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Hiroshi Hirai

Publications and source records attributed to Hiroshi Hirai.

At least 37 records · Page 2Linked to original sources
arXiv2020

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 Lovász-Cherkassky theorem to the node-capacity setting. Our results build on discrete convexity in the node-connectivity terminal backup problem.

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arXiv2020

Compression of M${}^\natural$-convex Functions -- Flag Matroids and Valuated Permutohedra

Murota (1998) and Murota and Shioura (1999) introduced concepts of M-convex function and M${}^\natural$-convex function as discrete convex functions, which are generalizations of valuated matroids due to Dress and Wenzel (1992). In the present paper we consider a new operation defined by a convolution of sections of an M${}^\natural$-convex function that transforms the given M${}^\natural$-convex function to an M-convex function, which we call a compression of an M${}^\natural$-convex function. For the class of valuated generalized matroids, which are special M${}^\natural$-convex functions, the compression induces a valuated permutohedron together with a decomposition of the valuated generalized matroid into flag-matroid strips, each corresponding to a maximal linearity domain of the induced valuated permutohedron. We examine the details of the structure of flag-matroid strips and the induced valuated permutohedron by means of discrete convex analysis of Murota.

math.CO↗
arXiv2020

On a weighted linear matroid intersection algorithm by deg-det computation

In this paper, we address the weighted linear matroid intersection problem from the computation of the degree of the determinants of a symbolic matrix. We show that a generic algorithm computing the degree of noncommutative determinants, proposed by the second author, becomes an $O(mn^3 \log n)$ time algorithm for the weighted linear matroid intersection problem, where two matroids are given by column vectors $n \times m$ matrices $A,B$. We reveal that our algorithm is viewed as a "nonstandard" implementation of Frank's weight splitting algorithm for linear matroids. This gives a linear algebraic reasoning to Frank's algorithm. Although our algorithm is slower than existing algorithms in the worst case estimate, it has a notable feature: Contrary to existing algorithms, our algorithm works on different matroids represented by another "sparse" matrices $A^0,B^0$, which skips unnecessary Gaussian eliminations for constructing residual graphs.

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arXiv2019

Uniform modular lattices and affine buildings

In this paper, we present a simple lattice-theoretic characterization for affine buildings of type A. We introduce a class of modular lattices, called uniform modular lattices, and show that uniform modular lattices and affine buildings of type A constitute the same object. This is an affine counterpart of the well-known equivalence between projective geometries ($\simeq$ complemented modular lattices) and spherical buildings of type A.

math.CO↗
arXiv2019

Computing the degree of determinants via discrete convex optimization on Euclidean buildings

In this paper, we consider the computation of the degree of the Dieudonné determinant of a linear symbolic matrix $A = A_0 + A_1 x_1 + \cdots + A_m x_m$, where each $A_i$ is an $n \times n$ polynomial matrix over $\mathbb{K}[t]$ and $x_1,x_2,\ldots,x_m$ are pairwise "non-commutative" variables. This quantity is regarded as a weighted generalization of the non-commutative rank (nc-rank) of a linear symbolic matrix, and its computation is shown to be a generalization of several basic combinatorial optimization problems, such as weighted bipartite matching and weighted linear matroid intersection problems. Based on the work on nc-rank by Fortin and Rautenauer (2004), and Ivanyos, Qiao, and Subrahmanyam (2018), we develop a framework to compute the degree of the Dieudonné determinant of a linear symbolic matrix. We show that the deg-det computation reduces to a discrete convex optimization problem on the Euclidean building for ${\rm SL}(\mathbb{K}(t)^n)$. To deal with this optimization problem, we introduce a class of discrete convex functions on the building. This class is a natural generalization of L-convex functions in discrete convex analysis (DCA). We develop a DCA-oriented algorithm (steepest descent algorithm) to compute the degree of determinants. Our algorithm works with matrix computation on $\mathbb{K}$, and uses a subroutine to compute a certificate vector subspace for the nc-rank, where the number of calls of the subroutine is sharply estimated. Our algorithm enhances some classical combinatorial optimization algorithms with new insights, and is also understood as a variant of the combinatorial relaxation algorithm, which was developed earlier by Murota for computing the degree of the (ordinary) determinant.

