  We consider a general class of combinatorial optimization problems including
among others allocation, multiple knapsack, matching or travelling salesman
problems. The standard version of those problems is the maximum weight
optimization problem where a sum of values is optimized. However, the sum is
not a good aggregation function when the fairness of the distribution of those
values (corresponding for example to different agents' utilities or criteria)
is important. In this paper, using the generalized Gini index (GGI), a
well-known inequality measure, instead of the sum to model fairness, we
formulate a new general problem, that we call fair combinatorial optimization.
Although GGI is a non-linear aggregating function, a $0,1$-linear program (IP)
can be formulated for finding a GGI-optimal solution by exploiting a
linearization of GGI proposed by Ogryczak and Sliwinski. However, the time
spent by commercial solvers (e.g., CPLEX, Gurobi...) for solving (IP) increases
very quickly with instances' size and can reach hours even for relatively
small-sized ones. As a faster alternative, we propose a heuristic for solving
(IP) based on a primal-dual approach using Lagrangian decomposition. %We
experimentally evaluate our methods against the exact solution of (IP) by CPLEX
on several fair optimization problems related to matching to demonstrate the
efficiency of our proposition. We demonstrate the efficiency of our method by
evaluating it against the exact solution of (IP) by CPLEX on several fair
optimization problems related to matching. The numerical results show that our
method outputs in a very short time efficient solutions giving lower bounds
that CPLEX may take several orders of magnitude longer to obtain. Moreover, for
instances for which we know the optimal value, these solutions are
quasi-optimal with optimality gap less than 0.3%.
