Approximation algorithms for node-weighted prize-collecting Steiner tree problems on planar graphs

We study the prize-collecting version of the Node-weighted Steiner Tree problem (NWPCST) restricted to planar graphs. We give a new primal-dual Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for planar NWPCST. We then show a ($2.88 + \epsilon$)-approximation which establishes a new best approximation guarantee for planar NWPCST. This is done by combining our LMP algorithm with a threshold rounding technique and utilizing the 2.4-approximation of Berman and Yaroslavtsev for the version without penalties. We also give a primal-dual 4-approximation algorithm for the more general forest version using techniques introduced by Hajiaghay and Jain

On largest volume simplices and sub-determinants

We show that the problem of finding the simplex of largest volume in the convex hull of $n$ points in $\mathbb{Q}^d$ can be approximated with a factor of $O(\log d)^{d/2}$ in polynomial time. This improves upon the previously best known approximation guarantee of $d^{(d-1)/2}$ by Khachiyan. On the other hand, we show that there exists a constant $c>1$ such that this problem cannot be approximated with a factor of $c^d$, unless $P=NP$. % This improves over the $1.09$ inapproximability that was previously known. Our hardness result holds even if $n = O(d)$, in which case there exists a \bar c\,^{d}-approximation algorithm that relies on recent sampling techniques, where $\bar c$ is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a $d\times n$ matrix

Balanced Interval Coloring

We consider the discrepancy problem of coloring $n$ intervals with $k$ colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal. Somewhat surprisingly, a coloring with maximal difference at most one always exists. Furthermore, we give an algorithm with running time $O(n \log n + kn \log k)$ for its construction. This is in particular interesting because many known results for discrepancy problems are non-constructive. This problem naturally models a load balancing scenario, where $n$ tasks with given start- and endtimes have to be distributed among $k$ servers. Our results imply that this can be done ideally balanced. When generalizing to $d$-dimensional boxes (instead of intervals), a solution with difference at most one is not always possible. We show that for any $d \ge 2$ and any $k \ge 2$ it is NP-complete to decide if such a solution exists, which implies also NP-hardness of the respective minimization problem. In an online scenario, where intervals arrive over time and the color has to be decided upon arrival, the maximal difference in the size of color classes can become arbitrarily high for any online algorithm.Comment: Accepted at STACS 201

La obra poética de Hernando Colón

16 p.Cultivó también Hernando Colón, como nos cuentan sus biógrafos, la poesía. Poesías sobre las que se emiten juicios en general poco favorables y siempre de segunda mano, ya que los autores que se han ocupado de su obra confiesan desconocerlas o se limitan a juzgarlas por un par de ejemplos aislados. Por ello nos ha parecido oportuno reunir en este trabajo el corpus de poesías a él atribuidas: 18 poemas en castellano y uno en latín. De esta forma pensamos que quizá se atraerá la atención sobre esta actividad del cordobés, actividad que nunca ha sido objeto de interés para los estudiosos de nuestra literatura del siglo XVI y que ha sido pasada por alto por nuestros colombinistas.Peer reviewe

Primal-dual Approximation Algorithms for Node-Weighted Steiner Forest on Planar Graphs

NODE-WEIGHTED STEINER FOREST is the following problem: Given an undirected graph, a set of pairs of terminal vertices, a weight function on the vertices, find a minimum weight set of vertices that includes and connects each pair of terminals. We consider the restriction to planar graphs where the problem remains NP-complete. Demaine et al. showed that the generic primal-dual algorithm of Goemans and Williamson is a 6-approximation on planar graphs. We present (1) two different analyses to prove an approximation factor of 3, (2) show that our analysis is best possible for the chosen proof strategy, and (3) generalize this result to feedback problems on planar graphs. We give a simple proof for the first result using contraction techniques and following a standard proof strategy for the generic primal-dual algorithm. Given this proof strategy our analysis is best possible which implies that proving a better upper bound for this algorithm, if possible, would require different proof methods. Then, we give a reduction on planar graphs of FEEDBACK VERTEX SET to NODE-WEIGHTED STEINER TREE, and SUBSET FEEDBACK VERTEX SET to NODE-WEIGHTED STEINER FOREST. This generalizes our result to the feedback problems studied by Goemans and Williamson. For the opposite direction, we show how our constructions can be combined with the proof idea for the feedback problems to yield an alternative proof of the same approximation guarantee for NODE-WEIGHTED STEINER FOREST. (C) 2012 Elsevier Inc. All rights reserved

Node-weighted network design and maximum sub-determinants

We consider the Node-weighted Steiner Forest problem on planar graphs. Demaine et al. showed that a generic primal-dual algorithm gives a 6-approximation. We present two different proofs of an approximation factor of~$3$. Then, we draw a connection to Goemans and Williamson who apply the primal-dual algorithm to feedback problems on planar graphs. We reduce these problems to Node-weighted Steiner Forest and show that for the graphs in the reductions their respective linear programming relaxations are equivalent to each other. This explains why both type of problems can be approximated with primal-dual methods. We show that the largest sub-determinant of a matrix $A\in Q^{d\times n}$ can be approximated with a factor of $O(\log d)^{d/2}$ in polynomial time. This problem also subsumes the problem of finding a simplex of largest volume in the convex hull of $n$ points in $Q^d$ for which we obtain the same approximation guarantee. The previously best known approximation guarantee for both problems was $d^{(d-1)/2}$ by Khachiyan. We further show that it is NP-hard to approximate both problems with a factor of~$c^d$ for some explicit constant~$c$. We highlight the importance of sub-determinants in combinatorial optimization by showing their significance in two problems. First, the Stable Set problem asks for a maximum cardinality set of pairwise non-adjacent vertices. The problem is NP-hard to approximate with factor $n^{1-\epsilon}$ for any $\epsilon>0$, where $n$ is the number of vertices. We restrict to instances where the sub-determinants of the constraint matrix are bounded by a function in~$n$. This is equivalent to restricting to graphs with bounded odd cycle packing number $ocp$, the maximum number of vertex-disjoint odd cycles in the graph. We obtain a polynomial-time approximation scheme for graphs with $ocp=o(n/\log n)$. Further, we obtain an $\alpha$-approximation algorithm for general graphs where~$\alpha$ smoothly increases from a constant to $n$ as $ocp$ grows from $O(n/\log n)$ to $n/3$. In contrast, we show that assuming the exponential-time hypothesis, Stable Set cannot be solved in polynomial time if $ocp=\Omega(\log^{1+\epsilon} n)$ for some $\epsilon>0$. Second, we consider coloring $n$ intervals with $k$ colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is at most one. This problem induces a totally unimodular constraint matrix and is therefore efficiently solvable. We construct a fast algorithm with running time $O(n \log n + kn\log k)$