alleviation by a concatenation crossover in genetic programming}, volume = 816, year = 2020 }
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  Friedrich, Tobias; Kötzing, Timo; Lagodzinski, J. A. Gregor; Neumann, Frank; Schirneck, MartinAnalysis of the (1+1) EA on Subclasses of Linear Functions under Uniform and Linear Constraints. Theoretical Computer Science 2020: 3-19
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  Linear functions have gained great attention in the run time analysis of evolutionary computation methods. The corresponding investigations have provided many effective tools for analyzing more complex problems. So far, the runtime analysis of evolutionary algorithms has mainly focused on unconstrained problems, but problems occurring in applications frequently involve constraints. Therefore, there is a strong need to extend the methods for analyzing unconstrained problems to a setting involving constraints. In this paper, we consider the behavior of the classical (1+1) evolutionary algorithm on linear functions under linear constraint. We show tight bounds in the case where the constraint is given by the OneMax function and the objective function is given by either the OneMax or the BinVal function. For the general case we present upper and lower bounds.
  @article{friedrich2020analysis, abstract = {Linear functions have gained great attention in the run time analysis of evolutionary computation methods. The corresponding investigations have provided many effective tools for analyzing more complex problems. So far, the runtime analysis of evolutionary algorithms has mainly focused on unconstrained problems, but problems occurring in applications frequently involve constraints. Therefore, there is a strong need to extend the methods for analyzing unconstrained problems to a setting involving constraints. In this paper, we consider the behavior of the classical (1+1) evolutionary algorithm on linear functions under linear constraint. We show tight bounds in the case where the constraint is given by the OneMax function and the objective function is given by either the OneMax or the BinVal function. For the general case we present upper and lower bounds.}, author = {Friedrich, Tobias and Kötzing, Timo and Lagodzinski, J. A. Gregor and Neumann, Frank and Schirneck, Martin}, editor = {Oliveto, Pietro S. and Sutton, Andrew M.}, journal = {Theoretical Computer Science}, keywords = {frankneumann gregorlagodzinski martinschirneck tcs timokoetzing tobiasfriedrich year2018}, pages = {3-19}, title = {Analysis of the (1+1) EA on Subclasses of Linear Functions under Uniform and Linear Constraints}, volume = 832, year = 2020 }

2018 [ nach oben ]

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  Göbel, Andreas; Lagodzinski, J. A. Gregor; Seidel, KarenCounting Homomorphisms to Trees Modulo a Prime. Mathematical Foundations of Computer Science (MFCS) 2018: 49:1-49:13
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article we study the complexity of~\($\#_p\textsc{HomsTo}H$\), the problem of counting graph homomorphisms from an input graph to a graph \($H$\) modulo a prime number~\($p$\). Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo~\($2$\), however, the influence of the structure of~\($H$\) on the tractability was shown to persist, which yields similar dichotomies. Our main result states that for every tree~\($H$\) and every prime~\($p$\) the problem \($\#_p\textsc{HomsTo}H$\) is either polynomial time computable or \($\#_p\mathsf{P}$\)-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph \($H$\) when counting modulo~2. In contrast to previous results on modular counting, the tractable cases of \($\#_p\textsc{HomsTo}H$\) are essentially the same for all values of the modulo when \($H$\) is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime~\($p$\). These results are the first suggesting that such dichotomies hold not only for the one-bit functions of the modulo~2 case but also for the modular counting functions of all primes~\($p$\).
  @inproceedings{gobel2018counting, abstract = {Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article we study the complexity of~$\#_p\textsc{HomsTo}H$, the problem of counting graph homomorphisms from an input graph to a graph $H$ modulo a prime number~$p$. Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo~$2$, however, the influence of the structure of~$H$ on the tractability was shown to persist, which yields similar dichotomies. Our main result states that for every tree~$H$ and every prime~$p$ the problem $\#_p\textsc{HomsTo}H$ is either polynomial time computable or $\#_p\mathsf{P}$-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph $H$ when counting modulo~2. In contrast to previous results on modular counting, the tractable cases of $\#_p\textsc{HomsTo}H$ are essentially the same for all values of the modulo when $H$ is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime~$p$. These results are the first suggesting that such dichotomies hold not only for the one-bit functions of the modulo~2 case but also for the modular counting functions of all primes~$p$.}, author = {Göbel, Andreas and Lagodzinski, J. A. Gregor and Seidel, Karen}, booktitle = {Mathematical Foundations of Computer Science (MFCS)}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, keywords = {andreasgoebel gregorlagodzinski karenseidel mfcs year2018}, pages = {49:1-49:13}, publisher = {Leibniz International Proceedings in Informatics (LIPIcs)}, title = {Counting Homomorphisms to Trees Modulo a Prime}, volume = 117, year = 2018 }
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  Bläsius, Thomas; Eube, Jan; Feldtkeller, Thomas; Friedrich, Tobias; Krejca, Martin S.; Lagodzinski, J. A. Gregor; Rothenberger, Ralf; Severin, Julius; Sommer, Fabian; Trautmann, JustinMemory-restricted Routing With Tiled Map Data. Systems, Man, and Cybernetics (SMC) 2018: 3347-3354
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Modern routing algorithms reduce query time by depending heavily on preprocessed data. The recently developed Navigation Data Standard (NDS) enforces a separation between algorithms and map data, rendering preprocessing inapplicable. Furthermore, map data is partitioned into tiles with respect to their geographic coordinates. With the limited memory found in portable devices, the number of tiles loaded becomes the major factor for run time. We study routing under these restrictions and present new algorithms as well as empirical evaluations. Our results show that, on average, the most efficient algorithm presented uses more than 20 times fewer tile loads than a normal A*.
