8 research outputs found
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
On duality and fractionality of multicommodity flows in directed networks
In this paper we address a topological approach to multiflow (multicommodity
flow) problems in directed networks. Given a terminal weight , we define a
metrized polyhedral complex, called the directed tight span , and
prove that the dual of -weighted maximum multiflow problem reduces to a
facility location problem on . Also, in case where the network is
Eulerian, it further reduces to a facility location problem on the tropical
polytope spanned by . By utilizing this duality, we establish the
classifications of terminal weights admitting combinatorial min-max relation
(i) for every network and (ii) for every Eulerian network. Our result includes
Lomonosov-Frank theorem for directed free multiflows and
Ibaraki-Karzanov-Nagamochi's directed multiflow locking theorem as special
cases.Comment: 27 pages. Fixed minor mistakes and typos. To appear in Discrete
Optimizatio
Minimum 0-Extension Problems on Directed Metrics
For a metric on a finite set , the minimum 0-extension problem
0-Ext is defined as follows: Given and , minimize
subject to , where the
sum is taken over all unordered pairs in . This problem generalizes several
classical combinatorial optimization problems such as the minimum cut problem
or the multiterminal cut problem. The complexity dichotomy of 0-Ext was
established by Karzanov and Hirai, which is viewed as a manifestation of the
dichotomy theorem for finite-valued CSPs due to Thapper and \v{Z}ivn\'{y}.
In this paper, we consider a directed version -Ext
of the minimum 0-extension problem, where and are not assumed to be
symmetric. We extend the NP-hardness condition of 0-Ext to
-Ext: If cannot be represented as the shortest
path metric of an orientable modular graph with an orbit-invariant ``directed''
edge-length, then -Ext is NP-hard. We also show a
partial converse: If is a directed metric of a modular lattice with an
orbit-invariant directed edge-length, then -Ext is
tractable. We further provide a new NP-hardness condition characteristic of
-Ext, and establish a dichotomy for the case where
is a directed metric of a star
Combinatorial Optimization
Combinatorial Optimization is a very active field that benefits from bringing together ideas from different areas, e.g., graph theory and combinatorics, matroids and submodularity, connectivity and network flows, approximation algorithms and mathematical programming, discrete and computational geometry, discrete and continuous problems, algebraic and geometric methods, and applications. We continued the long tradition of triannual Oberwolfach workshops, bringing together the best researchers from the above areas, discovering new connections, and establishing new and deepening existing international collaborations