Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

Approximating the generalized terminal backup problem via half-integral multiflow relaxation

We consider a network design problem called the generalized terminal backup problem. Whereas earlier work investigated the edge-connectivity constraints only, we consider both edge- and node-connectivity constraints for this problem. A major contribution of this paper is the development of a strongly polynomial-time 4/3-approximation algorithm for the problem. Specifically, we show that a linear programming relaxation of the problem is half-integral, and that the half-integral optimal solution can be rounded to a 4/3-approximate solution. We also prove that the linear programming relaxation of the problem with the edge-connectivity constraints is equivalent to minimizing the cost of half-integral multiflows that satisfy flow demands given from terminals. This observation presents a strongly polynomial-time algorithm for computing a minimum cost half-integral multiflow under flow demand constraints

Node-Connectivity Terminal Backup, Separately-Capacitated Multiflow, and Discrete Convexity

The terminal backup problems (Anshelevich and Karagiozova (2011)) form a class of network design problems: Given an undirected graph with a requirement on terminals, the goal is to find a minimum cost subgraph satisfying the connectivity requirement. The node-connectivity terminal backup problem requires a terminal to connect other terminals with a number of node-disjoint paths. This problem is not known whether is NP-hard or tractable. Fukunaga (2016) gave a $4/3$-approximation algorithm based on LP-rounding scheme using a general LP-solver. In this paper, we develop a combinatorial algorithm for the relaxed LP to find a half-integral optimal solution in $O(m\log (nUA)\cdot \operatorname{MF}(kn,m+k^2n))$ time, where $n$ is the number of nodes, $m$ is the number of edges, $k$ is the number of terminals, $A$ is the maximum edge-cost, $U$ is the maximum edge-capacity, and $\operatorname{MF}(n',m')$ is the time complexity of a max-flow algorithm in a network with $n'$ nodes and $m'$ edges. The algorithm implies that the $4/3$-approximation algorithm for the node-connectivity terminal backup problem is also efficiently implemented. For the design of algorithm, we explore a connection between the node-connectivity terminal backup problem and a new type of a multiflow, called a separately-capacitated multiflow. We show a min-max theorem which extends Lov\'{a}sz-Cherkassky theorem to the node-capacity setting. Our results build on discrete convexity in the node-connectivity terminal backup problem.Comment: A preliminary version of this paper was appeared in the proceedings of the 47th International Colloquium on Automata, Languages and Programming (ICALP 2020

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures