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

- 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*.

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

MinStanding(

S):

Instance:A triple (G,c,r) where \(G=(V,A)\) is a directed multigraph, \(c:V \rightarrow {\mathbb R}\) describes vertex capacities, andris an integer.Question:Does there exist an assignment \(p:A \rightarrow \{0, \dots , k\}\) such that the number of vertices inVviolating the inequalityis at most$$\begin{aligned} \sum _{a=(v,u)\in A}\alpha _{p(a)}+\sum _{a=(u,v)\in A}\beta _{p(a)} \le c(v) \end{aligned}$$(1)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.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.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.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.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