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Abstract. In this note, we prove that all cop-win graphs G in the game in which the robber and the cop move Since any δ-hyperbolic graph is cop-win for s = 2r and s = r + 2δ for any r > 0, this establishes a new -game-theoretical-characterization of Gromov hyperbolicity. We also show that for weakly modular graphs the dependency between δ and s is linear for any s < s. Using these results, we describe a simple constant-factor approximation of the hyperbolicity δ of a graph on n vertices in O(n 2 ) time when the graph is given by its distance-matrix.

A sum of affine powers is an expression of the formAlthough quite simple, this model is a generalization of two well-studied models: Waring decomposition and Sparsest Shift. For these three models there are natural extensions to several variables, but this paper is mostly focused on univariate polynomials. We present structural results which compare the expressive power of the three models; and we propose algorithms that find the smallest decomposition of f in the first model (sums of affine powers) for an input polynomial f given in dense representation. We also begin a study of the multivariate case. This work could be extended in several directions. In particular, just as for Sparsest Shift and Waring decomposition, one could consider extensions to "supersparse" polynomials and attempt a fuller study of the multivariate case. We also point out that the basic univariate problem studied in the present paper is far from completely solved: our algorithms all rely on some assumptions for the exponents e i in a decomposition of f , and some algorithms also rely on a distinctness assumption for the shifts a i . It would be very interesting to weaken these assumptions, or even to remove them entirely. Another related and poorly understood issue is that of the bit size of the constants a i , α i in an optimal decomposition: is it always polynomially related to the bit size of the input polynomial f given in dense representation?

A sum of affine powers is an expression of the form5668 Ecole Normale Supérieure de Lyon, CNRS, UCBL, INRIA. The authors are supported by ANR project CompA (code ANR-13-BS02-0001-01). Email: [Pascal.Koiran, Timothee.Pecatte]@ens-lyon.fr, iggarcia@ull.es.