Game saturation of intersecting families

We consider the following combinatorial game: two players, Fast and Slow, claim $k$-element subsets of $[n]=\{1,2,...,n\}$ alternately, one at each turn, such that both players are allowed to pick sets that intersect all previously claimed subsets. The game ends when there does not exist any unclaimed $k$-subset that meets all already claimed sets. The score of the game is the number of sets claimed by the two players, the aim of Fast is to keep the score as low as possible, while the aim of Slow is to postpone the game's end as long as possible. The game saturation number is the score of the game when both players play according to an optimal strategy. To be precise we have to distinguish two cases depending on which player takes the first move. Let $gsat_F(\mathbb{I}_{n,k})$ and $gsat_S(\mathbb{I}_{n,k})$ denote the score of the saturation game when both players play according to an optimal strategy and the game starts with Fast's or Slow's move, respectively. We prove that $\Omega_k(n^{k/3-5}) \le gsat_F(\mathbb{I}_{n,k}),gsat_S(\mathbb{I}_{n,k}) \le O_k(n^{k-\sqrt{k}/2})$ holds

Planar tautologies hard for resolution.

We prove exponential lower bounds on the resolution proofs of some tautologies, based on rectangular grid graphs. More specifically, we show a 2/sup /spl Omega/(n)/ lower bound for any resolution proof of the mutilated chessboard problem on a 2n/spl times/2n chessboard as well as for the Tseitin tautology (G. Tseitin, 1968) based on the n/spl times/n rectangular grid graph. The former result answers a 35 year old conjecture by J. McCarthy (1964)

Thecomposition of semi finished inventories at a solid board plant

A solid board factory produces rectangular sheets of cardboard in two different formats, namely large formats and small formats. The production process consists of two stages separated by an inventory point. In the first stage a cardboard machine produces the large formats. In the second stage a part of the large formats is cut into small formats by a separate rotary cut machine. Due to very large setup times, technical restrictions, and trim losses, the cardboard machine is not able to produce these small formats. The company follows two policies to satisfy customer demands for rotary cut format orders. When the company applies the first policy, then for each customer order an ‘optimal’ large format (with respect to trim loss) is determined and produced on the cardboard machine. In case of the second policy, a stock of a restricted number of large formats is determined in such a way that the expected trim loss is minimal. The rotary cut format order then uses the most suitable standard large format from the stock. Currently, the dimensions of the standard large formats in the semi finished inventory are based on intuitive motives, with an accent on minimizing trim losses. From the trim loss perspective it is most efficient to produce each rotary cut format from a specific large format. On the other hand, if there is only one large format in each caliper, the variety is minimal, but the trim loss might be inacceptably high. On average, the first policy results in a lower trim loss. In order to make efficiently use of the two machines and to meet customer’s due times the company applies both policies. In this paper we concentrate on the second policy, taking into account the various objectives and restrictions of the company. The purpose of the company is to have not too many different types of large formats and an acceptable amount of trim loss. The problem is formulated as a minimum clique covering problem with alternatives (MCCA), which is presumed to be NP-hard. We solve the problem by using an appropriate heuristic, which is built into a decision support system. Based on a set of real data, the actual composition of semi finished inventories is determined. The paper concludes with computational experiments.