math.OC↗
arXiv2019

A Cost-Scaling Algorithm for Minimum-Cost Node-Capacitated Multiflow Problem

In this paper, we address the minimum-cost node-capacitated multiflow problem in an undirected network. For this problem, Babenko and Karzanov (2012) showed strongly polynomial-time solvability via the ellipsoid method. Our result is the first combinatorial weakly polynomial-time algorithm for this problem. Our algorithm finds a half-integral minimum-cost maximum multiflow in $O(m \log(nCD)\mathrm{SF}(kn, m, k))$ time, where $n$ is the number of nodes, $m$ is the number of edges, $k$ is the number of terminals, $C$ is the maximum node capacity, $D$ is the maximum edge cost, and $\mathrm{SF}(n', m', η)$ is the time complexity of solving the submodular flow problem in a network of $n'$ nodes, $m'$ edges, and a submodular function with $η$-time-computable exchange capacity. Our algorithm is built on discrete convex analysis on graph structures and the concept of reducible bisubmodular flows.

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arXiv2019

A tractable class of binary VCSPs via M-convex intersection

A binary VCSP is a general framework for the minimization problem of a function represented as the sum of unary and binary cost functions. An important line of VCSP research is to investigate what functions can be solved in polynomial time. Cooper and Živný classified the tractability of binary VCSP instances according to the concept of "triangle," and showed that the only interesting tractable case is the one induced by the joint winner property (JWP). Recently, Iwamasa, Murota, and Živný made a link between VCSP and discrete convex analysis, showing that a function satisfying the JWP can be transformed into a function represented as the sum of two quadratic M-convex functions, which can be minimized in polynomial time via an M-convex intersection algorithm if the value oracle of each M-convex function is given. In this paper, we give an algorithmic answer to a natural question: What binary finite-valued CSP instances can be represented as the sum of two quadratic M-convex functions and can be solved in polynomial time via an M-convex intersection algorithm? We solve this problem by devising a polynomial-time algorithm for obtaining a concrete form of the representation in the representable case. Our result presents a larger tractable class of binary finite-valued CSPs, which properly contains the JWP class.

cs.DM↗
arXiv2019

A Nonpositive Curvature Property of Modular Semilattices

The orthoscheme complex of a graded poset is a metrization of its order complex such that the simplex of each maximal chain is isometric to the Euclidean simplex of vertices $0, e_1,e_1+e_2,\ldots, e_1+e_2+ \cdots + e_n$. This notion was introduced by Brady and McCammond in geometric group theory, and has applications in discrete optimization and submodularity theory. We address a question of what posets to yield the orthoscheme complex having CAT(0) property. The orthoscheme complex of a modular lattice is shown to be CAT(0), and it is conjectured that this is the case for a modular semilattice. In this paper, we prove this conjecture affirmatively. This result implies that a larger class of weakly modular graphs yields CAT(0) complexes.

math.MG↗
arXiv2019

Uniform semimodular lattices and valuated matroids

In this paper, we present a lattice-theoretic characterization for valuated matroids, which is an extension of the well-known cryptomorphic equivalence between matroids and geometric lattices ($=$ atomistic semimodular lattices). We introduce a class of semimodular lattices, called uniform semimodular lattices, and establish a cryptomorphic equivalence between integer-valued valuated matroids and uniform semimodular lattices. Our result includes a coordinate-free lattice-theoretic characterization of integer points in tropical linear spaces, incorporates the Dress-Terhalle completion process of valuated matroids, and establishes a smooth connection with Euclidean buildings of type A.

math.CO↗
arXiv2018

A dual descent algorithm for node-capacitated multiflow problems and its applications

