arXiv subjects

Yoshiko Wakabayashi

Publications and source records attributed to Yoshiko Wakabayashi.

12 recordsLinked to original sources

arXiv2026

Improved Upper Bounds for Dynamic Bin Packing of General, Unit-Fraction, and Power-Fraction Squares

This paper presents significant upper-bound improvements for dynamic 2D square bin packing, where square items arrive and depart over time and the objective is to minimize the peak number of concurrent active unit bins. In our model, repacking is permitted only within a destination bin upon item arrival; migration between active bins is strictly forbidden. By introducing a streamlined two-list algorithm and proving a tight $5/16$ occupied-area bound for Next-Fit Decreasing Height, we reduce the upper bound on the asymptotic competitive ratio for arbitrary squares from 4.2154 down to 3.918, breaking a longstanding theoretical ceiling. For restricted variants, we establish asymptotic competitive ratios of at most 3.356 for unit-fraction side lengths and 2.211 for power-fraction side lengths.

cs.DS↗

arXiv2026

Optimal and quasi-optimal locating-dominating densities in the infinite hexagonal grid with a finite number of rows

A set of vertices $S$ of a graph $G$ is locating-dominating if $S$ is dominating and, for each pair of distinct vertices not in $S$, their neighborhoods in $S$ are distinct. We present results on the minimum density of such sets in the infinite hexagonal grid with a finite number of rows $k$, also known as the hexagonal strip of width $k$, which we denote by $H_k$. For each $k\geq 2$, we present either an optimal solution or a quasi-optimal solution for $H_k$ that is within $1.3\%$ of the optimum. We describe an exact exponential-time algorithm for fixed k, which we implemented to find optimal solutions for $k \leq 5$. As the infinite grid $H_{k}$ always admits a periodic optimal solution, to deal with larger values of $k$, we present an integer linear program that finds an optimal periodic solution for $H_{k}$ for each fixed period. This program yields high-quality feasible solutions for $H_7$ and $H_8$, which we then combine with an optimal solution for $H_3$ to obtain quasi-optimal solutions for all $k\geq 6$. All these solutions admit a very short description.

math.CO↗

arXiv2026

The complexity of minimum-density locating-dominating set in infinite periodic graphs

A dominating set $S$ of a graph $G$ is a locating-dominating set (LDS) if, for each pair of distinct vertices not in~$S$, their neighbourhoods in $S$ are distinct. Finding a minimum-cardinality LDS in finite graphs is a well-known NP-hard problem. On infinite graphs, this problem naturally generalises to finding an LDS of minimum density. While density bounds have been widely studied for specific infinite regular grids, no computational complexity results exist for infinite graphs. We prove that the minimum-density LDS problem in infinite $\mathbb{Z}$-periodic graphs with a finite period is NP-hard. This result bridges the gap between cardinality minimization on finite graphs and density minimization on infinite graphs via a rigorous periodic reduction. Furthermore, our approach can be adapted to establish NP-hardness for related structural problems on infinite periodic graphs.

math.CO↗

arXiv2025

Boundedness for proper conflict-free and odd colorings

The proper conflict-free chromatic number, $χ_{pcf}(G)$, of a graph $G$ is the least $k$ such that $G$ has a proper $k$-coloring in which for each non-isolated vertex there is a color appearing exactly once among its neighbors. The proper odd chromatic number, $χ_{o}(G)$, of $G$ is the least $k$ such that $G$ has a proper coloring in which for every non-isolated vertex there is a color appearing an odd number of times among its neighbors. We say that a graph class $\mathcal{G}$ is $χ_{pcf}$-bounded ($χ_{o}$-bounded) if there is a function $f$ such that $χ_{pcf}(G) \leq f(χ(G))$ ($χ_{o}(G) \leq f(χ(G))$) for every $G \in \mathcal{G}$. Caro et al. (2022) asked for classes that are linearly $χ_{pcf}$-bounded ($χ_{pcf}$-bounded), and as a starting point, they showed that every claw-free graph $G$ satisfies $χ_{pcf}(G) \le 2Δ(G)+1$, which implies $χ_{pcf}(G) \le 4χ(G)+1$. In this paper, we improve the bound for claw-free graphs to a nearly tight bound by showing that such a graph $G$ satisfies $χ_{pcf}(G) \le Δ(G)+6$, and even $χ_{pcf}(G) \le Δ(G)+4$ if it is a quasi-line graph. These results also give evidence for a conjecture by Caro et al. Moreover, we show that convex-round graphs and permutation graphs are linearly $χ_{pcf}$-bounded. For these last two results, we prove a lemma that reduces the problem of deciding if a hereditary class is linearly $χ_{pcf}$-bounded to deciding if the bipartite graphs in the class are $χ_{pcf}$-bounded by an absolute constant. This lemma complements a theorem of Liu (2022) and motivates us to study boundedness in bipartite graphs. In particular, we show that biconvex bipartite graphs are $χ_{pcf}$-bounded while convex bipartite graphs are not even $χ_o$-bounded, and exhibit a class of bipartite circle graphs that is linearly $χ_o$-bounded but not $χ_{pcf}$-bounded.

