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Algorithmica

, Volume 80, Issue 11, pp 3428–3430 | Cite as

Correction to: A Connection Between Sports and Matroids: How Many Teams Can We Beat?

  • Ildikó Schlotter
  • Katarína Cechlárová
Correction
  • 208 Downloads

1 Correction to: Algorithmica DOI 10.1007/s00453-016-0256-2

2 Background

Our article “A connection between sports and matroids: How many teams can we beat?” [4] deals with a problem we called MinStanding(S). The motivation behind this problem comes from the following situation. Given an ongoing sports competition among a set T of teams, with each team having a current score and some matches left to be played, we ask whether it is possible for our distinguished team \(t \in T\) to obtain a final standing with at most r teams finishing before t.

For a general model that can be applied to various sports competitions, we denoted by S the set of all possible outcomes of a match, where each outcome is a pair (\(p_1,p_2\)) of non-negative reals corresponding to the situation where the match ends with the home team obtaining \(p_1\) points and the away team obtaining \(p_2\) points. If S contains pairs \((\alpha ,0)\) and \((0,\beta )\) for some positive \(\alpha \) and \(\beta \), then we say that S is well based.

Given a set S of outcomes with \(|S|=k+1\), the above question boils down to the following graph labelling problem:

MinStanding(S):

Instance: A triple (Gcr) where \(G=(V,A)\) is a directed multigraph, \(c:V \rightarrow {\mathbb R}\) describes vertex capacities, and r is an integer.

Question: Does there exist an assignment \(p:A \rightarrow \{0, \dots , k\}\) such that the number of vertices in V violating the inequality
$$\begin{aligned} \sum _{a=(v,u)\in A}\alpha _{p(a)}+\sum _{a=(u,v)\in A}\beta _{p(a)} \le c(v) \end{aligned}$$
(1)
is at most r?

3 The Error and its Correction

In Theorem 3 of our article [4], we incorrectly stated that “MinStanding(S) is \(\mathsf {W}[1]\)-hard with parameter \(|V(G)|-r\) for any well-based set S of outcomes, even if the (undirected version of the) input graph G is claw-free”.

The presented (erroneous) proof gave an FPT reduction from the \(\mathsf {W}[1]\)-hard Independent Set problem. Although the reduction itself is correct, we erroneously claimed that Independent Set is \(\mathsf {W}[1]\)-hard on claw-free graphs. However, this is not true, since Independent Set can be solved in polynomial time on claw-free graphs [2, 3]. What holds true is that Independent Set is \(\mathsf {W}[1]\)-hard on \(K_{1,4}\)-free graphs, as proved by Hermelin, Mnich, and Van Leeuwen [1]. So the term “claw-free” in the above statement (Theorem 3 of our article [4]) should be replaced by “\(K_{1,4}\)-free”.

The correct statement of the theorem is thus the following.

Theorem 1

MinStanding(S) is \(\mathsf {W}[1]\)-hard with parameter \(|V(G)|-r\) for any well-based set S of outcomes, even if the (undirected version of the) input graph G is \(K_{1,4}\)-free.

Proof

We give a simple FPT reduction from the \(\mathsf {W}[1]\)-hard Independent Set problem, which is known to be \(\mathsf {W}[1]\)-hard even on \(K_{1,4}\)-free graphs [1]. Let G be the input graph and \(\ell \) the parameter given. The constructed instance of MinStanding(S) will be \((\overrightarrow{G},c,|V(G)|-\ell )\) where \(\overrightarrow{G}\) is an arbitrarily oriented version of G, and c is the constant zero function.

Now, it is easy to see that a set X of vertices in G is independent if and only if there is a score assignment on \(\overrightarrow{G}\) in which vertices of X are not violating. Note that here we make use of the fact that S is well based. \(\square \)

Notes

Acknowledgements

We are grateful to Matthias Mnich who kindly pointed out our error.

References

  1. 1.
    Hermelin, D., van Leeuwen, E.J.: Parameterized complexity of induced graph matching on claw-free graphs. Algorithmica 70(3), 513–530 (2014)MathSciNetMATHGoogle Scholar
  2. 2.
    Minty, G.J.: On maximal independent sets of vertices in claw-free graphs. J. Comb. Theory Ser. B 28(3), 284–304 (1980)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Sbihi, N.: Algorithme de recherche d’un stable de cardinalité maximum dans un graphe sans étoile. Discrete Math. 29(1), 53–76 (1980)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Schlotter, I., Cechlárová, K.: A connection between sports and matroids: How many teams can we beat? Algorithmica (to appear). doi: 10.1007/s00453-016-0256-2

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Budapest University of Technology and EconomicsBudapestHungary
  2. 2.Institute of Mathematics, Faculty of ScienceP.J. Šafárik UniversityKošiceSlovakia

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