In this paper, we develop an $O((m \log k) {\rm MSF} (n,m,1))$-time algorithm to find a half-integral node-capacitated multiflow of the maximum total flow-value in a network with $n$ nodes, $m$ edges, and $k$ terminals, where ${\rm MSF} (n',m',γ)$ denotes the time complexity of solving the maximum submodular flow problem in a network with $n'$ nodes, $m'$ edges, and the complexity $γ$ of computing the exchange capacity of the submodular function describing the problem. By using Fujishige-Zhang algorithm for submodular flow, we can find a maximum half-integral multiflow in $O(m n^3 \log k)$ time. This is the first combinatorial strongly polynomial time algorithm for this problem. Our algorithm is built on a developing theory of discrete convex functions on certain graph structures. Applications include "ellipsoid-free" combinatorial implementations of a 2-approximation algorithm for the minimum node-multiway cut problem by Garg, Vazirani, and Yannakakis.

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arXiv2018

Polyhedral Clinching Auctions for Two-sided Markets

In this paper, we present a new model and two mechanisms for auctions in two-sided markets of buyers and sellers, where budget constraints are imposed on buyers. Our model incorporates polymatroidal environments, and is applicable to a wide variety of models that include multiunit auctions, matching markets and reservation exchange markets. Our mechanisms are build on polymatroidal network flow model by Lawler and Martel, and enjoy various nice properties such as incentive compatibility of buyers, individual rationality, pareto optimality, strong budget balance. The first mechanism is a simple "reduce-to-recover" algorithm that reduces the market to be one-sided, applies the polyhedral clinching auction by Goel et al, and lifts the resulting allocation to the original two-sided market via polymatroidal network flow. The second mechanism is a two-sided generalization of the polyhedral clinching auction, which improves the first mechanism in terms of the fairness of revenue sharing on sellers. Both mechanisms are implemented by polymatroid algorithms. We demonstrate how our framework is applied to internet display ad auctions.

cs.GT↗
arXiv2018

A representation of antimatroids by Horn rules and its application to educational systems

We study a representation of an antimatroid by Horn rules, motivated by its recent application to computer-aided educational systems. We associate any set $\mathcal{R}$ of Horn rules with the unique maximal antimatroid $\mathcal{A}(\mathcal{R})$ that is contained in the union-closed family $\mathcal{K}(\mathcal{R})$ naturally determined by ${\cal R}$. We address algorithmic and Boolean function theoretic aspects on the association ${\cal R} \mapsto \mathcal{A}(\mathcal{R})$, where ${\cal R}$ is viewed as the input. We present linear time algorithms to solve the membership problem and the inference problem for ${\cal A}({\cal R})$. We also provide efficient algorithms for generating all members and all implicates of ${\cal A}({\cal R})$. We show that this representation is essentially equivalent to the Korte-Lovász representation of antimatroids by rooted sets. Based on the equivalence, we provide a quadratic time algorithm to construct the uniquely-determined minimal representation. % These results have potential applications to computer-aided educational systems, where an antimatroid is used as a model of the space of possible knowledge states of learners, and is constructed by giving Horn queries to a human expert.

math.CO↗
arXiv2018

Counting Integral Points in Polytopes via Numerical Analysis of Contour Integration

In this paper, we address the problem of counting integer points in a rational polytope described by $P(y) = \{ x \in \mathbb{R}^m \colon Ax = y, x \geq 0\}$, where $A$ is an $n \times m$ integer matrix and $y$ is an $n$-dimensional integer vector. We study the Z-transformation approach initiated by Brion-Vergne, Beck, and Lasserre-Zeron from the numerical analysis point of view, and obtain a new algorithm on this problem: If $A$ is nonnegative, then the number of integer points in $P(y)$ can be computed in $O(\mathrm{poly} (n,m, \|y\|_\infty) (\|y\|_\infty + 1)^n)$ time and $O(\mathrm{poly} (n,m, \|y\|_\infty))$ space.This improves, in terms of space complexity, a naive DP algorithm with $O((\|y\|_\infty + 1)^n)$-size DP table. Our result is based on the standard error analysis to the numerical contour integration for the inverse Z-transform, and establish a new type of an inclusion-exclusion formula for integer points in $P(y)$. We apply our result to hypergraph $b$-matching, and obtain a $O(\mathrm{poly}( n,m,\|b\|_\infty) (\|b\|_\infty +1)^{(1-1/k)n})$ time algorithm for counting $b$-matchings in a $k$-partite hypergraph with $n$ vertices and $m$ hyperedges. This result is viewed as a $b$-matching generalization of the classical result by Ryser for $k=2$ and its multipartite extension by Bj{ö}rklund-Husfeldt.

