Cost Allocation for Set Covering: the Happy Nucleolus

TL;DR

The paper tackles fair cost allocation for set covering by introducing the happy nucleolus, the lexicographically maximal excess allocation under the constraint of full cost coverage via the optimum fractional cover. It identifies a priori a small defining family $\mathcal{C}^*$ (and $\mathcal{C}^{**}$ when needed) of at most $|P||\mathcal{T}|$ coalition constraints that suffices to determine the happy nucleolus, enabling a polynomial-time computation via the Maschler scheme. The work proves the minimality and stability of $\mathcal{C}^*$ under a natural complement condition and discusses when this condition is necessary, demonstrating that the happy nucleolus reduces to the standard nucleolus when the core is nonempty. It also provides lower-bound results showing the tightness of the a posteriori coalition counts and discusses implications for fair, symmetry-respecting cost allocations in practical set-covering-inspired problems such as vehicle routing.

Abstract

We consider cost allocation for set covering problems. We allocate as much cost to the elements (players) as possible without violating the group rationality condition (no subset of players pays more than covering this subset would cost), and so that the excess vector is lexicographically maximized. This is identical to the well-known nucleolus if the core of the corresponding cooperative game is nonempty, i.e., if some optimum fractional cover is integral. In general, we call this the 'happy nucleolus'. Like for the nucleolus, the excess vector contains an entry for every subset of players, not only for the sets in the given set covering instance. Moreover, it is NP-hard to compute a single entry because this requires solving a set covering problem. Nevertheless, we give an explicit family of at most $mn$ subsets, each with a trivial cover (by a single set), such that the happy nucleolus is always completely determined by this proxy excess vector; here $m$ and $n$ denote the number of sets and the number of players in our set covering instance. We show that this is the unique minimal such family in a natural sense. While computing the nucleolus for set covering is NP-hard, our results imply that the happy nucleolus can be computed in polynomial time.

Figures (7)

• Figure 1: This simple vehicle routing instance asks for a set of tours that visit all customers a, b, c, d, e; here a tour is a cycle that starts and ends at the depot, the black square. Distances are given by the metric closure of the numbers on depicted edges. The optimum vehicle routing solution costs 9, and in this example we can allocate 9 units and satisfy group rationality. Two possible such cost allocations are shown in bold next to the vertices. The one on the left is not even symmetric: although a and b are completely symmetric, they pay different shares. Moreover, customer c pays more than customer d although it is definitely not harder to serve. The allocation on the right looks fairer, and this is in fact the (happy) nucleolus.
• Figure 2: This capacitated vehicle routing instance asks for a set of tours that visit all customers a, b, c, d; here one tour can serve up to two customers. Again, distances are given by the metric closure of the numbers on depicted edges. Optimum integral and fractional vehicle routing solutions have total cost 21 and 18, respectively. The cost allocation shown on the left is the nucleolus, the one on the right is the happy nucleolus. The table shows all excess values (up to symmetric coalitions); green fields highlight the first entries of the excess vector. The middle solution allocates the same amount as the nucleolus but optimizes the excess only for coalitions that can be served by a single tour (red fields indicate ignored coalitions). Note that this solution differs from the nucleolus. The happy nucleolus would stay the same if we considered only coalitions served by a single tour.
• Figure 3: A capacitated vehicle routing instance with 15 customers, each with coordinates in $\{0,\ldots,10\}^2$. One tour can serve up to five customers; all tours must begin and end at the depot (black square, at (6,9)). The cost of a tour is the total Euclidean distance traveled. We can also decide not to serve a customer $p$ and instead pay a dropping penalty $d(p)$, shown in the second column of the table. The third column shows (rounded values of) the happy nucleolus $y$, visualized by the color scale from 1 to 7.
• Figure 4: Instance with three players and only one set, covering them all. Equally distributing $y_p =1$ for all players $p$ yields the happy nucleolus. It is enough to consider singleton coalitions to determine it. But adding a 2-element coalition (and not all 2-element coalitions) will yield a different solution.
• Figure 5: Triangle instance of \ref{['example:triangle']}. For any cost function $c:\{T_1,T_2,T_3\}\to\mathbb{R}_{\ge 0}$, there is only one cost allocation $y\in\mathbb{R}^P_{\ge 0}$ with $y(P)=\mathrm{LP}(P,\mathcal{T},c)$ that makes the three coalitions $T_1,T_2,T_3$ happy.

Theorems & Definitions (18)

• Definition 1: excess vector
• Proposition 2: deng1999algorithmic
• proof
• Definition 3: happy nucleolus
• Definition 4: determining the happy nucleolus
• Theorem 5
• Corollary 5
• Lemma 6
• proof
• Theorem 6