Searcharxiv⌕ Search

arXiv subjects

Mathias Hauptmann

Publications and source records attributed to Mathias Hauptmann.

9 recordsLinked to original sources
arXiv2022

Randomized Approximation Schemes for the Tutte Polynomial and Random Clustering in Subdense and Superdense Graphs

Extending the work of Alon, Frieze abnd Welsh, we show that there are randomized polynomial time approximation schemes for computing the Tutte polynomial in subdense graphs with an minimal node degree of $\Omega\left ( \frac{n}{\sqrt{\log n}}\right )$ . The same holds for the partition function $Z$ in the random cluster model with uniform edge probabilities and for the associated distribution $\lambda (A),\: A \subseteq E$ whenever the underlying graph $G=(V,E)$ is $c\cdot\frac{n}{\sqrt{\log (n)}}$-subdense. In the superdense case with node degrees $n-o(n)$, we show that the Tutte polynomial $T_G(x,y)$ is asymptotically equal to $Q=(x-1)(y-1)$. Moreover, we briefly discuss the problem of approximating $Z$ in the case of $(\alpha, \beta )$-power law graphs.

cs.DS↗
arXiv2016

On Alternation and the Union Theorem

Under the assumption $P=Σ_2^p$, we prove a new variant of the Union Theorem of McCreight and Meyer for the class $Σ_2^p$. This yields a union function $F$ which is computable in time $F(n)^c$ for some constant $c$ and satisfies $P=DTIME(F)=Σ_2(F)=Σ_2^p$ with respect to a subfamily $(\tilde{S}_i)$ of $Σ_2$-machines. We show that this subfamily does not change the complexity classes $P$ and $Σ_2^p$. Moreover, a padding construction shows that this also implies $DTIME(F^c)=Σ_2(F^c)$. On the other hand, we prove a variant of Gupta's result who showed that $DTIME(t)\subsetneqΣ_2(t)$ for time-constructible functions $t(n)$. Our variant of this result holds with respect to the subfamily $(\tilde{S}_i)$ of $Σ_2$-machines. We show that these two results contradict each other. Hence the assumption $P=Σ_2^p$ cannot hold.

cs.CC↗
arXiv2016

Approximation Complexity of Max-Cut on Power Law Graphs

In this paper we study the MAX-CUT problem on power law graphs (PLGs) with power law exponent $β$. We prove some new approximability results on that problem. In particular we show that there exist polynomial time approximation schemes (PTAS) for MAX-CUT on PLGs for the power law exponent $β$ in the interval $(0,2)$. For $β>2$ we show that for some $ε>0$, MAX-CUT is NP-hard to approximate within approximation ratio $1+ε$, ruling out the existence of a PTAS in this case. Moreover we give an approximation algorithm with improved constant approximation ratio for the case of $β>2$.

cs.DS↗
arXiv2015

Approximability of TSP on Power Law Graphs

In this paper we study the special case of Graphic TSP where the underlying graph is a power law graph (PLG). We give a refined analysis of some of the current best approximation algorithms and show that an improved approximation ratio can be achieved for certain ranges of the power law exponent $β$. For the value of power law exponent $β=1.5$ we obtain an approximation ratio of $1.34$ for Graphic TSP. Moreover we study the $(1,2)$-TSP with the underlying graph of $1$-edges being a PLG. We show improved approximation ratios in the case of underlying deterministic PLGs for $β$ greater than $1.666$. For underlying random PLGs we further improve the analysis and show even better expected approximation ratio for the range of $β$ between $1$ and $3.5$. On the other hand we prove the first explicit inapproximability bounds for $(1,2)$-TSP for an underlying power law graph.

cs.DS↗
arXiv2015

On the Approximability of Independent Set Problem on Power Law Graphs

We give the first nonconstant lower bounds for the approximability of the Independent Set Problem on the Power Law Graphs. These bounds are of the form $n^ε$ in the case when the power law exponent satisfies $β<1$. In the case when $β=1$, the lower bound is of the form $\log (n)^ε$. The embedding technique used in the proof could also be of independent interest.

cs.DS↗
arXiv2012

Inapproximability of Dominating Set in Power Law Graphs

We give logarithmic lower bounds for the approximability of the Minimum Dominating Set problem in connected (alpha,beta)-Power Law Graphs. We give also a best up to now upper approximation bound on the problem for the case of the parameters beta>2. We develop also a new functional method for proving lower approximation bounds and display a sharp phase transition between approximability and inapproximability of the underlying problem. This method could also be of independent interest.

cs.CC↗
arXiv2012

Efficient Parallel Computation of Nearest Neighbor Interchange Distances

The nni-distance is a well-known distance measure for phylogenetic trees. We construct an efficient parallel approximation algorithm for the nni-distance in the CRCW-PRAM model running in O(log n) time on O(n) processors. Given two phylogenetic trees T1 and T2 on the same set of taxa and with the same multi-set of edge-weights, the algorithm constructs a sequence of nni-operations of weight at most O(log n) \cdot opt, where opt denotes the minimum weight of a sequence of nni-operations transforming T1 into T2 . This algorithm is based on the sequential approximation algorithm for the nni-distance given by DasGupta et al. (2000). Furthermore, we show that the problem of identifying so called good edge-pairs between two weighted phylogenies can be computed in O(log n) time on O(n log n) processors.

cs.DS↗
arXiv2012

Approximability of the Vertex Cover Problem in Power Law Graphs

In this paper we construct an approximation algorithm for the Minimum Vertex Cover Problem (Min-VC) with an expected approximation ratio of 2-f(beta) for random Power Law Graphs (PLG) in the (alpha,beta)-model of Aiello et. al., where f(beta) is a strictly positive function of the parameter beta. We obtain this result by combining the Nemhauser and Trotter approach for Min-VC with a new deterministic rounding procedure which achieves an approximation ratio of 3/2 on a subset of low degree vertices for which the expected contribution to the cost of the associated linear program is sufficiently large.

cs.DS↗