Fnd2012 Open Problems - Recent questions and answers
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Powered by Question2AnswerAnswered: How to remove Steiner points from planar metrics?
http://fnd2012.mimuw.edu.pl/qa/index.php?qa=2&qa_1=how-to-remove-steiner-points-from-planar-metrics&show=5#a5
<p>
Some related construction is in <a rel="nofollow" href="ftp://ftp.cs.brown.edu/pub/techreports/95/cs95-04.pdf">Subramianian PhD</a>. If you are given a planar graph where k terminals lie on the external face, then there exists an epsilon-approximate substitute of size O(epsilon^-1 k log k log D), where D is the diameter of the graph. This is described in Section 2.1.</p>http://fnd2012.mimuw.edu.pl/qa/index.php?qa=2&qa_1=how-to-remove-steiner-points-from-planar-metrics&show=5#a5Wed, 01 Aug 2012 11:28:09 +0000Approximating Star Cover Problems
http://fnd2012.mimuw.edu.pl/qa/index.php?qa=4&qa_1=approximating-star-cover-problems
<p>
(Sorry about the LaTeX symbols)</p>
<p>
Given a set of points $S$, a distance function $d$, an integer $k$ and a bound $B$ on maximum load of any facility, the problem is to decide whether we<br>
can select $k$ facilities in $S$ and serve other points by these facilities such<br>
that the maximum load of any facility is $B$. The load on a facility $f$<br>
is the sum of distances to the clients it serves. Alternately, since an open<br>
facility together with the assigned clients, forms a star and its load is exactly<br>
equal to the cost of the star, we can call this the Star Cover Decision Problem<br>
(SCDP). We can consider two optimization versions of the above stated decision problem:<br>
<br>
<strong>Minimum Load Star Cover or MLSC </strong>: Given $S$, $d$ and an integer $k$,<br>
minimize the maximum load on each facility.<br>
</p>
<p>
<strong>Minimum Cardinality Star Cover or MCSC </strong>: Given $S$, $d$ and a bound $B$ on<br>
maximum load, minimize the number of facilities to be opened, such that the load on each facility is at most $B$.<br>
<br>
We can also consider a bicriteria approximation for the decision problem i.e. given<br>
$S$, $d$, $k$ and $B$ find an $(\alpha,\beta)$-approximation, such that at most<br>
$\alpha k$ facilities are opened and maximum load on a facility is at most $\beta<br>
B$.<br>
<br>
<br>
<em>Known results</em><br>
<br>
Arkin, Hassin and Levin consider the Minimum Cardinality Star Cover problem, where distance $d$ is a metric and give a $(2\alpha+1)$-approximation for the problem, where $\alpha$ is the best approximation ratio of the $k$-median problem.<br>
<br>
Even, Garg, Konemann, Ravi and Sinha give a bicriteria approximation of $(4,4)$ for the case when $d$ is a metric i.e. for given $k$ and $B$, their algorithm opens at most $4k$ facilities and the completion time is at most $4B$. This was improved to a $(3+\epsilon,3+\epsilon)$ approximation by Arkin, Hassin and Levin.<br>
<br>
The star cover decision problem (SCDP) is NP-complete, even when the distance function $d$ is a line metric or a star metric. Furthermore, the problem remains hard even if the facilities to open are specified. The proofs of hardness for line and star metrics are by reductions from 3-PARTITION and MAKESPAN respectively.<br>
<br>
The Minimum Cardinality Star Cover or MCSC problem is $\Omega(\log n)$-hard to approximate if the distance function, $d$, is not a metric, by a reduction from set cover, where $|S|=n$. A similar analysis as that of greedy algorithm for set cover, gives a $\log n$-approximation for the MCSC problem in the general case. There is a $2$-approximation for the case when the distance function is a line metric.<br>
<br>
For the Minimum Load Star Cover or MLSC problem, the LP relaxation of a natural IP formulation has a large integrality gap. There is a $3$-approximation when the distance function is a star metric. In the case of distance function being a line metric, it can be shown that if every point is assigned to either the closest facility to the right or closest facility to the left then the maximum load can be a factor $k$ times worse than the optimum.<br>
<br>
<em><strong>OPEN PROBLEM</strong></em><br>
<br>
Give a nontrivial approximation for the MLSC problem or show a lower bound for the approximation factor. Even the case when the input is a line metric is open.<br>
<br>
<br>
<strong>References</strong><br>
<br>
E. Arkin, R. Hassin and A. Levin. Approximations for minimum and min-max vehicle<br>
routing problems, Journal of Algorithms, 59(1), 2006.<br>
<br>
G. Even, N. Garg, J. Koenemann, R.Ravi, A. Sinha. Covering Graphs using Trees and Stars. In Proc of the 6th International Workshop on Approximation Algorithms for<br>
Combinatorial Optimization Problems, 2003.<br>
</p>
<p>
<em>Acknowledgment:</em> Thanks to Vineet Goyal for working out some of the above observations.</p>http://fnd2012.mimuw.edu.pl/qa/index.php?qa=4&qa_1=approximating-star-cover-problemsTue, 31 Jul 2012 22:00:37 +0000Can one improve approximation algorithms for union and intersection covering problems?
http://fnd2012.mimuw.edu.pl/qa/index.php?qa=3&qa_1=approximation-algorithms-intersection-covering-problems
<p>
In <a rel="nofollow" href="http://drops.dagstuhl.de/opus/volltexte/2011/3321/">this paper</a> we have introduced union and intersection covering problems. In such problems we are given h layers - graphs with the same set of vertices. In each layer we need to solve some graph problem to cover vertices, e.g., connect vertices to the root by using a Steiner tree. In intersection problems we are asked to chose k-vertices of the graphs and cover them in all the layers, on the other hand in union problems, we are asked to cover all vertices but each vertex should be covered in at least one layer.<br>
<br>
There are two interesting open problems here:<br>
<br>
- We were able to give O(h) approximation algorithm for the union problem of, e.g., Steiner trees. <strong>Can on get a constant approximation that would be independent of h (for reasonably small h)?</strong><br>
<br>
- <strong>Can one close the gap between upper and lower bounds for intersection problems?</strong> In particular, our hardness results do not depend on h, while the approximation ratios deteriorate rather rapidly for increasing h.</p>http://fnd2012.mimuw.edu.pl/qa/index.php?qa=3&qa_1=approximation-algorithms-intersection-covering-problemsTue, 31 Jul 2012 21:45:39 +0000Does there exists a dual combinatorial interpretation of the matching problem in planar graphs?
http://fnd2012.mimuw.edu.pl/qa/index.php?qa=1&qa_1=exists-combinatorial-interpretation-matching-problem-planar
<p>
There exists very nice reduction of the bipartite perfect matching problem in planar graph to the shortest path problem in the dual graph. The reduction can be find in the paper by Miller and Naor: http://www.cs.cmu.edu/~glmiller/Publications/Papers/MillerNaor95.pdf. Combined with with the paper by <span class="f">Fakcharoenphol and Rao: </span>http://www.cs.berkeley.edu/~satishr/negcycle.ps, or with the paper by Mozes and Wulff-Nilsen http://www.springerlink.com/content/p216876051k1kr61/ this gives an almost linear time algorithm for finding perfect matchings in biparite planar graphs. However, this interpretation is restricted only to bipartite graphs. Does similar reduction work for general planar graphs as well?</p>http://fnd2012.mimuw.edu.pl/qa/index.php?qa=1&qa_1=exists-combinatorial-interpretation-matching-problem-planarTue, 31 Jul 2012 11:10:46 +0000