The switch operators and push-the-button games: a sequential compound over rulesets

We study operators that combine combinatorial games. This field was initiated by Sprague-Grundy (1930s), Milnor (1950s) and Berlekamp-Conway-Guy (1970-80s) via the now classical disjunctive sum operator on (abstract) games. The new class consists in operators for rulesets, dubbed the switch-operators. The ordered pair of rulesets (R 1 , R 2) is compatible if, given any position in R 1 , there is a description of how to move in R 2. Given compatible (R 1 , R 2), we build the push-the-button game R 1 R 2 , where players start by playing according to the rules R 1 , but at some point during play, one of the players must switch the rules to R 2 , by pushing the button ". Thus, the game ends according to the terminal condition of ruleset R 2. We study the pairwise combinations of the classical rulesets Nim, Wythoff and Euclid. In addition, we prove that standard periodicity results for Subtraction games transfer to this setting, and we give partial results for a variation of Domineering, where R 1 is the game where the players put the domino tiles horizontally and R 2 the game where they play vertically (thus generalizing the octal game 0.07).Comment: Journal of Theoretical Computer Science (TCS), Elsevier, A Para{\^i}tr

Compound Node-Kayles on Paths

In his celebrated book "On Number and Games" (Academic Press, New-York, 1976), J.H. Conway introduced twelve versions of compound games. We analyze these twelve versions for the Node-Kayles game on paths. For usual disjunctive compound, Node-Kayles has been solved for a long time under normal play, while it is still unsolved under mis\`ere play. We thus focus on the ten remaining versions, leaving only one of them unsolved.Comment: Theoretical Computer Science (2009) to appea

Building Nim

The game of nim, with its simple rules, its elegant solution and its historical importance is the quintessence of a combinatorial game, which is why it led to so many generalizations and modifications. We present a modification with a new spin: building nim. With given finite numbers of tokens and stacks, this two-player game is played in two stages (thus belonging to the same family of games as e.g. nine-men's morris): first building, where players alternate to put one token on one of the, initially empty, stacks until all tokens have been used. Then, the players play nim. Of course, because the solution for the game of nim is known, the goal of the player who starts nim play is a placement of the tokens so that the Nim-sum of the stack heights at the end of building is different from 0. This game is trivial if the total number of tokens is odd as the Nim-sum could never be 0, or if both the number of tokens and the number of stacks are even, since a simple mimicking strategy results in a Nim-sum of 0 after each of the second player's moves. We present the solution for this game for some non-trivial cases and state a general conjecture

Generic Strategies for Chemical Space Exploration

Computational approaches to exploring "chemical universes", i.e., very large sets, potentially infinite sets of compounds that can be constructed by a prescribed collection of reaction mechanisms, in practice suffer from a combinatorial explosion. It quickly becomes impossible to test, for all pairs of compounds in a rapidly growing network, whether they can react with each other. More sophisticated and efficient strategies are therefore required to construct very large chemical reaction networks. Undirected labeled graphs and graph rewriting are natural models of chemical compounds and chemical reactions. Borrowing the idea of partial evaluation from functional programming, we introduce partial applications of rewrite rules. Binding substrate to rules increases the number of rules but drastically prunes the substrate sets to which it might match, resulting in dramatically reduced resource requirements. At the same time, exploration strategies can be guided, e.g. based on restrictions on the product molecules to avoid the explicit enumeration of very unlikely compounds. To this end we introduce here a generic framework for the specification of exploration strategies in graph-rewriting systems. Using key examples of complex chemical networks from sugar chemistry and the realm of metabolic networks we demonstrate the feasibility of a high-level strategy framework. The ideas presented here can not only be used for a strategy-based chemical space exploration that has close correspondence of experimental results, but are much more general. In particular, the framework can be used to emulate higher-level transformation models such as illustrated in a small puzzle game

Coordination of Purchasing and Bidding Activities Across Markets

In both consumer purchasing and industrial procurement, combinatorial interdependencies among the items to be purchased are commonplace. E-commerce compounds the problem by providing more opportunities for switching suppliers at low costs, but also potentially eases the problem by enabling automated market decision-making systems, commonly referred to as trading agents, to make purchasing decisions in an integrated manner across markets. Most of the existing research related to trading agents assumes that there exists a combinatorial market mechanism in which buyers (or sellers) can bid (or sell) service or merchant bundles. Todayâ??s prevailing e-commerce practice, however, does not support this assumption in general and thus limits the practical applicability of these approaches. We are investigating a new approach to deal with the combinatorial interdependency challenges for online markets. This approach relies on existing commercial online market institutions such as posted-price markets and various online auctions that sell single items. It uses trading agents to coordinate a buyerâ??s purchasing and bidding activities across multiple online markets simultaneously to achieve the best overall procurement effectiveness. This paper presents two sets of models related to this approach. The first set of models formalizes optimal purchasing decisions across posted-price markets with fixed transaction costs. Flat shipping costs, a common e-tailing practice, are captured in these models. We observe that making optimal purchasing decisions in this context is NP-hard in the strong sense and suggest several efficient computational methods based on discrete location theory. The second set of models is concerned with the coordination of bidding activities across multiple online auctions. We study the underlying coordination problem for a collection of first or second-price sealed-bid auctions and derive the optimal coordination and bidding policies.

Networking chemical robots for reaction multitasking

The development of the internet of things has led to an explosion in the number of networked devices capable of control and computing. However, whilst common place in remote sensing, these approaches have not impacted chemistry due to difficulty in developing systems flexible enough for experimental data collection. Herein we present a simple and affordable (<$500) chemistry capable robot built with a standard set of hardware and software protocols that can be networked to coordinate many chemical experiments in real time. We demonstrate how multiple processes can be done with two internet connected robots collaboratively, exploring a set of azo-coupling reactions in a fraction of time needed for a single robot, as well as encoding and decoding information into a network of oscillating reactions. The system can also be used to assess the reproducibility of chemical reactions and discover new reaction outcomes using game playing to explore a chemical space

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application