User:Igngvs/draft on minimum maximal k-partial-matching problem

The minimum maximal k-partial matching (MMKPM) problem is a combinatorial optimization problem related to bipartite graphs that arises in the context of digital design. Informally, given a bipartite graph, the MMKPM problem require to form k sets by selecting for each left vertex all of its adjacent right vertices up to k, and adding each of them to one different set. The goal of the problem is to minimize the sum of cardinalities of the k sets.

When k = 1, the goal of MMKPM problem is to find the minimum cardinality subset of right-vertices that cover all of the left-vertices. This is equivalent to both the Hitting set and Set cover problems modelled with bipartite graphs. Therefore, the MMKPM problem is a generalization of these problems.

Formal definition
Given a bipartite graph $$G=(U \cup V,E)$$, we define a partial-matching as a set of edges, $$E' \subseteq E$$, such that no two edges in $$E'$$ share a common endpoint in $$U$$ (note that the edges in $$E'$$ can share common endpoints in $$V$$). A k-partial-matching is a collection $$P=\{P_1,P_2,\ldots,P_k\}$$ of k disjoint partial-matchings. A maximal k-partial-matching is a k-partial-matching that contains the maximum number of edges, i.e., a k-partial-matching $$P=\{P_1,P_2,\ldots,P_k\}$$ such that $$| \cup_{i=1}^k P_i|$$ is maximum.

Let the cost of a maximal-k-partial-matching $$P=\{P_1,P_2,\ldots,P_k\}$$ be defined as $$c(P) = \sum_{i=1}^k |V(P_i)|$$ where $$V(P_i)$$ denotes the set of vertices in $$V$$ which are incident with any edge of $$P_i \subseteq E$$. Let $$\Delta_U(G)$$ be the maximum degree of the vertices in $$U$$. In the MMKPM decision problem, the input is a bipartite graph $$G$$, a positive integer $$k \leq \Delta_U(G)$$ and an integer $$b$$; the question is whether there is a maximal k-partial-matching $$P$$ with cost $$c(P) < b$$. In the MMKPM optimization problem, the input is a bipartite graph $$G$$, and an integer $$k \leq \Delta_U(G)$$; the task is to find a maximal k-partial-matching which minimum cost.