arXiv subjects

Yuni Iwamasa

Publications and source records attributed to Yuni Iwamasa.

29 records · Page 2Linked to original sources

arXiv2022

Reconstructing phylogenetic trees from multipartite quartet systems

A phylogenetic tree is a graphical representation of an evolutionary history of taxa in which the leaves correspond to the taxa and the non-leaves correspond to speciations. One of important problems in phylogenetic analysis is to assemble a global phylogenetic tree from small phylogenetic trees, particularly, quartet trees. {\sc Quartet Compatibility} is the problem of deciding whether there is a phylogenetic tree inducing a given collection of quartet trees, and to construct such a phylogenetic tree if it exists. It is known that {\sc Quartet Compatibility} is NP-hard and that there are only a few results known for polynomial-time solvable subclasses. In this paper, we introduce two novel classes of quartet systems, called complete multipartite quartet system and full multipartite quartet system, and present polynomial-time algorithms for {\sc Quartet Compatibility} for these systems.

math.CO↗

arXiv2021

A combinatorial algorithm for computing the entire sequence of the maximum degree of minors of a generic partitioned polynomial matrix with $2 \times 2$ submatrices

In this paper, we consider the problem of computing the entire sequence of the maximum degree of minors of a block-structured symbolic matrix (a generic partitioned polynomial matrix) $A = (A_{αβ} x_{αβ} t^{d_{αβ}})$, where $A_{αβ}$ is a $2 \times 2$ matrix over a field $\mathbf{F}$, $x_{αβ}$ is an indeterminate, and $d_{αβ}$ is an integer for $α= 1,2,\dots, μ$ and $β= 1,2,\dots,ν$, and $t$ is an additional indeterminate. This problem can be viewed as an algebraic generalization of the maximum weight bipartite matching problem. The main result of this paper is a combinatorial $O(μν\min\{μ, ν\}^2)$-time algorithm for computing the entire sequence of the maximum degree of minors of a $(2 \times 2)$-type generic partitioned polynomial matrix of size $2μ\times 2ν$. We also present a minimax theorem, which can be used as a good characterization (NP $\cap$ co-NP characterization) for the computation of the maximum degree of minors of order $k$. Our results generalize the classical primal-dual algorithm (the Hungarian method) and minimax formula (Egerváry's theorem) for the maximum weight bipartite matching problem.

math.CO↗

arXiv2021

Monotone edge flips to an orientation of maximum edge-connectivity à la Nash-Williams

We initiate the study of $k$-edge-connected orientations of undirected graphs through edge flips for $k \geq 2$. We prove that in every orientation of an undirected $2k$-edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge-connectivity, and the final orientation is $k$-edge-connected. This yields an ``edge-flip based'' new proof of Nash-Williams' theorem: an undirected graph $G$ has a $k$-edge-connected orientation if and only if $G$ is $2k$-edge-connected. As another consequence of the theorem, we prove that the edge-flip graph of $k$-edge-connected orientations of an undirected graph $G$ is connected if $G$ is $(2k+2)$-edge-connected. This has been known to be true only when $k=1$.

math.CO↗

arXiv2021

A combinatorial algorithm for computing the rank of a generic partitioned matrix with $2 \times 2$ submatrices

In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) $A = (A_{αβ} x_{αβ})$, where $A_{αβ}$ is a $2 \times 2$ matrix over a field $\mathbf{F}$ and $x_{αβ}$ is an indeterminate for $α= 1,2,\dots, μ$ and $β= 1,2, \dots, ν$. This problem can be viewed as an algebraic generalization of the bipartite matching problem and was considered by Iwata and Murota (1995). Recent interests in this problem lie in the connection with non-commutative Edmonds' problem by Ivanyos, Qiao, and Subrahamanyam (2018) and Garg, Gurvits, Oliveiva, and Wigderson (2019), where a result by Iwata and Murota implicitly states that the rank and non-commutative rank (nc-rank) are the same for this class of symbolic matrices. The main result of this paper is a simple and combinatorial $O((μν)^2 \min \{ μ, ν\})$-time algorithm for computing the symbolic rank of a $(2 \times 2)$-type generic partitioned matrix of size $2μ\times 2ν$. Our algorithm is inspired by the Wong sequence algorithm by Ivanyos, Qiao, and Subrahamanyam for the nc-rank of a general symbolic matrix, and requires no blow-up operation, no field extension, and no additional care for bounding the bit-size. Moreover it naturally provides a maximum rank completion of $A$ for an arbitrary field $\mathbf{F}$.