cs.DM↗
arXiv2017

Weakly modular graphs and nonpositive curvature

This article investigates structural, geometrical, and topological characterizations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various `nonpositive curvature' and `local-to-global' properties and characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a far-reaching common generalization of median graphs (and more generally, of modular and orientable modular graphs), Helly graphs, bridged graphs, and dual polar graphs occurring under different disguises in several seemingly-unrelated fields of mathematics: Metric graph theory, Geometric group theory, Incidence geometries and buildings, Theoretical computer science and combinatorial optimization. We give a local-to-global characterization of weakly modular graphs and their subclasses in terms of simple connectedness of associated triangle-square complexes and specific local combinatorial conditions. In particular, we revisit characterizations of dual polar graphs by Cameron and by Brouwer-Cohen. We also show that (disk-)Helly graphs are precisely the clique-Helly graphs with simply connected clique complexes. With $l_1$-embeddable weakly modular and sweakly modular graphs we associate high-dimensional cell complexes, having several strong topological and geometrical properties (contractibility and the CAT(0) property). Their cells have a specific structure: they are basis polyhedra of even $\triangle$-matroids in the first case and orthoscheme complexes of gated dual polar subgraphs in the second case. We resolve some open problems concerning subclasses of weakly modular graphs: we prove a Brady-McCammond conjecture about CAT(0) metric on the orthoscheme complexes of modular lattices; we answer Chastand's question about prime graphs for pre-median graphs.

math.MG↗
arXiv2017

Maximum vanishing subspace problem, CAT(0)-space relaxation, and block-triangularization of partitioned matrix

In this paper, we address the following algebraic generalization of the bipartite stable set problem. We are given a block-structured matrix (partitioned matrix) $A = (A_{αβ})$, where $A_{αβ}$ is an $m_α$ by $n_β$ matrix over field ${\bf F}$ for $α=1,2,\ldots,μ$ and $β= 1,2,\ldots,ν$. The maximum vanishing subspace problem (MVSP) is to maximize $\sum_α \dim X_α + \sum_β \dim Y_β$ over vector subspaces $X_α \subseteq {\bf F}^{m_α}$ for $α=1,2,\ldots,μ$ and $Y_β \subseteq {\bf F}^{n_β}$ for $β= 1,2,\ldots,ν$ such that each $A_{αβ}$ vanishes on $X_α \times Y_β$ when $A_{αβ}$ is viewed as a bilinear form ${\bf F}^{m_α} \times {\bf F}^{n_β} \to {\bf F}$. This problem arises from a study of a canonical block-triangular form of $A$ by Ito, Iwata, and Murota~(1994), and is closely related to the noncommutative rank of a matrix with indeterminates. We prove that a weighted version (WMVP) of MVSP can be solved in psuedo polynomial time, provided arithmetic operations on ${\bf F}$ can be done in constant time. Our proof is a novel combination of submodular optimization on modular lattice and convex optimization on CAT(0)-space. We present implications of this result on block-triangularization of partitioned matrix.

math.OC↗
arXiv2017

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.

math.OC↗
arXiv2017

On integer network synthesis problem with tree-metric cost

Network synthesis problem (NSP) is the problem of designing a minimum-cost network (from the empty network) satisfying a given connectivity requirement. Hau, Hirai, and Tsuchimura showed that if the edge-cost is a tree metric, then a simple greedy-type algorithm solves NSP to obtain a half-integral optimal solution. This is a generalization of the classical result by Gomory and Hu for the uniform edge-cost. In this note, we present an integer version of Hau, Hirai, and Tsuchimura's result for integer network synthesis problem (INSP), where a required network must have an integer capacity. We prove that if each connectivity requirement is at least 2 and the edge-cost is a tree metrc, then INSP can be solved in polynomial time.

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