The sport teams grouping problem

Abstract

The sport teams grouping problem (STGP) concerns the assignment of sport teams to round-robin tournaments. The objective is to minimize the total travel distance of the participating teams while simultaneously respecting fairness constraints. The STGP is an NP-Hard combinatorial optimization problem highly relevant in practice. This paper investigates the performance of some complimentary optimization approaches to the STGP. Three integer programming formulations are presented and thoroughly analyzed: two compact formulations and another with an exponential number of variables, for which a branch-and-price algorithm is proposed. Additionally, a meta-heuristic method is applied to quickly generate feasible high-quality solutions for a set of real-world instances. By combining the different approaches’ results, solutions within 1.7% of the optimum values were produced for all feasible instances. Additionally, to support further research, the considered STGP instances and corresponding solutions files were shared online.

Appendix A

This appendix presents the proofs for Theorems 1 and 2, previously presented in Sect. 3.4.

Theorem1 The linear relaxation of $\mathcal {F}_3$ is stronger than the linear relaxation of $\mathcal {F}_1$.

Proof

Theorem 1 is proven true if conditions C1 and C2 are satisfied:

• C1: for each STGP instance I, $z_{\mathcal {F}_3}(I) \ge z_{\mathcal {F}_1}(I)$,

• C2: there exists a STGP instance I where $z_{\mathcal {F}_3}(I) > z_{\mathcal {F}_1}(I)$.

The validity of C2 is relatively simple to show; in fact, many of the instances considered in Sect. 7 satisfy this condition. C1 remains to be proven true. Consider $\lambda ^*_{\omega }$ a feasible solution of the linear relaxation of $\mathcal {F}_3$ on some instance I while also assuming $L \leftarrow \varOmega$ such that each $\omega \in \varOmega$ has an equivalent $\ell \in L$, meaning $\omega = \ell$. Given this solution and the set $L = \varOmega$, x-, y- and z-values can be constructed:

$$\begin{aligned} \text {for each } i \in T \text { and } \ell \in L: \;\;&x_{i, \ell } = x_{i, \omega } \leftarrow \lambda ^*_{\omega } \end{aligned}$$

(29)

$$\begin{aligned} \text {for each } (i,j) \in A^2: \;\;&y_{i,j} \leftarrow \sum _{\omega : (i,j) \in \omega } {\lambda ^*_\omega }. \end{aligned}$$

(30)

$$\begin{aligned} \text {for each } \ell \in L: \;\;&z_{\ell } = z_{\omega } \leftarrow \lambda ^*_{\omega } \end{aligned}$$

(31)

Next, it is shown how the resulting solution is a feasible solution of the linear relaxation of $\mathcal {F}_1$, indicating that it satisfies (2)–(6). Constraints (2) are always satisfied, since each team $i \in T$ is assigned to exactly one league [see Constraints (18)]:

$$\begin{aligned} \sum _{\ell \in L} x_{i, \ell } = \sum _{\omega \in \varOmega } x_{i, \omega } = \sum _{\omega : i \in \omega } \lambda ^*_{\omega } = 1 \end{aligned}$$

(32)

Since the leagues (cliques) $\omega \in \varOmega$ are feasible, $m^- \le |\omega | \le m^+ \; \forall \omega \in \varOmega$, inequalities in (3) follow:

$$\begin{aligned} \sum _{i \in T} x_{i, \ell } = \sum _{i \in T} x_{i, \omega } = |\omega | \lambda ^*_{\omega } = |\omega | z_{\omega } = |\omega | z_{\ell } \end{aligned}$$

(33)

Given that all leagues $\omega \in \varOmega$ are feasible it also follows that the number of teams from a club in any $\omega$ is less than or equal to $c^+$. Therefore, Constraints (4) are also satisfied since for all leagues $\ell \in L$ and clubs $c \in C$:

$$\begin{aligned} \sum _{i \in T_c} x_{i, \ell } \le c^+ \lambda ^*_{\omega } = c^+ z_{\ell } \end{aligned}$$

(34)

Constraints (5) also hold. If teams i and j are in the same league, they are therefore together in a league $\omega \in \varOmega$. Thus, for all $(i,j) \in A^2$ and leagues $\ell \in L$ (or $\omega \in \varOmega$) containing both i and j:

$$\begin{aligned} x_{i, \ell } + x_{j, \ell } = 2\lambda ^*_{\omega } \le y_{i,j} + 1 \end{aligned}$$

(35)

Observe that Constraints (5) are also satisfied when teams i and j are not in the same league $\ell$ since this implies $x_{i, \ell } + x_{j, \ell } \le 1$.

Finally, if teams i and j are not allowed in a league, leagues $\omega \in \varOmega$ will contain at most one of them (since all leagues $\omega$ are feasible). Hence, Constraints (6) also hold for all $(i,j) \in F^2$ and $\ell \in L$:

$$\begin{aligned} x_{i, \ell } + x_{j, \ell } \le \lambda ^*_{\omega } \le 1 \end{aligned}$$

(36)

It is thus proven that every solution I of the linear relaxation of $\mathcal {F}_3$ satisfy (2)–(6). Since both $\mathcal {F}_1$ and $\mathcal {F}_3$ have the same objective function, Theorem 1 is proven. $\square$

Theorem2 The linear relaxation of $\mathcal {F}_3$ is stronger than the linear relaxation of $\mathcal {F}_2$.

