Serge Gaspers

On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms

Abstract We present a time

Backdoors to satisfaction

\mathcal {O}(1.7548^{n})

On Independent Sets and Bicliques in Graphs

algorithm finding a minimum feedback vertex set in an undirected graph on n vertices. We also prove that a graph on n vertices can contain at most 1.8638n minimal feedback vertex sets and that there exist graphs having 105n/10≈1.5926n minimal feedback vertex sets.

Iterative Compression and Exact Algorithms

A backdoor set is a set of variables of a propositional formula such that fixing the truth values of the variables in the backdoor set moves the formula into some polynomial-time decidable class. If we know a small backdoor set we can reduce the question of whether the given formula is satisfiable to the same question for one or several easy formulas that belong to the tractable class under consideration. In this survey we review parameterized complexity results for problems that arise in the context of backdoor sets, such as the problem of finding a backdoor set of size at most k, parameterized by k. We also discuss recent results on backdoor sets for problems that are beyond NP.

An Exponential Time 2-Approximation Algorithm for Bandwidth

Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. In this paper we improve Prisner’s upper bound on the number of maximal bicliques (Combinatorica, 20, 109–117, 2000) and show that the maximum number of maximal bicliques in a graph on n vertices is Θ(3n/3). Our major contribution is an exact exponential-time algorithm. This branching algorithm computes the number of distinct maximal independent sets in a graph in time O(1.3642n), where n is the number of vertices of the input graph. We use this counting algorithm and previously known algorithms for various other problems related to independent sets to derive algorithms for problems related to bicliques via polynomial-time reductions.

Finding a minimum feedback vertex set in time O (1.7548 n )

Iterative compression has recently led to a number of breakthroughs in parameterized complexity. Here, we show that the technique can also be useful in the design of exact exponential time algorithms to solve NP-hard problems. We exemplify our findings with algorithms for the Maximum Independent Set problem, a parameterized and a counting version of d-Hitting Set and the Maximum Induced Cluster Subgraph problem.

Fair assignment of indivisible objects under ordinal preferences

The bandwidth of a graph G on n vertices is the minimum b such that the vertices of G can be labeled from 1 to n such that the labels of every pair of adjacent vertices differ by at most b. In this paper, we present a 2-approximation algorithm for the Bandwidth problem that takes worst-case

Strong Backdoors to Bounded Treewidth SAT

\mathcal{O}(1.9797^n)

Clean the graph before you draw it

= \mathcal{O}(3^{0.6217 n})