Abstract-Evolutionary algorithms use crossover to combine information from pairs of solutions and use selection to retain the best solutions. Ideally, crossover takes distinct good features from each of the two structures involved. This process creates a conflict: progress results from crossing over structures with different features, but crossover produces new structures that are like their parents and so reduces the diversity on which it depends. As evolution continues, the algorithm searches a smaller and smaller portion of the search space. Mutation can help maintain diversity but is not a panacea for diversity loss. This paper explores evolutionary algorithms that use combinatorial graphs to limit possible crossover partners. These graphs limit the speed and manner in which information can spread giving competing solutions time to mature. This use of graphs is a computationally inexpensive method of picking a global level of tradeoff between exploration and exploitation. The results of using 26 graphs with a diverse collection of graphical properties are presented. The test problems used are: one-max, the De Jong functions, the Griewangk function in three to seven dimensions, the self-avoiding random walk problem in 9, 12, 16, 20, 25, 30, and 36 dimensions, the plus-one-recall-store (PORS) problem with = 15 16 and 17, location of length-six one-error-correcting DNA barcodes, and solving a simple differential equation semi-symbolically.The choice of combinatorial graph has a significant effect on the time-to-solution. In the cases studied, the optimal choice of graph improved solution time as much as 63-fold with typical impact being in the range of 15% to 100% variation. The graph yielding superior performance is found to be problem dependent. In general, the optimal graph diameter increases and the optimal average degree decreases with the complexity and difficulty of the fitness landscape. The use of diverse graphs as population structures for a collection of problems also permits a classification of the problems. A phylogenetic analysis of the problems using normalized time to solution on each graph groups the numerical problems as a clade together with one-max; self-avoiding walks form a clade with the semisymbolic differential equation solution; and the PORS and DNA barcode problems form a superclade with the numerical problems but are substantially distinct from them. This novel form of analysis has the potential to aid researchers choosing problems for a test suite.Index Terms-Evolutionary algorithm, graph-based algorithms, population structure, test suite.

Abstract.A phylogenetic network is a rooted acyclic digraph with vertices corresponding to taxa. Let X denote a set of vertices containing the root, the leaves, and all vertices of outdegree 1. Regard X as the set of vertices on which measurements such as DNA can be made. A vertex is called normal if it has one parent, and hybrid if it has more than one parent. The network is called normal if it has no redundant arcs and also from every vertex there is a directed path to a member of X such that all vertices after the first are normal. This paper studies properties of normal networks.Under a simple model of inheritance that allows homoplasies only at hybrid vertices, there is essentially unique determination of the genomes at all vertices by the genomes at members of X if and only if the network is normal. This model is a limiting case of more standard models of inheritance when the substitution rate is sufficiently low.Various mathematical properties of normal networks are described. These properties include that the number of vertices grows at most quadratically with the number of leaves and that the number of hybrid vertices grows at most linearly with the number of leaves.

As phylogenetic networks grow increasingly complicated, systematic methods for simplifying them to reveal properties will become more useful. This paper considers how to modify acyclic phylogenetic networks into other acyclic networks by contracting specific arcs that include a set D. The networks need not be binary, so vertices in the networks may have more than two parents and/or more than two children. In general, in order to make the resulting network acyclic, additional arcs not in D must also be contracted. This paper shows how to choose D so that the resulting acyclic network is “pre-normal”. As a result, removal of all redundant arcs yields a normal network. The set D can be selected based only on the geometry of the network, giving a well-defined normal phylogenetic network depending only on the given network. There are CSD maps relating most of the networks. The resulting network can be visualized as a “wired lift” in the original network, which appears as the original network with each arc drawn in one of three ways.

Abstract-A rooted acyclic digraph N with labelled leaves displays a tree T when there exists a way to select a unique parent of each hybrid vertex resulting in the tree T . Let T r(N ) denote the set of all trees displayed by the network N . In general, there may be many other networks M such that T r(M ) = T r(N ). A network is regular if it is isomorphic with its cover digraph. If N is regular and D is a collection of trees displayed by N , this paper studies some procedures to try to reconstruct N given D. If the input is D = T r(N ), one procedure is described which will reconstruct N . Hence if N and M are regular networks and T r(N ) = T r(M ), it follows that N = M , proving that a regular network is uniquely determined by its displayed trees. If D is a (usually very much smaller) collection of displayed trees that satisfies certain hypotheses, modifications of the procedure will still reconstruct N given D.

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