Searcharxiv⌕ Search

arXiv subjects

Jens Vygen

Publications and source records attributed to Jens Vygen.

25 records · Page 2Linked to original sources
arXiv2017

Few Sequence Pairs Suffice: Representing All Rectangle Placements

We consider representations of general non-overlapping placements of rectangles by spatial relations (west, south, east, north) of pairs of rectangles. We call a set of representations complete if it contains a representation of every placement of $n$ rectangles. We prove a new upper bound of $\mathcal{O}(\frac{n!}{n^6} \cdot (\frac{11+5 \sqrt 5}{2})^n)$ and a new lower bound of $Ω(\frac{n!}{n^4} \cdot (4 + 2 \sqrt2)^n)$ on the minimum cardinality of complete sets of representations. A key concept in the proofs of these results are pattern-avoiding permutations. The new upper bound directly improves upon the well-known sequence pair representation, which has size $(n!)^2$, by only considering a restricted set of sequence pairs. It implies theoretically faster algorithms for VLSI placement problems.

math.CO↗
arXiv2017

On the Integrality Gap of the Prize-Collecting Steiner Forest LP

In the prize-collecting Steiner forest (PCSF) problem, we are given an undirected graph $G=(V,E)$, edge costs $\{c_e\geq 0\}_{e\in E}$, terminal pairs $\{(s_i,t_i)\}_{i=1}^k$, and penalties $\{π_i\}_{i=1}^k$ for each terminal pair; the goal is to find a forest $F$ to minimize $c(F)+\sum_{i: (s_i,t_i)\text{ not connected in }F}π_i$. The Steiner forest problem can be viewed as the special case where $π_i=\infty$ for all $i$. It was widely believed that the integrality gap of the natural (and well-studied) linear-programming (LP) relaxation for PCSF is at most 2. We dispel this belief by showing that the integrality gap of this LP is at least $9/4$. This holds even for planar graphs. We also show that using this LP, one cannot devise a Lagrangian-multiplier-preserving (LMP) algorithm with approximation guarantee better than $4$. Our results thus show a separation between the integrality gaps of the LP-relaxations for prize-collecting and non-prize-collecting (i.e., standard) Steiner forest, as well as the approximation ratios achievable relative to the optimal LP solution by LMP- and non-LMP- approximation algorithms for PCSF. For the special case of prize-collecting Steiner tree (PCST), we prove that the natural LP relaxation admits basic feasible solutions with all coordinates of value at most $1/3$ and all edge variables positive. Thus, we rule out the possibility of approximating PCST with guarantee better than $3$ using a direct iterative rounding method.

cs.DM↗
arXiv2016

Reassembling trees for the traveling salesman

Many recent approximation algorithms for different variants of the traveling salesman problem (asymmetric TSP, graph TSP, s-t-path TSP) exploit the well-known fact that a solution of the natural linear programming relaxation can be written as convex combination of spanning trees. The main argument then is that randomly sampling a tree from such a distribution and then completing the tree to a tour at minimum cost yields a better approximation guarantee than simply taking a minimum cost spanning tree (as in Christofides' algorithm). We argue that an additional step can help: reassembling the spanning trees before sampling. Exchanging two edges in a pair of spanning trees can improve their properties under certain conditions. We demonstrate the usefulness for the metric s-t-path TSP by devising a deterministic polynomial-time algorithm that improves on Sebő's previously best approximation ratio of 8/5.

cs.DM↗
arXiv2015

Better $s$-$t$-Tours by Gao Trees

We consider the $s$-$t$-path TSP: given a finite metric space with two elements $s$ and $t$, we look for a path from $s$ to $t$ that contains all the elements and has minimum total distance. We improve the approximation ratio for this problem from 1.599 to 1.566. Like previous algorithms, we solve the natural LP relaxation and represent an optimum solution $x^*$ as a convex combination of spanning trees. Gao showed that there exists a spanning tree in the support of $x^*$ that has only one edge in each narrow cut (i.e., each cut $C$ with $x^*(C)<2$). Our main theorem says that the spanning trees in the convex combination can be chosen such that many of them are such "Gao trees''.

cs.DM↗
arXiv2015

Dijkstra meets Steiner: a fast exact goal-oriented Steiner tree algorithm

We present a new exact algorithm for the Steiner tree problem in edge-weighted graphs. Our algorithm improves the classical dynamic programming approach by Dreyfus and Wagner. We achieve a significantly better practical performance via pruning and future costs, a generalization of a well-known concept to speed up shortest path computations. Our algorithm matches the best known worst-case run time and has a fast, often superior, practical performance: on some large instances originating from VLSI design, previous best run times are improved upon by orders of magnitudes. We are also able to solve larger instances of the $d$-dimensional rectilinear Steiner tree problem for $d \in \{3, 4, 5\}$, whose Hanan grids contain up to several millions of edges.

cs.DS↗
arXiv2012

Shorter Tours by Nicer Ears: 7/5-approximation for graphic TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs

We prove new results for approximating the graphic TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graphic TSP itself, we improve the approximation ratio to 7/5. For a generalization, the connected-$T$-join problem, we obtain the first nontrivial approximation algorithm, with ratio 3/2. This contains the graphic $s$-$t$-path-TSP as a special case. Our improved approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4/3. The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios.

cs.DM↗