arXiv subjects

Hiroshi Hirai

Publications and source records attributed to Hiroshi Hirai.

47 records · Page 3Linked to original sources

arXiv2017

Shortest (A+B)-path packing via hafnian

Björklund and Husfeldt developed a randomized polynomial time algorithm to solve the shortest two disjoint paths problem. Their algorithm is based on computation of permanents modulo 4 and the isolation lemma. In this paper, we consider the following generalization of the shortest two disjoint paths problem, and develop a similar algebraic algorithm. The shortest perfect $(A+B)$-path packing problem is: given an undirected graph $G$ and two disjoint node subsets $A,B$ with even cardinalities, find a shortest $|A|/2+|B|/2$ disjoint paths whose ends are both in $A$ or both in $B$. Besides its NP-hardness, we prove that this problem can be solved in randomized polynomial time if $|A|+|B|$ is fixed. Our algorithm basically follows the framework of Björklund and Husfeldt but uses a new technique: computation of hafnian modulo $2^k$ combined with Gallai's reduction from $T$-paths to matchings. We also generalize our technique for solving other path packing problems, and discuss its limitation.

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arXiv2017

A Compact Representation for Modular Semilattices and its Applications

A modular semilattice is a semilattice generalization of a modular lattice. We establish a Birkhoff-type representation theorem for modular semilattices, which says that every modular semilattice is isomorphic to the family of ideals in a certain poset with additional relations.This new poset structure, which we axiomatize in this paper, is called a PPIP (projective poset with inconsistent pairs). A PPIP is a common generalization of a PIP (poset with inconsistent pairs) and a projective ordered space. The former was introduced by Barthélemy and Constantin for establishing Birkhoff-type theorem for median semilattices, and the latter by Herrmann, Pickering, and Roddy for modular lattices. We show the $Θ(n)$ representation complexityand a construction algorithm for PPIP-representations of $(\wedge, \vee)$-closed sets in the product $L^n$ of modular semilattice $L$. This generalizes the results of Hirai and Oki for a special median semilattice $S_k$. We also investigate implicational bases for modular semilattices. Extending earlier results of Wild and Herrmann for modular lattices, we determine optimal implicational bases and develop a polynomial time recognition algorithm for modular semilattices. These results can be applied to retain the minimizer set of a submodular function on a modular semilattice.

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arXiv2017

A compact representation for minimizers of $k$-submodular functions

A $k$-submodular function is a generalization of submodular and bisubmodular functions. This paper establishes a compact representation for minimizers of a $k$-submodular function by a poset with inconsistent pairs (PIP). This is a generalization of Ando-Fujishige's signed poset representation for minimizers of a bisubmodular function. We completely characterize the class of PIPs (elementary PIPs) arising from $k$-submodular functions. We give algorithms to construct the elementary PIP of minimizers of a $k$-submodular function $f$ for three cases: (i) a minimizing oracle of $f$ is available, (ii) $f$ is network-representable, and (iii) $f$ arises from a Potts energy function. Furthermore, we provide an efficient enumeration algorithm for all maximal minimizers of a Potts $k$-submodular function. Our results are applicable to obtain all maximal persistent labelings in actual computer vision problems. We present experimental results for real vision instances.

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arXiv2016

L-convexity on graph structures

In this paper, we study classes of discrete convex functions: submodular functions on modular semilattices and L-convex functions on oriented modular graphs. They were introduced by the author in complexity classification of minimum 0-extension problems. We clarify the relationship to other discrete convex functions, such as $k$-submodular functions, skew-bisubmodular functions, L$^\natural$-convex functions, tree submodular functions, and UJ-convex functions. We show that they are actually viewed as submodular/L-convex functions in our sense. We also prove a sharp iteration bound of the steepest descent algorithm for minimizing our L-convex functions. The underlying structures, modular semilattices and oriented modular graphs, have rich connections to projective and polar spaces, Euclidean building, and metric spaces of global nonpositive curvature (CAT(0) spaces). By utilizing these connections, we formulate an analogue of the Lovász extension, introduce well-behaved subclasses of submodular/L-convex functions, and show that these classes can be characterized by the convexity of the Lovász extension. We demonstrate applications of our theory to combinatorial optimization problems that include multicommodity flow, multiway cut, and related labeling problems: these problems have been outside the scope of discrete convex analysis so far.

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arXiv2016

On k-Submodular Relaxation

$k$-submodular functions, introduced by Huber and Kolmogorov, are functions defined on $\{0, 1, 2, \dots, k\}^n$ satisfying certain submodular-type inequalities. $k$-submodular functions typically arise as relaxations of NP-hard problems, and the relaxations by $k$-submodular functions play key roles in design of efficient, approximation, or fixed-parameter tractable algorithms. Motivated by this, we consider the following problem: Given a function $f : \{1, 2, \dots, k\}^n \rightarrow \mathbb{R} \cup \{\infty\}$, determine whether $f$ is extended to a $k$-submodular function $g : \{0, 1, 2, \dots, k\}^n \rightarrow \mathbb{R} \cup \{\infty\}$, where $g$ is called a $k$-submodular relaxation of $f$. We give a polymorphic characterization of those functions which admit a $k$-submodular relaxation, and also give a combinatorial $O((k^n)^2)$-time algorithm to find a $k$-submodular relaxation or establish that a $k$-submodular relaxation does not exist. Our algorithm has interesting properties: (1) If the input function is integer valued, then our algorithm outputs a half-integral relaxation, and (2) if the input function is binary, then our algorithm outputs the unique optimal relaxation. We present applications of our algorithm to valued constraint satisfaction problems.

