91 research outputs found
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
The Quest for Strong Inapproximability Results with Perfect Completeness
The Unique Games Conjecture (UGC) has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the UGC. For the important case when the input CSP instance admits a satisfying assignment, it therefore remains wide open to understand how well it can be approximated.
This work is motivated by the pursuit of a better understanding of the inapproximability of perfectly satisfiable instances of CSPs. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover which we call "V label cover." Assuming a conjecture concerning the inapproximability of V label cover on perfectly satisfiable instances, we prove the following implications:
* There is an absolute constant c0 such that for k >= 3, given a satisfiable instance of Boolean k-CSP, it is hard to find an assignment satisfying more than c0 k^2/2^k fraction of the constraints.
* Given a k-uniform hypergraph, k >= 2, for all epsilon > 0, it is hard to tell if it is q-strongly colorable or has no independent set with an epsilon fraction of vertices, where q = ceiling[k + sqrt(k) - 0.5].
* Given a k-uniform hypergraph, k >= 3, for all epsilon > 0, it is hard to tell if it is (k-1)-rainbow colorable or has no independent set with an epsilon fraction of vertices.
We further supplement the above results with a proof that an ``almost Unique\u27\u27 version of Label Cover can be approximated within a constant factor on satisfiable instances
New Tools and Connections for Exponential-Time Approximation
In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and an integer r>1, and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of
1.
r for maximum independent set in Oâ(exp(O~(n/rlog2r+rlog2r)))
time,
2.
r for chromatic number in Oâ(exp(O~(n/rlogr+rlog2r)))
time,
3.
(2â1/r)
for minimum vertex cover in Oâ(exp(n/rΩ(r)))
time, and
4.
(kâ1/r)
for minimum k-hypergraph vertex cover in Oâ(exp(n/(kr)Ω(kr)))
time.
(Throughout, O~
and Oâ omit polyloglog(r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were Oâ(2n/r) (Bourgeois et al. in Discret Appl Math 159(17):1954â1970, 2011; Cygan et al. in Exponential-time approximation of hard problems, 2008). For maximum independent set and chromatic number, these bounds were complemented by exp(n1âo(1)/r1+o(1)) lower bounds (under the Exponential Time Hypothesis (ETH)) (Chalermsook et al. in Foundations of computer science, FOCS, pp. 370â379, 2013; Laekhanukit in Inapproximability of combinatorial problems in subexponential-time. Ph.D. thesis, 2014). Our results show that the naturally-looking Oâ(2n/r) bounds are not tight for all these problems. The key to these results is a sparsification procedure that reduces a problem to a bounded-degree variant, allowing the use of approximation algorithms for bounded-degree graphs. To obtain the first two results, we introduce a new randomized branching rule. Finally, we show a connection between PCP parameters and exponential-time approximation algorithms. This connection together with our independent set algorithm refute the possibility to overly reduce the size of Chanâs PCP (Chan in J. ACM 63(3):27:1â27:32, 2016). It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture (Dinur in Electron Colloq Comput Complex (ECCC) 23:128, 2016; Manurangsi and Raghavendra in A birthday repetition theorem and complexity of approximating dense CSPs, 2016)
A Tight Approximation Algorithm for the Cluster Vertex Deletion Problem
We give the first -approximation algorithm for the cluster vertex deletion
problem. This is tight, since approximating the problem within any constant
factor smaller than is UGC-hard. Our algorithm combines the previous
approaches, based on the local ratio technique and the management of true
twins, with a novel construction of a 'good' cost function on the vertices at
distance at most from any vertex of the input graph.
As an additional contribution, we also study cluster vertex deletion from the
polyhedral perspective, where we prove almost matching upper and lower bounds
on how well linear programming relaxations can approximate the problem.Comment: 23 pages, 3 figure
Randomisierte Approximation fĂŒr das Matching- und KnotenĂŒberdeckung Problem in Hypergraphen: KomplexitĂ€t und Algorithmen
This thesis studies the design and mathematical analysis of randomized approximation algorithms for the hitting set and b-matching problems in hypergraphs.
We present a randomized algorithm for the hitting set problem based on linear programming. The analysis of the randomized algorithm rests upon the probabilistic method, more precisely on some concentration inequalities for the sum of independent random variables plus some martingale based inequalities, as the bounded difference inequality, which is a derived from Azuma inequality.
In combination with combinatorial arguments we achieve some new results for different instance classes that improve upon the known approximation results for the problem (Krevilevich (1997), Halperin (2001)).
We analyze the complexity of the b-matching problem in hypergraphs and obtain two new results.
We give a polynomial time reduction from an instance of a suitable problem to an instance of the b-matching problem and prove a non-approximability ratio for the problem in l-uniform hypergraphs. This generalizes the result of Safra et al. (2006) from b=1 to b in O(l/log(l)). Safra et al. showed that the 1-matching problem in l-uniform hypergraphs can not be approximated in polynomial time within a ratio O(l/log(l)), unless P = NP.
Moreover, we show that the b-matching problem on l-uniform
hypergraphs with bounded vertex degree has no polynomial time
approximation scheme PTAS, unless P=NP.Diese Arbeit befasst sich mit dem Entwurf und der mathematischen
Analyse von randomisierten Approximationsalgorithmen fĂŒr das Hitting
Set Problem und das b-Matching Problem in Hypergraphen.
Zuerst prĂ€sentieren wir einen randomisierten Algorithmus fĂŒr das
Hitting Set Problem, der auf linearer Programmierung basiert. Mit
diesem Verfahren und einer Analyse, die auf der probabilistischen
Methode fuĂt, erreichen wir fĂŒr verschiedene Klassen von Instanzen
drei neue ApproximationsgĂŒten, die die bisher bekannten Ergebnisse
(Krevilevich [1997], Halperin [2001]) fĂŒr das Problem verbessern. Die Analysen beruhen auf Konzentrationsungleichungen fĂŒr Summen von
unabhÀngigen Zufallsvariablen aber auch Martingal-basierten Ungleichungen, wie die aus der Azuma-Ungleichung abgeleitete Bounded
Difference-Inequality, in Kombination mit kombinatorischen Argumenten.
FĂŒr das b-Matching Problem in Hypergraphen analysieren wir zunĂ€chst
seine KomplexitÀt und erhalten zwei neue Ergebnisse.
Wir geben eine polynomielle Reduktion von einer Instanz eines geeigneten Problems zu einer Instanz des b-Matching-Problems an und
zeigen ein Nicht-Approximierbarkeitsresultat fĂŒr das Problem in uniformen Hypergraphen. Dieses Resultat verallgemeinert das Ergebnis
von Safra et al. (2006) von b = 1 auf b in O(l/log(l))).
Safra et al. zeigten, dass es fĂŒr das 1-Matching Problem in uniformen Hypergraphen unter der Annahme P != NP keinen polynomiellen Approximationsalgorithmus mit einer Ratio O(l/log(l)) gibt.
Weiterhin beweisen wir, dass es in uniformen Hypergraphen mit beschrĂ€nktem Knoten-Grad kein PTAS fĂŒr das Problem gibt, es sei denn
P = NP
- âŠ