Parameterized TSP: Beating the Average

In the Travelling Salesman Problem (TSP), we are given a complete graph $K_n$ together with an integer weighting $w$ on the edges of $K_n$, and we are asked to find a Hamilton cycle of $K_n$ of minimum weight. Let $h(w)$ denote the average weight of a Hamilton cycle of $K_n$ for the weighting $w$. Vizing (1973) asked whether there is a polynomial-time algorithm which always finds a Hamilton cycle of weight at most $h(w)$. He answered this question in the affirmative and subsequently Rublineckii (1973) and others described several other TSP heuristics satisfying this property. In this paper, we prove a considerable generalisation of Vizing's result: for each fixed $k$, we give an algorithm that decides whether, for any input edge weighting $w$ of $K_n$, there is a Hamilton cycle of $K_n$ of weight at most $h(w)-k$ (and constructs such a cycle if it exists). For $k$ fixed, the running time of the algorithm is polynomial in $n$, where the degree of the polynomial does not depend on $k$ (i.e., the generalised Vizing problem is fixed-parameter tractable with respect to the parameter $k$)

Upper estimate of martingale dimension for self-similar fractals

We study upper estimates of the martingale dimension $d_m$ of diffusion processes associated with strong local Dirichlet forms. By applying a general strategy to self-similar Dirichlet forms on self-similar fractals, we prove that $d_m=1$ for natural diffusions on post-critically finite self-similar sets and that $d_m$ is dominated by the spectral dimension for the Brownian motion on Sierpinski carpets.Comment: 49 pages, 7 figures; minor revision with adding a referenc

Efficient domination in knights graphs

The influence of a vertex set S ⊆V(G) is I(S) = Σv∈S(1 + deg(v)) = Σv∈S |N[v]|, which is the total amount of domination done by the vertices in S. The efficient domination number F(G) of a graph G is equal to the maximum influence of a packing, that is, F(G) is the maximum number of vertices one can dominate under the restriction that no vertex gets dominated more than once. In this paper, we consider the efficient domination number of some finite and infinite knights chessboard graphs

Parameterized Traveling Salesman Problem:Beating the Average

In the traveling salesman problem (TSP), we are given a complete graph Kn together with an integer weighting w on the edges of Kn, and we are asked to find a Hamilton cycle of Kn of minimum weight. Let h(w) denote the average weight of a Hamilton cycle of Kn for the weighting w. Vizing in 1973 asked whether there is a polynomial-time algorithm which always finds a Hamilton cycle of weight at most h(w). He answered this question in the affirmative and subsequently Rublineckii, also in 1973, and others described several other TSP heuristics satisfying this property. In this paper, we prove a considerable generalization of Vizing’s result: for each fixed k, we give an algorithm that decides whether, for any input edge weighting w of Kn, there is a Hamilton cycle of Kn of weight at most h(w) − k (and constructs such a cycle if it exists). For k fixed, the running time of the algorithm is polynomial in n, where the degree of the polynomial does not depend on k (i.e., the generalized Vizing problem is fixed-parameter tractable with respect to the parameter k)