  @inproceedings{blasius2018memory, abstract = {Modern routing algorithms reduce query time by depending heavily on preprocessed data. The recently developed Navigation Data Standard (NDS) enforces a separation between algorithms and map data, rendering preprocessing inapplicable. Furthermore, map data is partitioned into tiles with respect to their geographic coordinates. With the limited memory found in portable devices, the number of tiles loaded becomes the major factor for run time. We study routing under these restrictions and present new algorithms as well as empirical evaluations. Our results show that, on average, the most efficient algorithm presented uses more than 20 times fewer tile loads than a normal A*.}, author = {Bläsius, Thomas and Eube, Jan and Feldtkeller, Thomas and Friedrich, Tobias and Krejca, Martin S. and Lagodzinski, J. A. Gregor and Rothenberger, Ralf and Severin, Julius and Sommer, Fabian and Trautmann, Justin}, booktitle = {Systems, Man, and Cybernetics (SMC)}, keywords = {fabiansommer gregorlagodzinski janeube juliusseverin justintrautmann martinskrejca ralfrothenberger smc thomasblaesius thomasfeldtkeller tobiasfriedrich year2018}, month = 10, pages = {3347-3354}, title = {Memory-restricted Routing With Tiled Map Data}, year = 2018 }
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  Kötzing, Timo; Lagodzinski, J. A. Gregor; Lengler, Johannes; Melnichenko, AnnaDestructiveness of Lexicographic Parsimony Pressure and Alleviation by a Concatenation Crossover in Genetic Programming. Parallel Problem Solving From Nature (PPSN) 2018: 42-54
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  For theoretical analyses there are two specifics distinguishing GP from many other areas of evolutionary computation. First, the variable size representations, in particular yielding a possible bloat (i.e. the growth of individuals with redundant parts). Second, the role and realization of crossover, which is particularly central in GP due to the tree-based representation. Whereas some theoretical work on GP has studied the effects of bloat, crossover had a surprisingly little share in this work. We analyze a simple crossover operator in combination with local search,where a preference for small solutions minimizes bloat (lexicographic parsimony pressure); the resulting algorithm is denoted ConcatenationCrossover GP. For this purpose three variants of the well-studied Majority test function with large plateaus are considered. We show that the Concatenation Crossover GP can efficiently optimize these test functions,while local search cannot be efficient for all three variants independent of employing bloat control.