math.CO↗

arXiv2021

Optimal matroid bases with intersection constraints: Valuated matroids, M-convex functions, and their applications

For two matroids $M_1$ and $M_2$ with the same ground set $V$ and two cost functions $w_1$ and $w_2$ on $2^V$, we consider the problem of finding bases $X_1$ of $M_1$ and $X_2$ of $M_2$ minimizing $w_1(X_1)+w_2(X_2)$ subject to a certain cardinality constraint on their intersection $X_1 \cap X_2$. For this problem, Lendl, Peis, and Timmermans (2019) discussed modular cost functions: they reduced the problem to weighted matroid intersection for the case where the cardinality constraint is $|X_1 \cap X_2|\le k$ or $|X_1 \cap X_2|\ge k$; and designed a new primal-dual algorithm for the case where the constraint is $|X_1 \cap X_2|=k$. The aim of this paper is to generalize the problems to have nonlinear convex cost functions, and to comprehend them from the viewpoint of discrete convex analysis. We prove that each generalized problem can be solved via valuated independent assignment, valuated matroid intersection, or $\mathrm{M}$-convex submodular flow, to offer a comprehensive understanding of weighted matroid intersection with intersection constraints. We also show the NP-hardness of some variants of these problems, which clarifies the coverage of discrete convex analysis for those problems. Finally, we present applications of our generalized problems in the recoverable robust matroid basis problem, combinatorial optimization problems with interaction costs, and matroid congestion games.

math.CO↗

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↗

arXiv2018

Discrete Convexity in Joint Winner Property

In this paper, we reveal a relation between joint winner property (JWP) in the field of valued constraint satisfaction problems (VCSPs) and M${}^\natural$-convexity in the field of discrete convex analysis (DCA). We introduce the M${}^\natural$-convex completion problem, and show that a function $f$ satisfying the JWP is Z-free if and only if a certain function $\overline{f}$ associated with $f$ is M${}^\natural$-convex completable. This means that if a function is Z-free, then the function can be minimized in polynomial time via M${}^\natural$-convex intersection algorithms. Furthermore we propose a new algorithm for Z-free function minimization, which is faster than previous algorithms for some parameter values.

cs.DM↗

arXiv2018

The quadratic M-convexity testing problem

M-convex functions, which are a generalization of valuated matroids, play a central role in discrete convex analysis. Quadratic M-convex functions constitute a basic and important subclass of M-convex functions, which has a close relationship with phylogenetics as well as valued constraint satisfaction problems. In this paper, we consider the quadratic M-convexity testing problem (QMCTP), which is the problem of deciding whether a given quadratic function on $\{0,1\}^n$ is M-convex. We show that QMCTP is co-NP-complete in general, but is polynomial-time solvable under a natural assumption. Furthermore, we propose an $O(n^2)$-time algorithm for solving QMCTP in the polynomial-time solvable case.

math.OC↗

arXiv2017

On a general framework for network representability in discrete optimization

In discrete optimization, representing an objective function as an $s$-$t$ cut function of a network is a basic technique to design an efficient minimization algorithm. A network representable function can be minimized by computing a minimum $s$-$t$ cut of a directed network, which is a very easy and fastly solved problem. Hence it is natural to ask what functions are network representable. In the case of pseudo Boolean functions (functions on $\{0,1\}^n$), it is known that any submodular function on $\{0,1\}^3$ is network representable. Živný--Cohen--Jeavons showed by using the theory of expressive power that a certain submodular function on $\{0,1\}^4$ is not network representable. In this paper, we introduce a general framework for the network representability of functions on $D^n$, where $D$ is an arbitrary finite set. We completely characterize network representable functions on $\{0,1\}^n$ in our new definition. We can apply the expressive power theory to the network representability in the proposed definition. We prove that some ternary bisubmodular function and some binary $k$-submodular function are not network representable.

math.OC↗

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.

math.OC↗

arXiv2014

Networks maximizing the consensus time of voter models

We explore the networks that yield the largest mean consensus time of voter models under different update rules. By analytical and numerical means, we show that the so-called lollipop graph, barbell graph, and double star graph maximise the mean consensus time under the update rules called the link dynamics, voter model, and invasion process, respectively. For each update rule, the largest mean consensus time scales as O(N^3), where N is the number of nodes in the network.

cond-mat.dis-nn↗