On approximating metric 1-median in sublinear time

This paper focuses on approximating the metric 1-median problem with sublinear number of queries and time complexity. We first show a simper proof of the currently best 4-approximation algorithm, and then propose a recursive algorithm. For any integer h?2$h?2$, the algorithm finds a 2h  -approximation solution with O(n^1+1/h)$O(n^{1+1/h})$ queries and time complexity.     

Finding the nucleolus of any n-person cooperative game by a single linear program

In this paper we show a new method for calculating the nucleolus by solving a unique minimization linear program with O(4ⁿ)$O(4^n)$ constraints whose coefficients belong to {−1,0,1}$\{-1,0,1\}$. We discuss the need of having all these constraints and empirically prove that they can be reduced to O(_kmax2ⁿ)$O(k_{\mathit{max}}{2}^n)$, where k_max is a positive integer comparable with the number of players. A computational experience shows the applicability of our method over (pseudo)random transferable utility cooperative games with up to 18 players.     

Computing the pathwidth of directed graphs with small vertex cover

We give an algorithm that computes the pathwidth of a given directed graph D   in 3^τ(D)n^O(1)$3^{\tau (D)}{n}^{O(1)}$ time where n is the number of vertices of D   and τ(D)$\tau (D)$ is the number of vertices of a minimum vertex cover of the underlying undirected graph of D. This result extends that of [5] for undirected graphs to directed graphs. Moreover, our algorithm is based on a standard dynamic programming with a simple tree-pruning trick, which is extremely simple and easy to implement. An additional advantage of our algorithm is that a minimum vertex cover appears in the analysis but not in the algorithm itself, in contrast to the algorithm of [5] which needs to compute a minimum vertex cover.     

A new heuristic recursive algorithm for the strip rectangular packing problem

A fast and new heuristic recursive algorithm to find a minimum height for two-dimensional strip rectangular packing problem is presented. This algorithm is mainly based on heuristic strategies and a recursive structure, and its average running time is T(n)=θ(n³)$T(n)=\theta (n^3)$. The computational results on a class of benchmark problems have shown that this algorithm not only finds shorter height than the known meta-heuristic ones, but also runs in shorter time. Especially for large test problems, it performs better.     

Faster two-dimensional pattern matching with rotations

The most efficient currently known algorithms for two-dimensional pattern matching with rotations have a worst case time complexity of O(n²m³)$\mathrm{O}(n^2{m}^3)$, where the size of the text is n×n$n\times n$ and the size of the pattern is m×m$m\times m$. In this paper we present a new algorithm for the problem whose running time is O(n²m²)$\mathrm{O}(n^2{m}^2)$.     

An exact exponential time algorithm for counting bipartite cliques

We present a simple exact algorithm for counting bicliques of given size in a bipartite graph on n   vertices. We achieve running time of O(1.2491ⁿ)$O({1.2491}^n)$, improving upon known exact algorithms for finding and counting bipartite cliques.     

Optimal fault-tolerant embedding of paths in twisted cubes

The twisted cube is an important variation of the hypercube. It possesses many desirable properties for interconnection networks. In this paper, we study fault-tolerant embedding of paths in twisted cubes. Let _TQn(V,E)${\mathit{TQ}}_n(V,E)$ denote the n-dimensional twisted cube. We prove that a path of length l   can be embedded between any two distinct nodes with dilation 1 for any faulty set F⊂V(_TQn)∪E(_TQn)$F\subset V({\mathit{TQ}}_n)\cup E({\mathit{TQ}}_n)$ with |F|?n-3$\left|F\right|?n-3$ and any integer l   with 2^n-1-1?l?|V(_TQn-F)|-1$2^{n-1}-1?l?|V({\mathit{TQ}}_n-F)|-1$ (n?3$n?3$). This result is optimal in the sense that the embedding has the smallest dilation 1. The result is also complete in the sense that the two bounds on path length l   and faulty set size |F|$\left|F\right|$ for a successful embedding are tight. That is, the result does not hold if l?2^n-1-2$l?2^{n-1}-2$ or |F|?n-2$\left|F\right|?n-2$. We also extend the result on (n-3)$(n-3)$-Hamiltonian connectivity of _TQn${\mathit{TQ}}_n$ in the literature.     

Optimum sweeps of simple polygons with two guards

A polygon P admits a sweep if two mobile guards can detect an unpredictable, moving target inside P  , no matter how fast the target moves. Two guards move on the polygon boundary and are required to always be mutually visible. The objective of this study is to find an optimum sweep such that the sum of the distances travelled by the two guards in the sweep is minimized. We present an O(n²)$O(n^2)$ time and O(n)$O(n)$ space algorithm for optimizing this metric, where n   is the number of vertices of the given polygon. Our result is obtained by reducing this problem to finding a shortest path between two nodes in a graph of size O(n)$O(n)$.     

The worst-case time complexity for generating all maximal cliques and computational experiments

We present a depth-first search algorithm for generating all maximal cliques of an undirected graph, in which pruning methods are employed as in the Bron–Kerbosch algorithm. All the maximal cliques generated are output in a tree-like form. Subsequently, we prove that its worst-case time complexity is O(3^n/3)$\mathrm{O}(3^{n/3})$ for an n-vertex graph. This is optimal as a function of n  , since there exist up to 3^n/3$3^{n/3}$ maximal cliques in an n-vertex graph. The algorithm is also demonstrated to run very fast in practice by computational experiments.