Abstract:
We revisit the exact shortest unique substring (SUS) finding problem, and propose its approximate version where mismatches are allowed, due to its applications in subfields such as computational biology. We design a generic in-place framework that fits to solve both the exact and approximate $k$-mismatch SUS finding, using the minimum $2n$ memory words plus $n$ bytes space, where $n$ is the input string size. By using the in-place framework, we can find the exact and approximate $k$-mismatch SUS for every string position using a total of $O(n)$ and $O(n^2)$ time, respectively, regardless of the value of $k$. Our framework does not involve any compressed or succinct data structures and thus is practical and easy to implement.

Abstract:
Let $\D = $$ \{d_1,d_2,...d_D\}$ be a given set of $D$ string documents of total length $n$, our task is to index $\D$, such that the $k$ most relevant documents for an online query pattern $P$ of length $p$ can be retrieved efficiently. We propose an index of size $|CSA|+n\log D(2+o(1))$ bits and $O(t_{s}(p)+k\log\log n+poly\log\log n)$ query time for the basic relevance metric \emph{term-frequency}, where $|CSA|$ is the size (in bits) of a compressed full text index of $\D$, with $O(t_s(p))$ time for searching a pattern of length $p$ . We further reduce the space to $|CSA|+n\log D(1+o(1))$ bits, however the query time will be $O(t_s(p)+k(\log \sigma \log\log n)^{1+\epsilon}+poly\log\log n)$, where $\sigma$ is the alphabet size and $\epsilon >0$ is any constant.

Abstract:
We present several results about position heaps, a relatively new alternative to suffix trees and suffix arrays. First, we show that, if we limit the maximum length of patterns to be sought, then we can also limit the height of the heap and reduce the worst-case cost of insertions and deletions. Second, we show how to build a position heap in linear time independent of the size of the alphabet. Third, we show how to augment a position heap such that it supports access to the corresponding suffix array, and vice versa. Fourth, we introduce a variant of a position heap that can be simulated efficiently by a compressed suffix array with a linear number of extra bits.

Abstract:
Let $k \in \mathbb{N}$ and let $G$ be a graph. A function $f: V(G) \rightarrow 2^{[k]}$ is a rainbow function if, for every vertex $x$ with $f(x)=\emptyset$, $f(N(x)) =[k]$. The rainbow domination number $\gamma_{kr}(G)$ is the minimum of $\sum_{x \in V(G)} |f(x)|$ over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs.

Abstract:
The number of the non-shared edges of two phylogenies is a basic measure of the dissimilarity between the phylogenies. The non-shared edges are also the building block for approximating a more sophisticated metric called the nearest neighbor interchange (NNI) distance. In this paper, we give the first subquadratic-time algorithm for finding the non-shared edges, which are then used to speed up the existing approximating algorithm for the NNI distance from $O(n^2)$ time to $O(n \log n)$ time. Another popular distance metric for phylogenies is the subtree transfer (STT) distance. Previous work on computing the STT distance considered degree-3 trees only. We give an approximation algorithm for the STT distance for degree-$d$ trees with arbitrary $d$ and with generalized STT operations.

Abstract:
Solitaire {\sc Flood-it}, or {\sc Honey-Bee}, is a game played on a colored graph. The player resides in a source vertex. Originally his territory is the maximal connected, monochromatic subgraph that contains the source. A move consists of calling a color. This conquers all the nodes of the graph that can be reached by a monochromatic path of that color from the current territory of the player. It is the aim of the player to add all vertices to his territory in a minimal number of moves. We show that the minimal number of moves can be computed in polynomial time when the game is played on AT-free graphs.

Abstract:
We study convexity properties of graphs. In this paper we present a linear-time algorithm for the geodetic number in tree-cographs. Settling a 10-year-old conjecture, we prove that the Steiner number is at least the geodetic number in AT-free graphs. Computing a maximal and proper monophonic set in $\AT$-free graphs is NP-complete. We present polynomial algorithms for the monophonic number in permutation graphs and the geodetic number in $P_4$- sparse graphs.

Abstract:
We show that there are polynomial-time algorithms to compute maximum independent sets in the categorical products of two cographs and two splitgraphs. The ultimate categorical independence ratio of a graph G is defined as lim_{k --> infty} \alpha(G^k)/n^k. The ultimate categorical independence ratio is polynomial for cographs, permutation graphs, interval graphs, graphs of bounded treewidth and splitgraphs. When G is a planar graph of maximal degree three then alpha(G \times K_4) is NP-complete. We present a PTAS for the ultimate categorical independence ratio of planar graphs. We present an O^*(n^{n/3}) exact, exponential algorithm for general graphs. We prove that the ultimate categorical independent domination ratio for complete multipartite graphs is zero, except when the graph is complete bipartite with color classes of equal size (in which case it is 1/2).

Abstract:
Let G be a graph. The independence-domination number is the maximum over all independent sets I in G of the minimal number of vertices needed to dominate I. In this paper we investigate the computational complexity of independence domination for graphs in several graph classes related to cographs. We present an exact exponential algorithm. We also present a PTAS for planar graphs.

Abstract:
A function f: V \rightarrow \{-1,0,1\} is a minus-domination function of a graph G=(V,E) if the values over the vertices in each closed neighborhood sum to a positive number. The weight of f is the sum of f(x) over all vertices x \in V. The minus-domination number \gamma^{-}(G) is the minimum weight over all minus-domination functions. The size of a minus domination is the number of vertices that are assigned 1. In this paper we show that the minus-domination problem is fixed-parameter tractable for d-degenerate graphs when parameterized by the size of the minus-dominating set and by d. The minus-domination problem is polynomial for graphs of bounded rankwidth and for strongly chordal graphs. It is NP-complete for splitgraphs. Unless P=NP there is no fixed-parameter algorithm for minus-domination. 79,1 5%