8 research outputs found

    Discrete Convex Functions on Graphs and Their Algorithmic Applications

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    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

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    In this paper we address a topological approach to multiflow (multicommodity flow) problems in directed networks. Given a terminal weight μ\mu, we define a metrized polyhedral complex, called the directed tight span TμT_{\mu}, and prove that the dual of μ\mu-weighted maximum multiflow problem reduces to a facility location problem on TμT_{\mu}. Also, in case where the network is Eulerian, it further reduces to a facility location problem on the tropical polytope spanned by μ\mu. 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

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    For a metric μ\mu on a finite set TT, the minimum 0-extension problem 0-Ext[μ][\mu] is defined as follows: Given V⊇TV\supseteq T and  c:(V2)→Q+\ c:{V \choose 2}\rightarrow \mathbf{Q_+}, minimize ∑c(xy)μ(γ(x),γ(y))\sum c(xy)\mu(\gamma(x),\gamma(y)) subject to γ:V→T, γ(t)=t (∀t∈T)\gamma:V\rightarrow T,\ \gamma(t)=t\ (\forall t\in T), where the sum is taken over all unordered pairs in VV. 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[μ][\mu] 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 0→\overrightarrow{0}-Ext[μ][\mu] of the minimum 0-extension problem, where μ\mu and cc are not assumed to be symmetric. We extend the NP-hardness condition of 0-Ext[μ][\mu] to 0→\overrightarrow{0}-Ext[μ][\mu]: If μ\mu cannot be represented as the shortest path metric of an orientable modular graph with an orbit-invariant ``directed'' edge-length, then 0→\overrightarrow{0}-Ext[μ][\mu] is NP-hard. We also show a partial converse: If μ\mu is a directed metric of a modular lattice with an orbit-invariant directed edge-length, then 0→\overrightarrow{0}-Ext[μ][\mu] is tractable. We further provide a new NP-hardness condition characteristic of 0→\overrightarrow{0}-Ext[μ][\mu], and establish a dichotomy for the case where μ\mu is a directed metric of a star

    Combinatorial Optimization

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    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

    Folder Complexes and Multiflow Combinatorial Dualities

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