6 research outputs found
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)
iterations with bloat control on ORDER as well as
MAJORITY; and (ii) and
(and for )
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 to a fixed (quantum) graph in any finite field of prime
order . The subproblem with graph 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 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
-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 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