math.CO↗

arXiv2021

Approximation and parameterized algorithms to find balanced connected partitions of graphs

Partitioning a connected graph into $k$~vertex-disjoint connected subgraphs of similar (or given) orders is a classical problem that has been intensively investigated since late seventies. Given a connected graph $G=(V,E)$ and a weight function $w : V \to \mathbb{Q}_\geq$, a connected $k$-partition of $G$ is a partition of $V$ such that each class induces a connected subgraph. The balanced connected $k$-partition problem consists in finding a connected $k$-partition in which every class has roughly the same weight. To model this concept of balance, one may seek connected $k$-partitions that either maximize the weight of a lightest class $(\text{max-min BCP}_k)$ or minimize the weight of a heaviest class $(\text{min-max BCP}_k)$. Such problems are equivalent when $k=2$, but they are different when $k\geq 3$. In this work, we propose a simple pseudo-polynomial $\frac{k}{2}$-approximation algorithm for $\text{min-max BCP}_k$ which runs in time $\mathcal{O}(W|V||E|)$, where $W = \sum_{v \in V} w(v)$. Based on this algorithm and using a scaling technique, we design a (polynomial) $(\frac{k}{2} +\varepsilon)$-approximation for the same problem with running-time $\mathcal{O}(|V|^3|E|/\varepsilon)$, for any fixed $\varepsilon>0$. Additionally, we propose a fixed-parameter tractable algorithm based on integer linear programming for the unweighted $\text{max-min BCP}_k$ parameterized by the size of a vertex cover.

cs.DS↗

arXiv2019

Integer Programming Approaches to Balanced Connected $k$-Partition

We address the problem of partitioning a vertex-weighted connected graph into $k$ connected subgraphs that have similar weights, for a fixed integer $k\geq 2$. This problem, known as the \emph{balanced connected $k$-partition problem} ($BCP_k$), is defined as follows. Given a connected graph $G$ with nonnegative weights on the vertices, find a partition $\{V_i\}_{i=1}^k$ of $V(G)$ such that each class $V_i$ induces a connected subgraph of $G$, and the weight of a class with the minimum weight is as large as possible. It is known that $BCP_k$ is $NP$-hard even on bipartite graphs and on interval graphs. It has been largely investigated under different approaches and perspectives. On the practical side, $BCP_k$ is used to model many applications arising in police patrolling, image processing, cluster analysis, operating systems and robotics. We propose three integer linear programming formulations for the balanced connected $k$-partition problem. The first one contains only binary variables and a potentially large number of constraints that are separable in polynomial time. Some polyhedral results on this formulation, when all vertices have unit weight, are also presented. The other formulations are based on flows and have a polynomial number of constraints and variables. Preliminary computational experiments have shown that the proposed formulations outperform the other formulations presented in the literature.

cs.DM↗

arXiv2017

A tight lower bound for an online hypercube packing problem and bounds for prices of anarchy of a related game

We prove a tight lower bound on the asymptotic performance ratio $ρ$ of the bounded space online $d$-hypercube bin packing problem, solving an open question raised in 2005. In the classic $d$-hypercube bin packing problem, we are given a sequence of $d$-dimensional hypercubes and we have an unlimited number of bins, each of which is a $d$-dimensional unit hypercube. The goal is to pack (orthogonally) the given hypercubes into the minimum possible number of bins, in such a way that no two hypercubes in the same bin overlap. The bounded space online $d$-hypercube bin packing problem is a variant of the $d$-hypercube bin packing problem, in which the hypercubes arrive online and each one must be packed in an open bin without the knowledge of the next hypercubes. Moreover, at each moment, only a constant number of open bins are allowed (whenever a new bin is used, it is considered open, and it remains so until it is considered closed, in which case, it is not allowed to accept new hypercubes). Epstein and van Stee [SIAM J. Comput. 35 (2005), no. 2, 431-448] showed that $ρ$ is $Ω(\log d)$ and $O(d/\log d)$, and conjectured that it is $Θ(\log d)$. We show that $ρ$ is in fact $Θ(d/\log d)$. To obtain this result, we elaborate on some ideas presented by those authors, and go one step further showing how to obtain better (offline) packings of certain special instances for which one knows how many bins any bounded space algorithm has to use. Our main contribution establishes the existence of such packings, for large enough $d$, using probabilistic arguments. Such packings also lead to lower bounds for the prices of anarchy of the selfish $d$-hypercube bin packing game. We present a lower bound of $Ω(d/\log d)$ for the pure price of anarchy of this game, and we also give a lower bound of $Ω(\log d)$ for its strong price of anarchy.

