Zsolt Tuza

Bernadett Acs, Gabor Szederkenyi, Zsolt Tuza et al.

In this paper an algorithm is given to determine all possible structurally different linearly conjugate realizations of a given kinetic polynomial system. The solution is based on the iterative search for constrained dense realizations using linear programming. Since there might exist exponentially many different reaction graph structures, we cannot expect to have a polynomial-time algorithm, but we can organize the computation in such a way that polynomial time is elapsed between displaying any two consecutive realizations. The correctness of the algorithm is proved, and possibilities of a parallel implementation are discussed. The operation of the method is shown on two illustrative examples.

Peter Horak, Zsolt Tuza

The "Gluing Algorithm" of Semaev [Des.\ Codes Cryptogr.\ 49 (2008), 47--60] --- that finds all solutions of a sparse system of linear equations over the Galois field $GF(q)$ --- has average running time $O(mq^{\max \left\vert \cup_{1}^{k}X_{j}\right\vert -k}), $ where $m$ is the total number of equations, and $\cup_{1}^{k}X_{j}$ is the set of all unknowns actively occurring in the first $k$ equations. Our goal here is to minimize the exponent of $q$ in the case where every equation contains at most three unknowns. %Applying hypergraph-theoretic methods we prove The main result states that if the total number $\left\vert \cup_{1}^{m}X_{j}\right\vert$ of unknowns is equal to $m$, then the best achievable exponent is between $c_1m$ and $c_2m$ for some positive constants $c_1$ and $c_2.$