
Göbel, Andreas; Lagodzinski, J. A. Gregor; Seidel, Karen Counting Homomorphisms to Trees Modulo a Prime. International Symposium on Mathematical Foundations of Computer Science (MFCS) 2018: 49:149:13
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 nonmodular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in nonmodular 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 onebit functions of the modulo~2 case but also for the modular counting functions of all primes~\($p$\).

Bläsius, Thomas; Eube, Jan; Feldtkeller, Thomas; Friedrich, Tobias; Krejca, Martin S.; Lagodzinski, J. A. Gregor; Rothenberger, Ralf; Severin, Julius; Sommer, Fabian; Trautmann, Justin Memoryrestricted Routing With Tiled Map Data. IEEE International Conference on Systems, Man, and Cybernetics (SMC) 2018
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*.

Doer, Benjaminr; Kötzing, Timo; Lagodzinski, J. A. Gregor; Lengler, Johannes Bounding Bloat in Genetic Programming. arxiv 2018
While many optimization problems work with a fixed number of decision variables and thus a fixedlength representation of possible solutions, genetic programming (GP) works on variablelength 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 mutationbased genetic programming for the two test functions ORDER and MAJORITY. We overcome previous assumptions on the magnitude of bloat and give matching or closetomatching upper and lower bounds for the expected optimization time. In particular, we show that the (1+1) GP takes (i) \(\Theta(T_init + n \log n)\) iterations with bloat control on ORDER as well as MAJORITY; and (ii) \(O(T_{init log T_init + n (\log n)^3)\) and \(\Omega(T_init + n \log n)\) (and \(\Omega(T_init \log T_{init})\) for \(n=1\)) iterations without bloat control on MAJORITY.

Kötzing, Timo; Lagodzinski, J. A. Gregor; Lengler, Johannes; Melnichenko, Anna Destructiveness of Lexicographic Parsimony Pressure and Alleviation by a Concatenation Crossover in Genetic Programming. arxiv 2018
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 treebased 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 (\emph{lexicographic parsimony pressure}); the resulting algorithm is denoted \emph{Concatenation Crossover GP. For this purpose three variants of the wellstudied 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.

Kötzing, Timo; Lagodzinski, J. A. Gregor; Lengler, Johannes; Melnichenko, Anna Destructiveness of Lexicographic Parsimony Pressure and Alleviation by a Concatenation Crossover in Genetic Programming. Parallel Problem Solving From Nature (PPSN) 2018

Friedrich, Tobias; Kötzing, Timo; Lagodzinski, J. A. Gregor; Neumann, Frank; Schirneck, Martin Analysis of the (1+1) EA on Subclasses of Linear Functions under Uniform and Linear Constraints. Theoretical Computer Science 2018
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.

Göbel, Andreas; Lagodzinski, J. A. Gregor; Seidel, Karen Counting Homomorphisms to Trees Modulo a Prime. arXiv 2018

Doerr, Benjamin; Kötzing, Timo; Lagodzinski, J. A. Gregor; Lengler, Johannes Bounding Bloat in Genetic Programming. Genetic and Evolutionary Computation Conference (GECCO) 2017: 921928
While many optimization problems work with a fixed number of decision variables and thus a fixedlength representation of possible solutions, genetic programming (GP) works on variablelength 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 mutationbased genetic programming for the two test functions ORDER and MAJORITY. We overcome previous assumptions on the magnitude of bloat and give matching or closetomatching 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.

Friedrich, Tobias; Kötzing, Timo; Lagodzinski, J. A. Gregor; Neumann, Frank; Schirneck, Martin Analysis of the (1+1) EA on Subclasses of Linear Functions under Uniform and Linear Constraints. Foundations of Genetic Algorithms (FOGA) 2017: 4554
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.