cs.DS↗

arXiv2017

On path-cycle decompositions of triangle-free graphs

In this work, we study conditions for the existence of length-constrained path-cycle decompositions, that is, partitions of the edge set of a graph into paths and cycles of a given minimum length. Our main contribution is the characterization of the class of all triangle-free graphs with odd distance at least $3$ that admit a path-cycle decomposition with elements of length at least $4$. As a consequence, it follows that Gallai's conjecture on path decomposition holds in a broad class of sparse graphs.

math.CO↗

arXiv2017

Prices of anarchy of selfish 2D bin packing games

We consider a game-theoretical problem called selfish 2-dimensional bin packing game, a generalization of the 1-dimensional case already treated in the literature. In this game, the items to be packed are rectangles, and the bins are unit squares. The game starts with a set of items arbitrarily packed in bins. The cost of an item is defined as the ratio between its area and the total occupied area of the respective bin. Each item is a selfish player that wants to minimize its cost. A migration of an item to another bin is allowed only when its cost is decreased. We show that this game always converges to a Nash equilibrium (a stable packing where no single item can decrease its cost by migrating to another bin). We show that the pure price of anarchy of this game is unbounded, so we address the particular case where all items are squares. We show that the pure price of anarchy of the selfish square packing game is at least 2.3634 and at most 2.6875. We also present analogous results for the strong Nash equilibrium (a stable packing where no nonempty set of items can simultaneously migrate to another common bin and decrease the cost of each item in the set). We show that the strong price of anarchy when all items are squares is at least 2.0747 and at most 2.3605.

cs.GT↗

arXiv2015

Decomposing highly edge-connected graphs into paths of any given length

In 2006, Barát and Thomassen posed the following conjecture: for each tree $T$, there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(G)|$ is divisible by $|E(T)|$, then $G$ admits a decomposition into copies of $T$. This conjecture was verified for stars, some bistars, paths of length $3$, $5$, and $2^r$ for every positive integer $r$. We prove that this conjecture holds for paths of any fixed length.

math.CO↗

arXiv2015

Decompositions of highly connected graphs into paths of length five

We study the Decomposition Conjecture posed by Barát and Thomassen (2006), which states that for every tree $T$ there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(T)|$ divides $|E(G)|$, then $G$ admits a decomposition into copies of $T$. In a series of papers, Thomassen verified this conjecture for stars, some bistars, paths of length $3$, and paths whose length is a power of $2$. We verify the Decomposition Conjecture for paths of length $5$.

math.CO↗

arXiv2014

Polynomial-Time Approximation Schemes for Circle and Other Packing Problems

We give an asymptotic approximation scheme (APTAS) for the problem of packing a set of circles into a minimum number of unit square bins. To obtain rational solutions, we use augmented bins of height $1+γ$, for some arbitrarily small number $γ> 0$. Our algorithm is polynomial on $\log 1/γ$, and thus $γ$ is part of the problem input. For the special case that $γ$ is constant, we give a (one dimensional) resource augmentation scheme, that is, we obtain a packing into bins of unit width and height $1+γ$ using no more than the number of bins in an optimal packing. Additionally, we obtain an APTAS for the circle strip packing problem, whose goal is to pack a set of circles into a strip of unit width and minimum height. These are the first approximation and resource augmentation schemes for these problems. Our algorithm is based on novel ideas of iteratively separating small and large items, and may be extended to a wide range of packing problems that satisfy certain conditions. These extensions comprise problems with different kinds of items, such as regular polygons, or with bins of different shapes, such as circles and spheres. As an example, we obtain APTAS's for the problems of packing d-dimensional spheres into hypercubes under the $L_p$-norm.

cs.DS↗