We consider the following network design problem, that we call the Generalized Terminal Backup Problem: given a graph (or a hypergraph) G 0 = (V, E 0), a set of (at least 2) terminals T ⊆ V and a requirement r(t) for every t ∈ T , nd a multigraph G = (V, E) such that λ G 0 +G (t, T − t) ≥ r(t) for any t ∈ T. In the minimum cost version the objective is to nd G minimizing the total cost c(E) = uv∈E c(uv), given also costs c(uv) ≥ 0 for every pair u, v ∈ V. In the degree-specied version the question is to decide whether such a G exists, satisfying that the number of edges is a prescribed value m(v) at each node v ∈ V. The Terminal Backup Problem solved in [1] is the special case where G 0 is the empty graph and r(t) = 1 for every terminal t ∈ T. We solve the Generalized Terminal Backup Problem in the following two cases. In the rst case we solve the degree-specied version by a splitting-o theorem. This splitting-o theorem in turn provides the solution for the minimum cost version in the case when c is node-induced, that is c(uv) = w(u) + w(v) for some node weights w : V → R +. In the second solved case we turn to the general minimum cost version, and we are able to solve it when G 0 is the empty graph. This includes the Terminal Backup Problem [1] (r ≡ 1) and the Maximum-Weight b-matching Problem (T = V). The solution depends on an interesting new variant of a theorem of Lovász and Cherkassky, and on the solution of the so-called Simplex Matching problem [1]. Our algorithms run in strongly polynomial time for both problems.

As a generalization of the Edmonds arborescence packing theorem, Kamiyama-Katoh-Takizawa (2009) gave a good characterization of directed graphs that contain arc-disjoint arborescences spanning the set of vertices reachable from each root. Fortier-Király-Léonard-Szigeti-Talon (2018) asked whether the result can be extended to mixed graphs by allowing both directed arcs and undirected edges. In this paper, we solve this question by developing a polynomial-time algorithm for finding a collection of edge and arc-disjoint arborescences spanning the set of vertices reachable from each root in a given mixed graph.

Reconfiguration problems require finding a step-by-step transformation between a pair of feasible solutions for a particular problem. The primary concern in Theoretical Computer Science has been revealing their computational complexity for classical problems.This paper presents an initial study on reconfiguration problems derived from a submodular function, which has more of a flavor of Data Mining. Our submodular reconfiguration problems request to find a solution sequence connecting two input solutions such that each solution has an objective value above a threshold in a submodular function 𝑓 : 2 [𝑛] → R + and is obtained from the previous one by applying a simple transformation rule. We formulate three reconfiguration problems: Monotone Submodular Reconfiguration (MSReco), which applies to influence maximization, and two versions of Unconstrained Submodular Reconfiguration (USReco), which apply to determinantal point processes. Our contributions are summarized as follows:• We prove that MSReco and USReco are both PSPACE-complete.• We design a 1 2 -approximation algorithm for MSReco and a 1 𝑛approximation algorithm for (one version of) USReco.• We devise inapproximability results that approximating the optimum value of MSReco within a (1− 1+𝜖 𝑛 2 )-factor is PSPACE-hard, and we cannot find a ( 5 6 + 𝜖)-approximation for USReco.• We conduct numerical study on the reconfiguration version of influence maximization and determinantal point processes using real-world social network and movie rating data. CCS CONCEPTS• Information systems → Data mining; • Theory of computation → Approximation algorithms analysis.

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Deploying caches on a network is an effective way to reduce the amount of data transmitted in a network. Recently, in an academic backbone network such as SINET (the Science Information Network) in Japan, the amount of transmitted data has significantly increased. It is desired to design an efficient mechanism to allocate caches in an optimal way. In this paper, we begin by formulating a discrete optimization model to find a cache allocation that minimizes the total transmission cost. We then design two efficient algorithms to solve our proposed model. The first one makes use of the fact that a backbone network has small treewidth. The algorithm runs in polynomial time when the number of items is fixed and a graph has a bounded treewidth. The other one reduces the problem to the minimum-cost flow problem under the practical assumption that each item has at most one copy. This yields a polynomial-time combinatorial algorithm. Our numerical experiments on the real SINET network show that our algorithms can solve the cache placement problem efficiently in practice.