Bounding Bloat in Genetic Programming

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) $\Theta(T_{init} + n \log n)$ 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.Comment: An extended abstract has been published at GECCO 201

Counting Homomorphisms to Trees Modulo a Prime

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 #_pHomsToH, 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 #_pHomsToH is either polynomial time computable or #_pP-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 #_pHomsToH 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

On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order

We study the problem of counting the number of homomorphisms from an input graph $G$ to a fixed (quantum) graph $\bar{H}$ in any finite field of prime order $\mathbb{Z}_p$. The subproblem with graph $H$ was introduced by Faben and Jerrum~[ToC'15] and its complexity is still uncharacterised despite active research, e.g. the very recent work of Focke, Goldberg, Roth, and Zivn\'y~[SODA'21]. Our contribution is threefold. First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph $\bar{H}$ collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite $(K_{3,3}\backslash\{e\},\, {domino})$-free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with $p=2$ this establishes new results.Comment: 84 pages, revised title and mainly the Introduction and the section on partially surjective homomorphism

Destructiveness of Lexicographic Parsimony Pressure and Alleviation by a Concatenation Crossover in Genetic Programming

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 Concatenation Crossover 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.Comment: to appear in PPSN 201

Fixed-Parameter Sensitivity Oracles

We combine ideas from distance sensitivity oracles (DSOs) and fixed-parameter tractability (FPT) to design sensitivity oracles for FPT graph problems. An oracle with sensitivity $f$ for an FPT problem $\Pi$ on a graph $G$ with parameter $k$ preprocesses $G$ in time $O(g(f,k) \cdot \textsf{poly}(n))$. When queried with a set $F$ of at most $f$ edges of $G$, the oracle reports the answer to the $\Pi$-with the same parameter $k$-on the graph $G-F$, i.e., $G$ deprived of $F$. The oracle should answer queries in a time that is significantly faster than merely running the best-known FPT algorithm on $G-F$ from scratch. We mainly design sensitivity oracles for the $k$-Path and the $k$-Vertex Cover problem. Following our line of research connecting fault-tolerant FPT and shortest paths problems, we also introduce parameterization to the computation of distance preservers. We study the problem, given a directed unweighted graph with a fixed source $s$ and parameters $f$ and $k$, to construct a polynomial-sized oracle that efficiently reports, for any target vertex $v$ and set $F$ of at most $f$ edges, whether the distance from $s$ to $v$ increases at most by an additive term of $k$ in $G-F$.Comment: 19 pages, 1 figure, abstract shortened to meet ArXiv requirements; accepted at ITCS'2

Analysis of the (1 + 1) EA on subclasses of linear functions under uniform and linear constraints

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 current analyses and used methods for analyzing unconstrained problems to a setting involving constraints. In this paper, we consider the behavior of the classical 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.Tobias Friedrich, Timo Kötzing, J.A. Gregor Lagodzinski, Frank Neumann, Martin Schirnec