On the k-restricted structure ratio in planar and outerplanar graphs

Graphs and Algorithm

Noname manuscript No. (will be inserted by the editor) Maximum Series-Parallel Subgraph

Abstract Consider the NP-hard problem of, given a simple graph G, to find a seriesparallel subgraph of G with the maximum number of edges. The algorithm that, given a connected graph G, outputs a spanning tree of G, is a 1 2-approximation. Indeed, if n is the number of vertices in G, any spanning tree in G has n−1 edges and any seriesparallel graph on n vertices has at most 2n−3 edges. We present a 7 12-approximation for this problem and results showing the limits of our approach

Reconfiguration of list edge-colorings in a graph

11th International Symposium, WADS 2009, Banff, Canada, August 21-23, 2009. ProceedingsWe study the problem of reconfiguring one list edge-coloring of a graph into another list edge-coloring by changing one edge color at a time, while at all times maintaining a list edge-coloring, given a list of allowed colors for each edge. First we show that this problem is PSPACE-complete, even for planar graphs of maximum degree 3 and just six colors. Then we consider the problem restricted to trees. We show that any list edge-coloring can be transformed into any other under the sufficient condition that the number of allowed colors for each edge is strictly larger than the degrees of both its endpoints. This sufficient condition is best possible in some sense. Our proof yields a polynomial-time algorithm that finds a transformation between two given list edge-colorings of a tree with n vertices using O(n [superscript 2]) recolor steps. This worst-case bound is tight: we give an infinite family of instances on paths that satisfy our sufficient condition and whose reconfiguration requires Ω(n [superscript 2]) recolor steps

On reconfiguration of disks in the plane and related problems

We revisit two natural reconfiguration models for systems of disjoint objects in the plane: translation and sliding. Consider a set of n pairwise interior-disjoint objects in the plane that need to be brought from a given start (initial) configuration S into a desired goal (target) configuration T, without causing collisions. In the translation model, in one move an object is translated along a fixed direction to another position in the plane. In the sliding model, one move is sliding an object to another location in the plane by means of an arbitrarily complex continuous motion (that could involve rotations). We obtain various combinatorial and computational results for these two models: (I) For systems of n congruent disks in the translation model, Abellanas et al. showed that 2n − 1 moves always suffice and ⌊8n/5 ⌋ moves are sometimes necessary for transforming the start configuration into the target configuration. Here we further improve the lower bound to ⌊5n/3 ⌋ − 1, and thereby give a partial answer to one of their open problems. (II) We show that the reconfiguration problem with congruent disks in the translation model is NPhard, in both the labeled and unlabeled variants. This answers another open problem of Abellanas et al. (III) We also show that the reconfiguration problem with congruent disks in the sliding model is NP-hard, in both the labeled and unlabeled variants. (IV) For the reconfiguration with translations of n arbitrary convex bodies in the plane, 2n moves are always sufficient and sometimes necessary

Geometric Multicut: Shortest Fences for Separating Groups of Objects in the Plane

We study the following separation problem: Given a collection of pairwise disjoint coloured objects in the plane with k different colours, compute a shortest “fence” F, i.e., a union of curves of minimum total length, that separates every pair of objects of different colours. Two objects are separated if F contains a simple closed curve that has one object in the interior and the other in the exterior. We refer to the problem as GEOMETRIC k-CUT, as it is a geometric analog to the well-studied multicut problem on graphs. We first give an O(n4log3n)-time algorithm that computes an optimal fence for the case where the input consists of polygons of two colours with n corners in total. We then show that the problem is NP-hard for the case of three colours. Finally, we give a randomised 4/3⋅1.2965-approximation algorithm for polygons and any number of colours

Carbon materials from conventional/unconventional technologies for electrochemical energy storage devices

In the last years our society has shown a growing interest on the development of both new sources of clean energy and advanced devices able to store it. In this context supercapacitors (SCs) and hybrid systems have emerged to cover the power and energy demands. Most of these electrochemical devices use carbon materials as electrodes being the activated carbons (ACs) the most commonly ones. Nonetheless graphene (G) has emerged as a promising electrode either by itself or combined with ACs in composites. This work investigates the use of a low added value coal-derived liquid (anthracene oil, AO) for the production of pitch-like carbon precursors to synthesize suitable active electrode materials (ACs, G, AC/G) in SCs and hybrid systems. In addition to the well-known oxidative thermal polymerization of AO, a new alternative based on the use of microwave heating is presented as a promising clean route to obtain such carbon precursors resulting in energy saving, shortening time and specific nonthermal effects. The characteristics of the carbon materials obtained from both conventional/ unconventional technologies are compared mainly in terms of their specific surface area, surface chemistry and electrical conductivity which would allow the design of energy storage devices with an improved electrochemical performance

Hardness and approximation for the geodetic set problem in some graph classes

In this paper, we study the computational complexity of finding the \emph{geodetic number} of graphs. A set of vertices $S$ of a graph $G$ is a \emph{geodetic set} if any vertex of $G$ lies in some shortest path between some pair of vertices from $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality. In this paper, we prove that solving the \textsc{MGS} problem is NP-hard on planar graphs with a maximum degree six and line graphs. We also show that unless $P=NP$, there is no polynomial time algorithm to solve the \textsc{MGS} problem with sublogarithmic approximation factor (in terms of the number of vertices) even on graphs with diameter $2$. On the positive side, we give an $O\left(\sqrt[3]{n}\log n\right)$-approximation algorithm for the \textsc{MGS} problem on general graphs of order $n$. We also give a $3$-approximation algorithm for the \textsc{MGS} problem on the family of solid grid graphs which is a subclass of planar graphs

On the Discrete Unit Disk Cover Problem

Abstract. Given a set P of n points and a set D of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in P is covered by at least one disk in D or not and (ii) if so, then find a minimum cardinality subset D ∗ ⊆ D such that unit disks in D ∗ cover all the points in P. The discrete unit disk cover problem is a geometric version of the general set cover problem which is NP-hard [14]. The general set cover problem is not approx-imable within c log |P|, for some constant c, but the DUDC problem was shown to admit a constant factor approximation. In this paper, we pro-vide an algorithm with constant approximation factor 18. The running time of the proposed algorithm is O(n log n+m logm+mn). The previ-ous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time O(m2n4).

A Comparative Study of Modern Inference Techniques for Structured Discrete Energy Minimization Problems

International audienceSzeliski et al. published an influential study in 2006 on energy minimization methods for Markov Random Fields (MRF). This study provided valuable insights in choosing the best optimization technique for certain classes of problems. While these insights remain generally useful today, the phenomenal success of random field models means that the kinds of inference problems that have to be solved changed significantly. Specifically , the models today often include higher order interactions, flexible connectivity structures, large label-spaces of different car-dinalities, or learned energy tables. To reflect these changes, we provide a modernized and enlarged study. We present an empirical comparison of more than 27 state-of-the-art optimization techniques on a corpus of 2,453 energy minimization instances from diverse applications in computer vision. To ensure reproducibility, we evaluate all methods in the OpenGM 2 framework and report extensive results regarding runtime and solution quality. Key insights from our study agree with the results of Szeliski et al. for the types of models they studied. However, on new and challenging types of models our findings disagree and suggest that polyhedral methods and integer programming solvers are competitive in terms of runtime and solution quality over a large range of model types