23 research outputs found
Characterizations of the set of integer points in an integral bisubmodular polyhedron
In this note, we provide two characterizations of the set of integer points
in an integral bisubmodular polyhedron. Our characterizations do not require
the assumption that a given set satisfies the hole-freeness, i.e., the set of
integer points in its convex hull coincides with the original set. One is a
natural multiset generalization of the exchange axiom of a delta-matroid, and
the other comes from the notion of the tangent cone of an integral bisubmodular
polyhedron.Comment: 9 page
Reconstructing Phylogenetic Tree From Multipartite Quartet System
A phylogenetic tree is a graphical representation of an evolutionary history in a set of taxa in which the leaves correspond to taxa and the non-leaves correspond to speciations. One of important problems in phylogenetic analysis is to assemble a global phylogenetic tree from smaller pieces of phylogenetic trees, particularly, quartet trees. Quartet Compatibility is to decide 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 Quartet Compatibility is NP-hard but 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 Quartet Compatibility for these systems. We also see that complete/full multipartite quartet systems naturally arise from a limited situation of block-restricted measurement
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 \v{Z}ivn\'{y} 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 \v{Z}ivn\'{y} 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.Comment: Full version of a STACS'18 pape
Quantaloidal approach to constraint satisfaction
The constraint satisfaction problem (CSP) is a computational problem that
includes a range of important problems in computer science. We point out that
fundamental concepts of the CSP, such as the solution set of an instance and
polymorphisms, can be formulated abstractly inside the 2-category
of finite sets and sets of functions between them.
The 2-category is a quantaloid, and the
formulation relies mainly on structure available in any quantaloid. This
observation suggests a formal development of generalisations of the CSP and
concomitant notions of polymorphism in a large class of quantaloids. We extract
a class of optimisation problems as a special case, and show that their
computational complexity can be classified by the associated notion of
polymorphism.Comment: 17 page
Finding a Maximum Restricted -Matching via Boolean Edge-CSP
The problem of finding a maximum -matching without short cycles has
received significant attention due to its relevance to the Hamilton cycle
problem. This problem is generalized to finding a maximum -matching which
excludes specified complete -partite subgraphs, where is a fixed
positive integer. The polynomial solvability of this generalized problem
remains an open question. In this paper, we present polynomial-time algorithms
for the following two cases of this problem: in the first case the forbidden
complete -partite subgraphs are edge-disjoint; and in the second case the
maximum degree of the input graph is at most . Our result for the first
case extends the previous work of Nam (1994) showing the polynomial solvability
of the problem of finding a maximum -matching without cycles of length four,
where the cycles of length four are vertex-disjoint. The second result expands
upon the works of B\'{e}rczi and V\'{e}gh (2010) and Kobayashi and Yin (2012),
which focused on graphs with maximum degree at most . Our algorithms are
obtained from exploiting the discrete structure of restricted -matchings and
employing an algorithm for the Boolean edge-CSP.Comment: 20 pages, 2 figure