  @inproceedings{kotzing2018destructiveness, abstract = {For theoretical analyses there are two specifics distinguishing GP from many other areas of evolutionary computation. First, the variable size representations, in particular yielding a possible bloat (i.e. the growth of individuals with redundant parts). Second, the role and realization of crossover, which is particularly central in GP due to the tree-based representation. Whereas some theoretical work on GP has studied the effects of bloat, crossover had a surprisingly little share in this work. We analyze a simple crossover operator in combination with local search,where a preference for small solutions minimizes bloat (lexicographic parsimony pressure); the resulting algorithm is denoted ConcatenationCrossover GP. For this purpose three variants of the well-studied Majority test function with large plateaus are considered. We show that the Concatenation Crossover GP can efficiently optimize these test functions,while local search cannot be efficient for all three variants independent of employing bloat control.}, author = {Kötzing, Timo and Lagodzinski, J. A. Gregor and Lengler, Johannes and Melnichenko, Anna}, booktitle = {Parallel Problem Solving From Nature (PPSN)}, keywords = {annamelnichenko gregorlagodzinski johanneslengler ppsn timokoetzing year2018}, pages = {42-54}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, title = {Destructiveness of Lexicographic Parsimony Pressure and Alleviation by a Concatenation Crossover in Genetic Programming}, volume = 11102, year = 2018 }

2017 [ nach oben ]

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  Doerr, Benjamin; Kötzing, Timo; Lagodzinski, J. A. Gregor; Lengler, JohannesBounding Bloat in Genetic Programming. Genetic and Evolutionary Computation Conference (GECCO) 2017: 921-928
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  While many optimization problems work with a fixed number of decision variables and thus a fixed-length representation of possible solutions, genetic programming (GP) works on variable-length representations. A naturally occurring problem is that of bloat (unnecessary growth of solutions) slowing down optimization. Theoretical analyses could so far not bound bloat and required explicit assumptions on the magnitude of bloat. In this paper we analyze bloat in mutation-based genetic programming for the two test functions ORDER and MAJORITY. We overcome previous assumptions on the magnitude of bloat and give matching or close-to-matching upper and lower bounds for the expected optimization time. In particular, we show that the \((1+1)\) GP takes (i) \(\Theta(T_\text{init + n\, \log n)\) iterations with bloat control on ORDER as well as MAJORITY; and (ii) \(O(T_{\text{init ,log \,T_{\text{init + n(\log n)^3)\) and \(\Omega(T_\text{init + n \,\log n)\) (and \(\Omega(T_\text{init \,\log \,T_{\text{init}})\) for \(n = 1\)) iterations without bloat control on MAJORITY.
  @inproceedings{doerr2017bounding, abstract = {While many optimization problems work with a fixed number of decision variables and thus a fixed-length representation of possible solutions, genetic programming (GP) works on variable-length representations. A naturally occurring problem is that of bloat (unnecessary growth of solutions) slowing down optimization. Theoretical analyses could so far not bound bloat and required explicit assumptions on the magnitude of bloat. In this paper we analyze bloat in mutation-based genetic programming for the two test functions ORDER and MAJORITY. We overcome previous assumptions on the magnitude of bloat and give matching or close-to-matching upper and lower bounds for the expected optimization time. In particular, we show that the \((1+1)\) GP takes (i) \(\Theta(T_{\text{init}} + n\, \log n)\) iterations with bloat control on ORDER as well as MAJORITY; and (ii) \(O(T_{\text{init}} \,\log \,T_{\text{init}} + n(\log n)^3)\) and \(\Omega(T_{\text{init}} + n \,\log n)\) (and \(\Omega(T_{\text{init}} \,\log \,T_{\text{init}})\) for \(n = 1\)) iterations without bloat control on MAJORITY.}, author = {Doerr, Benjamin and Kötzing, Timo and Lagodzinski, J. A. Gregor and Lengler, Johannes}, booktitle = {Genetic and Evolutionary Computation Conference (GECCO)}, keywords = {benjamindoerr gecco gregorlagodzinski johanneslengler timokoetzing year2017}, pages = {921-928}, title = {Bounding Bloat in Genetic Programming}, year = 2017 }
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  Friedrich, Tobias; Kötzing, Timo; Lagodzinski, J. A. Gregor; Neumann, Frank; Schirneck, MartinAnalysis of the (1+1) EA on Subclasses of Linear Functions under Uniform and Linear Constraints. Foundations of Genetic Algorithms (FOGA) 2017: 45-54
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Linear functions have gained a lot of attention in the area of run time analysis of evolutionary computation methods and the corresponding analyses have provided many effective tools for analyzing more complex problems. In this paper, we consider the behavior of the classical (1+1) Evolutionary Algorithm for linear functions under linear constraint. We show tight bounds in the case where both the objective function and the constraint is given by the OneMax function and present upper bounds as well as lower bounds for the general case. Furthermore, we also consider the LeadingOnes fitness function.
  @inproceedings{DBLP:conf/foga/0001KLNS17, abstract = {Linear functions have gained a lot of attention in the area of run time analysis of evolutionary computation methods and the corresponding analyses have provided many effective tools for analyzing more complex problems. In this paper, we consider the behavior of the classical (1+1) Evolutionary Algorithm for linear functions under linear constraint. We show tight bounds in the case where both the objective function and the constraint is given by the OneMax function and present upper bounds as well as lower bounds for the general case. Furthermore, we also consider the LeadingOnes fitness function.}, author = {Friedrich, Tobias and Kötzing, Timo and Lagodzinski, J. A. Gregor and Neumann, Frank and Schirneck, Martin}, booktitle = {Foundations of Genetic Algorithms (FOGA)}, keywords = {foga frankneumann gregorlagodzinski martinschirneck timokoetzing tobiasfriedrich year2017}, pages = {45-54}, publisher = {ACM Press}, title = {Analysis of the {(1+1)} EA on Subclasses of Linear Functions under Uniform and Linear Constraints}, year = 2017 }