Max-Leaves Spanning Tree is APX-hard for Cubic Graphs

We consider the problem of finding a spanning tree with maximum number of leaves (MaxLeaf). A 2-approximation algorithm is known for this problem, and a 3/2-approximation algorithm when restricted to graphs where every vertex has degree 3 (cubic graphs). MaxLeaf is known to be APX-hard in general, and NP-hard for cubic graphs. We show that the problem is also APX-hard for cubic graphs. The APX-hardness of the related problem Minimum Connected Dominating Set for cubic graphs follows

On the Complexity of the Single Individual SNP Haplotyping Problem

We present several new results pertaining to haplotyping. These results concern the combinatorial problem of reconstructing haplotypes from incomplete and/or imperfectly sequenced haplotype fragments. We consider the complexity of the problems Minimum Error Correction (MEC) and Longest Haplotype Reconstruction (LHR) for different restrictions on the input data. Specifically, we look at the gapless case, where every row of the input corresponds to a gapless haplotype-fragment, and the 1-gap case, where at most one gap per fragment is allowed. We prove that MEC is APX-hard in the 1-gap case and still NP-hard in the gapless case. In addition, we question earlier claims that MEC is NP-hard even when the input matrix is restricted to being completely binary. Concerning LHR, we show that this problem is NP-hard and APX-hard in the 1-gap case (and thus also in the general case), but is polynomial time solvable in the gapless case.Comment: 26 pages. Related to the WABI2005 paper, "On the Complexity of Several Haplotyping Problems", but with more/different results. This papers has just been submitted to the IEEE/ACM Transactions on Computational Biology and Bioinformatics and we are awaiting a decision on acceptance. It differs from the mid-August version of this paper because here we prove that 1-gap LHR is APX-hard. (In the earlier version of the paper we could prove only that it was NP-hard.

Flip Distance Between Triangulations of a Planar Point Set is APX-Hard

In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one edge and adding another one, such that the resulting graph is again a triangulation. Flips are a major way of locally transforming triangular meshes. We show that, given a point set $S$ in the Euclidean plane and two triangulations $T_1$ and $T_2$ of $S$, it is an APX-hard problem to minimize the number of edge flips to transform $T_1$ to $T_2$.Comment: A previous version only showed NP-completeness of the corresponding decision problem. The current version is the one of the accepted manuscrip

On Approximability of Steiner Tree in ℓp\ell_p-metrics

In the Continuous Steiner Tree problem (CST), we are given as input a set of points (called terminals) in a metric space and ask for the minimum-cost tree connecting them. Additional points (called Steiner points) from the metric space can be introduced as nodes in the solution. In the Discrete Steiner Tree problem (DST), we are given in addition to the terminals, a set of facilities, and any solution tree connecting the terminals can only contain the Steiner points from this set of facilities. Trevisan [SICOMP'00] showed that CST and DST are APX-hard when the input lies in the $\ell_1$-metric (and Hamming metric). Chleb\'ik and Chleb\'ikov\'a [TCS'08] showed that DST is NP-hard to approximate to factor of $96/95\approx 1.01$ in the graph metric (and consequently $\ell_\infty$-metric). Prior to this work, it was unclear if CST and DST are APX-hard in essentially every other popular metric! In this work, we prove that DST is APX-hard in every $\ell_p$-metric. We also prove that CST is APX-hard in the $\ell_{\infty}$-metric. Finally, we relate CST and DST, showing a general reduction from CST to DST in $\ell_p$-metrics. As an immediate consequence, this yields a $1.39$-approximation polynomial time algorithm for CST in $\ell_p$-metrics.Comment: Abstract shortened due to arxiv's requirement