Algorithmica

, Volume 80, Issue 11, pp 3428–3430

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

• Ildikó Schlotter
• Katarína Cechlárová
Correction

## 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)
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)
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)
4. 4.
Schlotter, I., Cechlárová, K.: A connection between sports and matroids: How many teams can we beat? Algorithmica (to appear). doi: