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 nonbipartite 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, nonterminal 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 two 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.

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 ouralgorithm. 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 two members, such as project assignment and variants of facility location.

We consider questions about vertex cuts in graphs, random walks in metric spaces, and dimension reduction in L1 and L2; these topics are intimately connected because they can each be reduced to the existence ofvarious families of real-valued Lipschitz maps on certain metric spaces. We view these issues through the lens of shortest-path metricson series-parallel graphs, and we discussthe implications for a variety of well-known open problems. Our main results follow.

We study the multicommodity buy-at-bulk network design problem in which we seek to design a network that satisfies the demands between terminals from a given set of source-sink pairs. The key characteristic of this problem is the fact that the cost functions associated with the edges of the graph are sub-additive monotone and hence experience economies of scale. In the non-uniform case, each edge has its own cost function -- possibly different from other edges. Special cases of this problem have been studied extensively: there are approximation algorithms when the edge cost functions are identical or when all source-sink pairs share the same source. We present the first non-trivial approximation algorithm for the general case. Our algorithm is an extremely simple randomized greedy algorithm and has an approximation guarantee of exp(O√ln n ln ln n)) when the instance has at most n source-sink pairs with unit demands. In the case of general demands, this yields an approximation factor of exp(O √ln N ln ln N)), where N is the sum of all demands.