The representative k-median problem generalizes the classical clustering formulations in that it partitions the data points into several disjoint demographic groups and poses a lower-bound constraint on the number of opened facilities from each group, such that all the groups are fairly represented by the opened facilities. Due to its simplicity, the local-search heuristic that optimizes an initial solution by iteratively swapping at most a constant number of closed facilities for the same number of opened ones (denoted by the O(1)-swap heuristic) has been frequently used in the representative k-median problem. Unfortunately, despite its good performance exhibited in experiments, whether the O(1)-swap heuristic has provable approximation guarantees for the case where the number of groups is more than 2 remains an open question for a long time. As an answer to this question, we show that the O(1)-swap heuristic (1) is guaranteed to yield a constant-factor approximation solution if the number of groups is a constant, and (2) has an unbounded approximation ratio otherwise. Our main technical contribution is a new approach for theoretically analyzing local-search heuristics, which derives the approximation ratio of the O(1)-swap heuristic via linearly combining the increased clustering costs induced by a set of hierarchically organized swaps.

Phylogenetic trees are unable to represent the evolutionary process for a collection of species if reticulation events happened, and a generalized model named phylogenetic network was introduced consequently. However, the representation of the evolutionary process for one gene is actually a phylogenetic tree that is ‘`contained’' in the phylogenetic network for the considered species containing the gene. Thus a fundamental computational problem named Tree Containment problem arises, which asks whether a phylogenetic tree is contained in a phylogenetic network. The previous research on the problem mainly focused on its rooted version of which the considered tree and network are rooted, and several algorithms were proposed when the considered network is binary or structure-restricted. There is almost no algorithm for its unrooted version except the recent fixed-parameter algorithm with runtime$O(4^kn^2)$, where k and n are the reticulation number and size of the considered unrooted binary phylogenetic network$N$, respectively. As the runtime is a little expensive when considering big values of k, we aim to improve it and successfully propose a fixed-parameter algorithm with runtime$O(2.594^kn^2)$in the paper. Additionally, we experimentally show its effectiveness on biological data and simulated data.

In 1990, Hendry conjectured that all Hamiltonian chordal graphs are cycle extendable. After a series of papers confirming the conjecture for a number of graph classes, the conjecture is yet refuted by Lafond and Seamone in 2015. Given that their counterexamples are not strongly chordal graphs and they are all only 2-connected, Lafond and Seamone asked the following two questions: (1) Are Hamiltonian strongly chordal graphs cycle extendable? (2) Is there an integer $k$ such that all $k$-connected Hamiltonian chordal graphs are cycle extendable? Later, a conjecture stronger than Hendry's is proposed. In this paper, we resolve all these questions in the negative. On the positive side, we add to the list of cycle-extendable graphs two more graph classes, namely, Hamiltonian 4-fan-free chordal graphs, where every induced $K_5 - e$ has true twins, and Hamiltonian $\{4{\sc -fan}, \overline{A} \}$-free chordal graphs.

Graph search, the process of visiting vertices in a graph in a specific order, has demonstrated magical powers in many important algorithms. But a systematic study was only initiated by Corneil et al. a decade ago, and only by then we started to realize how little we understand it. Even the apparently naïve question "which vertex can be the last visited by a graph search algorithm," known as the end vertex problem, turns out to be quite elusive. We give a full picture of all maximum cardinality searches on chordal graphs, which implies a polynomial-time algorithm for the end vertex problem of maximum cardinality search. It is complemented by a proof of NP-completeness of the same problem on weakly chordal graphs. We also show linear-time algorithms for deciding end vertices of breadth-first searches on interval graphs, and end vertices of lexicographic depth-first searches on chordal graphs. Finally, we present 2^n * n^O(1)-time algorithms for deciding the end vertices of breadth-first searches, depth-first searches, and maximum cardinality searches on general graphs.