Summary: TERMINAL BACKUP, 3D MATCHING, AND COVERING CUBIC GRAPHS
ELLIOT ANSHELEVICH AND ADRIANA KARAGIOZOVA
Abstract. We define a problem called Simplex Matching, and show that it is solvable in polynomial time. While
Simplex Matching is interesting in its own right as a nontrivial extension of non-bipartite min-cost matching, its main
value lies in many (seemingly very different) problems that can be solved using our algorithm. For example, suppose that
we are given a graph with terminal nodes, non-terminal nodes, and edge costs. Then, the Terminal Backup problem,
which consists of finding the cheapest forest connecting every terminal to at least one other terminal, is reducible to
Simplex Matching. Simplex Matching is also useful for various tasks that involve forming groups of at least 2 members,
such as project assignment and variants of facility location.
In an instance of Simplex Matching, we are given a hypergraph H with edge costs, and edge size at most 3. We
show how to find the min-cost perfect matching of H efficiently, if the edge costs obey a simple and realistic inequality
that we call the Simplex Condition. The algorithm we provide is relatively simple to understand and implement, but
difficult to prove correct. In the process of this proof we show some powerful new results about covering cubic graphs
with simple combinatorial objects.
Key words. graph packing, simplex matching, cycle cover, network design
AMS subject classifications. 05C70, 68Q25, 68R10
1. Introduction. Matching theory, as well as its extensions, is both extremely important and
well-studied. Perhaps surprisingly, there still remain basic matching problems that can be solved
efficiently, and yet are not solvable using existing matching algorithms and techniques [12, 15, 21, 22].
In this paper, we address one such problem that we call Simplex Matching and show how to solve it