Analytic Algorithmics and Combinatorics 

Abstract: We describe a variant of the Bellman--Ford algorithm for single-source shortest paths in graphs with negative edges but no negative cycles that randomly permutes the vertices and uses this randomized order to process the vertices within each pass of the algorithm The modification reduces the worst-case expected number of relaxation steps of the algorithm, compared to the previously-best variant by Yen (1970), by a factor of 2/3 with high probability We also use our high probability bound to add negative cycle detection to the randomized algorithm

Abstract: The main contribution of this paper is a new approach for enumerating Hamilton cycles in bounded degree graphs -- deriving thereby extremal bounds. We describe an algorithm which enumerates all Hamilton cycles of a given 3-regular n-vertex graph in time O(1.276n), improving on Eppstein's previous bound. The resulting new upper bound of O(1.276n) for the maximum number of Hamilton cycles in 3-regular n-vertex graphs gets close to the best known lower bound of Ω(1.259n). Our method differs from Eppstein's in that he considers in each step a new graph and modifies it, while we fix (at the very beginning) one Hamilton cycle C and then proceed around C, succesively producing partial Hamilton cycles. Our approach can also be used to show that the number of Hamilton cycles of a 4-regular n-vertex graph is at most O(18n/5) ≤ O(1.783n), which improves a previous bound by Sharir and Welzl. This result is complemented by a lower bound of 48n/8 ≥ 1.622n. Then we present an algorithm which finds the minimum weight Hamilton cycle of a given 4-regular graph in time √3n · poly(n) = O(1.733n), improving a previous result of Eppstein. This algorithm can be modified to compute the number of Hamilton cycles in the same time bound and to enumerate all Hamilton cycles in time (√3n +hc(G))·poly(n) with hc(G) denoting the number of Hamilton cycles of the given graph G. So our upper bound of O(1.783n) for the number of Hamilton cycles serves also as a time bound for enumeration. Using similar techniques as in the 3-regular case we establish upper bounds for the number of Hamilton cycles in 5-regular graphs and in graphs of average degree 3, 4, and 5.

Abstract: We are given a graph G with n vertices, where a random subset of k vertices has been made into a clique, and the remaining edges are chosen independently with probability 1/2. This random graph model is denoted G(n, 1/2, k). The hidden clique problem is to design an algorithm that finds the k-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov [3] uses spectral techniques to find the hidden clique with high probability when k = c√n for a sufficiently large constant c > 0. Recently, an algorithm that solves the same problem was proposed by Feige and Ron [14]. It has the advantages of being simpler and more intuitive, and of an improved running time of O(n2). However, the analysis in [14] gives success probability of only 2/3. In this paper we present a new algorithm for finding hidden cliques that both runs in time O(n2), and has a failure probability that is less than polynomially small.

Abstract: In this paper we study the NP-complete problem of finding small k-dominating sets in general graphs, which allow k---1 nodes to fail and still dominate the graph. The classic minimum dominating set problem is a special case with k = 1. We show that the approach of having at least k dominating nodes in the neighborhood of every node is not optimal. For each α > 1 it can give solutions k/α times larger than a minimum k-dominating set. We also study lower bounds on possible approximation ratios. We show that it is NP-hard to approximate the minimum k-dominating set problem with a factor better than (0.2267/k) ln(n/k). Furthermore, a result for special finite sums allows us to use a greedy approach for k-domination with an approximation ratio of ln(Δ + k) + 1 < ln(Δ) + 1.7, with Δ being the maximum node-degree. We also achieve an approximation ratio of ln(n)+1.7 for h-step k-domination, where nodes do not need to be direct neighbors of dominating nodes, but can be h steps away.