Topological Additive Numbering of Directed Acyclic Graphs

We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let $D$ be a digraph and $f$ a labeling of its vertices with positive integers; denote by $S(v)$ the sum of labels over all neighbors of each vertex $v$. The labeling $f$ is called \emph{topological additive numbering} if $S(u) < S(v)$ for each arc $(u,v)$ of the digraph. The problem asks to find the minimum number $k$ for which $D$ has a topological additive numbering with labels belonging to $\{ 1, \ldots, k \}$, denoted by $\eta_t(D)$. We characterize when a digraph has topological additive numberings, give a lower bound for $\eta_t(D)$, and provide an integer programming formulation for our problem, characterizing when its coefficient matrix is totally unimodular. We also present some families for which $\eta_t(D)$ can be computed in polynomial time. Finally, we prove that this problem is \np-Hard even when its input is restricted to planar bipartite digraphs

The inapproximability for the (0,1)-additive number

An {\it additive labeling} of a graph $G$ is a function $\ell :V(G) \rightarrow\mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$ ($x \sim y$ means that $x$ is joined to $y$). The {\it additive number} of $G$, denoted by $\eta(G)$, is the minimum number $k$ such that $G$ has a additive labeling $\ell :V(G) \rightarrow \mathbb{N}_k$. The {\it additive choosability} of a graph $G$, denoted by $\eta_{\ell}(G)$, is the smallest number $k$ such that $G$ has an additive labeling for any assignment of lists of size $k$ to the vertices of $G$, such that the label of each vertex belongs to its own list. Seamone (2012) \cite{a80} conjectured that for every graph $G$, $\eta(G)= \eta_{\ell}(G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $\eta_{\ell}(G)- \eta(G) \geq k$. A {\it $(0,1)$-additive labeling} of a graph $G$ is a function $\ell :V(G) \rightarrow\{0,1\}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$. A graph may lack any $(0,1)$-additive labeling. We show that it is $\mathbf{NP}$-complete to decide whether a $(0,1)$-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive number of $G$ is defined as $\sigma_{1} (G) = \min_{\ell \in \Gamma}\sum_{v\in V(G)}\ell(v)$ where $\Gamma$ is the set of $(0,1)$-additive labelings of $G$. We prove that given a planar graph that admits a $(0,1)$-additive labeling, for all $\varepsilon >0$, approximating the $(0,1)$-additive number within $n^{1-\varepsilon}$ is $\mathbf{NP}$-hard.Comment: 14 pages, 3 figures, Discrete Mathematics & Theoretical Computer Scienc

Sigma Partitioning: Complexity and Random Graphs

A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors in $P_i$. The $\textit{ sigma number}$ of a graph $G$, denoted by $\sigma(G)$, is the minimum number $k$ such that $G$ has a sigma partitioning $P_1, \ldots, P_k$. Also, a $\textit{ lucky labeling}$ of a graph $G$ is a function $\ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$ ($x \sim y$ means that $x$ and $y$ are adjacent). The $\textit{ lucky number}$ of $G$, denoted by $\eta(G)$, is the minimum number $k$ such that $G$ has a lucky labeling $\ell :V(G) \rightarrow \mathbb{N}_k$. It was conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is $\mathbf{NP}$-complete to decide whether $\eta(G)=2$ for a given 3-regular graph $G$. In this work, we prove this conjecture. Among other results, we give an upper bound of five for the sigma number of a uniformly random graph

Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm

Many practical problems in almost all scientific and technological disciplines have been classified as computationally hard (NP-hard or even NP-complete). In life sciences, combinatorial optimization problems frequently arise in molecular biology, e.g., genome sequencing; global alignment of multiple genomes; identifying siblings or discovery of dysregulated pathways.In almost all of these problems, there is the need for proving a hypothesis about certain property of an object that can be present only when it adopts some particular admissible structure (an NP-certificate) or be absent (no admissible structure), however, none of the standard approaches can discard the hypothesis when no solution can be found, since none can provide a proof that there is no admissible structure. This article presents an algorithm that introduces a novel type of solution method to "efficiently" solve the graph 3-coloring problem; an NP-complete problem. The proposed method provides certificates (proofs) in both cases: present or absent, so it is possible to accept or reject the hypothesis on the basis of a rigorous proof. It provides exact solutions and is polynomial-time (i.e., efficient) however parametric. The only requirement is sufficient computational power, which is controlled by the parameter $\alpha\in\mathbb{N}$. Nevertheless, here it is proved that the probability of requiring a value of $\alpha>k$ to obtain a solution for a random graph decreases exponentially: $P(\alpha>k) \leq 2^{-(k+1)}$, making tractable almost all problem instances. Thorough experimental analyses were performed. The algorithm was tested on random graphs, planar graphs and 4-regular planar graphs. The obtained experimental results are in accordance with the theoretical expected results.Comment: Working pape

Improved Local Search for Geometric Hitting Set

International audienceOver the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set D of geometric objects, compute the minimum-sized subset of P that hits all objects in D. For the case where D is a set of disks in the plane, the 30-year quest for a PTAS, starting from the seminal work of Hochbaum [19], was finally achieved in 2010. Surprisingly, the algorithm to achieve the PTAS is simple: local-search. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others). Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with hopeless running times; e.g., a 3-approximation for the geometric hitting takes more than n 66 time [15] for the geometric hitting set problem for disks in the plane! That leaves open the question of whether a better understanding – both combinatorial and algorithmic – of local search and the problem can give a better approximation ratio in a more reasonable time. In this paper, we investigate this question for one of the fundamental problems, hitting sets for disks in the plane. We present tight approximation bounds for (3, 2)-local search 1 and give an (8 + algorithm with running time O(n 2.34); the previous-best result achieving a similar approximation ratio gave a 10-approximation in time O(n 15) – that too just for unit disks. The techniques and ideas generalize to (4, 3) local search. Furthermore, as mentioned earlier, local-search has been used for several other geometric optimization problems; for all these problems our results show that (3, 2) local search gives an 8-approximation and no better 2. Similarly (4, 3)-local search gives a 5-approximation for all these problems