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arXiv2016

On uncrossing games for skew-supermodular functions

In this note, we consider the uncrossing game for a skew-supermodular function $f$, which is a two-player game with players, Red and Blue, and abstracts the uncrossing procedure in the cut-covering linear program associated with $f$. Extending the earlier results by Karzanov for $\{0,1\}$-valued skew-supermodular functions, we present an improved polynomial time strategy for Red to win, and give a strongly polynomial time uncrossing procedure for dual solutions of the cut-covering LP as its consequence. We also mention its implication on the optimality of laminar solutions.

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arXiv2015

L-extendable functions and a proximity scaling algorithm for minimum cost multiflow problem

In this paper, we develop a theory of new classes of discrete convex functions, called L-extendable functions and alternating L-convex functions, defined on the product of trees. We establish basic properties for optimization: a local-to-global optimality criterion, the steepest descend algorithm by successive $k$-submodular function minimizations, the persistency property, and the proximity theorem. Our theory is motivated by minimum cost free multiflow problem. To this problem, Goldberg and Karzanov gave two combinatorial weakly polynomial time algorithms based on capacity and cost scalings, without explicit running time. As an application of our theory, we present a new simple polynomial proximity scaling algorithm to solve minimum cost free multiflow problem in $O(n \log (n AC) {\rm MF}(kn, km))$ time, where $n$ is the number of nodes, $m$ is the number of edges, $k$ is the number of terminals, $A$ is the maximum of edge-costs, $C$ is the total sum of edge-capacities, and ${\rm MF}(n',m')$ denotes the time complexity to find a maximum flow in a network of $n'$ nodes and $m'$ edges. Our algorithm is designed to solve, in the same time complexity, a more general class of multiflow problems, minimum cost node-demand multiflow problem, and is the first combinatorial polynomial time algorithm to this class of problems. We also give an application to network design problem.

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arXiv2014

A Combinatorial Formula for Principal Minors of a Matrix with Tree-metric Exponents and Its Applications

Let $T$ be a tree with a vertex set $\{ 1,2,\dots, N \}$. Denote by $d_{ij}$ the distance between vertices $i$ and $j$. In this paper, we present an explicit combinatorial formula of principal minors of the matrix $(t^{d_{ij}})$, and its applications to tropical geometry, study of multivariate stable polynomials, and representation of valuated matroids. We also give an analogous formula for a skew-symmetric matrix associated with $T$.

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arXiv2014

Discrete Convexity and Polynomial Solvability in Minimum 0-Extension Problems

For a graph G and a set V containing the vertex set of G, a 0-extension of G is a metric d on V such that d extends the shortest path metric of G and for all x in V there exists a vertex s in G with d(x, s) = 0. The minimum 0-extension problem 0-Ext[G] on G is: given a set V containing V(G) and a nonnegative cost function c defined on the set of all pairs of V, find a 0-extension d of G with \sum c(xy)d(x, y) minimum. The 0-extension problem generalizes a number of basic combinatorial optimization problems, such as minimum (s,t)-cut problem and multiway cut problem. Karzanov proved the polynomial solvability of 0-Ext[G] for a certain large class of modular graphs G, and raised the question: What are the graphs G for which 0-Ext[G] can be solved in polynomial time? He also proved that 0-Ext[G] is NP-hard if G is not modular or not orientable (in a certain sense). In this paper, we prove the converse: if G is orientable and modular, then 0-Ext[G] can be solved in polynomial time. This completes the classification of graphs G for which 0-Ext[G] is tractable. To prove our main result, we develop a theory of discrete convex functions on orientable modular graphs, analogous to discrete convex analysis by Murota, and utilize a recent result of Thapper and Zivny on valued CSP.

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arXiv2011

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 $μ$, we define a metrized polyhedral complex, called the directed tight span $T_μ$, and prove that the dual of $μ$-weighted maximum multiflow problem reduces to a facility location problem on $T_μ$. Also, in case where the network is Eulerian, it further reduces to a facility location problem on the tropical polytope spanned by $μ$. 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.

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arXiv2010

On tight spans and tropical polytopes for directed distances

An extension $(V,d)$ of a metric space $(S,μ)$ is a metric space with $S \subseteq V$ and $d|_S = μ$, and is said to be tight if there is no other extension $(V,d')$ of $(S,μ)$ with $d' \leq d$. Isbell and Dress independently found that every tight extension is isometrically embedded into a certain metrized polyhedral complex associated with $(S,μ)$, called the tight span. This paper develops an analogous theory for directed metrics, which are "not necessarily symmetric" distance functions satisfying the triangle inequality. We introduce a directed version of the tight span and show that it has such a universal embedding property for tight extensions. Also we newly introduce another natural class of extensions, called cyclically tight extensions, and show that (a fiber of) the tropical polytope, introduced by Develin and Sturmfels, has a universal embedding property for cyclically tight extensions. As an application, we prove the following directed version of tree metric theorem: directed metric $μ$ is a directed tree metric if and only if the tropical rank of $-μ$ is at most two. Also we describe how tight spans and tropical polytopes are applied to the study in multicommodity flows in directed networks.

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