Proof

Theorem 2 is proven if conditions C3 and C4 are satisfied:

• C3: for each STGP instance I, $z_{\mathcal {F}_3}(I) \ge z_{\mathcal {F}_2}(I)$,

• C4: there exists a STGP instance I where $z_{\mathcal {F}_3}(I) > z_{\mathcal {F}_2}(I)$.

C4 is proven true by the experiments reported in Sect. 7. It is, however, required to mathematically prove that C3 is true. Consider again that $\lambda ^*_{\omega }$ is a feasible solution of the linear relaxation of $\mathcal {F}_3$ for an instance I; y-values may be easily constructed:

$$\begin{aligned} \text {for each } (i,j) \in A^2: \;\;&y_{i,j} \leftarrow \sum _{\omega : (i,j) \in \omega } {\lambda ^*_\omega }. \end{aligned}$$

(37)

Let $\bar{e}^k_{i,j}$ be the sum of $\lambda ^*_\omega$ for all $\omega \in \varOmega$ which include both i and j but not k, and $e_{i,j,k}$ be the sum of $\lambda ^*_\omega$ for all $\omega \in \varOmega$ which include i, j and k. More compactly:

$$\begin{aligned} \bar{e}_{i,j}^k= & {} \sum _{\omega \in \varOmega : \{i,j\} \in \omega \wedge k \notin \omega } {\lambda ^*_\omega } \end{aligned}$$

(38)

$$\begin{aligned} e_{i,j,k}= & {} \sum _{\omega \in \varOmega : \{i,j,k\} \in \omega } {\lambda ^*_\omega } \end{aligned}$$

(39)

Clearly, for each $k \in T$, $y_{i,j} = \bar{e}_{i,j}^k + e_{i,j,k}$. Equivalently, $y_{i,k} = \bar{e}^j_{i,k} + e_{i,j,k}$ and $y_{j,k} = \bar{e}^i_{j,k} + e_{i,j,k}$. Therefore, for each $(i,j,k) \in A^3$:

$$\begin{aligned}&y_{i,j}+y_{i,k}-y_{j,k} = \bar{e}^k_{i,j} + e_{i,j,k}+ \bar{e}^j_{i,k} + e_{i,j,k} -\bar{e}^i_{j,k} - e_{i,j,k} \le \nonumber \\&\quad \bar{e}^k_{i,j} + \bar{e}^j_{i,k} + e_{i,j,k} = \sum _{\omega \in \varOmega : i \in \omega } {\lambda ^*_\omega } = 1. \end{aligned}$$

(40)

The first equality in (40) follows by definition; the first inequality from the non-negativity of all y-variables; the second equality from the fact that terms $\bar{e}_{i,j}^k, \bar{e}^j_{i,k}$, and $e_{i,j,k}$ correspond to disjoint sets of cliques whose union are all cliques containing i; and the final equality follows from (18). Therefore, y satisfies (12).

Furthermore, observe that $e_{i,j,k} = 0 \ \forall (i,j,k) \in F^3$. This follows from i, j and k not being permitted in the same league, and therefore there exists no league $\omega \in \varOmega$ containing these three teams. Thus, modifying (40) shows that y also satisfies (13).

Consider Constraint (14). Observe that for each $i \in T$:

$$\begin{aligned} \sum _{j \in A_i} y_{i,j} = \sum _{j \in A_i} \sum _{\omega :(i,j) \in \omega } \lambda ^*_{\omega } = \sum _j \sum _{\omega :(i,j) \in \omega } \lambda ^*_{\omega } = \sum _{\omega :i \in \omega } (|\omega |-1)\lambda ^*_{\omega }. \end{aligned}$$

(41)

Indeed, the first equality follows from (30); the second equality from how clique containing i cannot contain some team $j \notin A_i$; and the final equality results from counting how many times the value $\lambda ^*_{\omega }$ is present in this term.

Next, given that each clique $\omega$ is feasible, $m^- \le |\omega | \le m^+$. Combining this with (41) results in:

$$\begin{aligned} \sum _{j \in A_i} y_{i,j}= & {} \sum _{\omega :i \in \omega } (|\omega |-1)\lambda ^*_{\omega } \le \sum _{\omega :i \in \omega } (m^+-1)\lambda ^*_{\omega } \nonumber \\= & {} (m^+-1) \sum _{\omega : i \in \omega } \lambda ^*_{\omega } = m^+-1\end{aligned}$$

(42)

$$\begin{aligned} \sum _{j \in A_i} y_{i,j}= & {} \sum _{\omega :i \in \omega } (|\omega |-1)\lambda ^*_{\omega } \le \sum _{\omega :i \in \omega } (m^+-1)\lambda ^*_{\omega } \nonumber \\= & {} (m^+-1) \sum _{\omega : i \in \omega } \lambda ^*_{\omega } = m^+-1. \end{aligned}$$

(43)

It follows that y satisfies (14).

Finally, consider (15). Observe that for each team $i \in T$:

$$\begin{aligned} \sum _{j \in A_i \cap T_i} y_{i,j} = \sum _{j \in A_i \cap T_i} \sum _{\omega :(i,j) \in \omega } \lambda ^*_{\omega } \le (c^+-1) \sum _{\omega :i \in \omega } \lambda ^*_{\omega } = c^+-1. \end{aligned}$$

(44)

Again, the first equality follows from (37) and the inequality from how each clique $\omega$ contains at most $c^+$ nodes from the same club as team i. Hence the value $\lambda _{\omega }$ occurs at most $c^+-1$ times in each clique $\omega$.

It is therefore proven that y satisfies (15).

Finally, note that both formulations have an equal objective function, and hence Theorem 2 is proven.