arXiv subjects

Naomi Nishimura

Publications and source records attributed to Naomi Nishimura.

At least 19 recordsLinked to original sources

arXiv2025

On Algorithmic Meta-Theorems for Solution Discovery: Tractability and Barriers

Solution discovery asks whether a given (infeasible) starting configuration to a problem can be transformed into a feasible solution using a limited number of transformation steps. This paper investigates meta-theorems for solution discovery for graph problems definable in monadic second-order logic (MSO$_1$ and MSO$_2$) and first-order logic (FO) where the transformation step is to slide a token to an adjacent vertex, focusing on parameterized complexity and structural graph parameters that do not involve the transformation budget $b$. We present both positive and negative results. On the algorithmic side, we prove that MSO$_2$-Discovery is in XP when parameterized by treewidth and that MSO$_1$-Discovery is fixed-parameter tractable when parameterized by neighborhood diversity. On the hardness side, we establish that FO-Discovery is W[1]-hard when parameterized by modulator to stars, modulator to paths, as well as twin cover, numbers. Additionally, we prove that MSO$_1$-Discovery is W[1]-hard when parameterized by bandwidth. These results complement the straightforward observation that solution discovery for the studied problems is fixed-parameter tractable when the budget $b$ is included in the parameter (in particular, parameterized by cliquewidth$+b$, where the cliquewidth of a graph is at most any of the studied parameters), and provide a near-complete (fixed-parameter tractability) meta-theorems investigation for solution discovery problems for MSO- and FO-definable graph problems and structural parameters larger than cliquewidth.

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arXiv2025

On the complexity of constrained reconfiguration and motion planning

Coordinating the motion of multiple agents in constrained environments is a fundamental challenge in robotics, motion planning, and scheduling. A motivating example involves $n$ robotic arms, each represented as a line segment. The objective is to rotate each arm to its vertical orientation, one at a time (clockwise or counterclockwise), without collisions nor rotating any arm more than once. This scenario is an example of the more general $k$-Compatible Ordering problem, where $n$ agents, each capable of $k$ state-changing actions, must transition to specific target states under constraints encoded as a set $\mathcal{G}$ of $k$ pairs of directed graphs. We show that $k$-Compatible Ordering is $\mathsf{NP}$-complete, even when $\mathcal{G}$ is planar, degenerate, or acyclic. On the positive side, we provide polynomial-time algorithms for cases such as when $k = 1$ or $\mathcal{G}$ has bounded treewidth. We also introduce generalized variants supporting multiple state-changing actions per agent, broadening the applicability of our framework. These results extend to a wide range of scheduling, reconfiguration, and motion planning applications in constrained environments.

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arXiv2025

Efficient algorithms to solve atom reconfiguration problems. III. The bird and batching algorithms and other parallel implementations on GPUs

We present efficient implementations of atom reconfiguration algorithms for both CPUs and GPUs, along with a batching routine to merge displacement operations for parallel execution. Leveraging graph-theoretic methods, our approach derives improved algorithms that achieve reduced time complexity and faster operational running times. First, we introduce an enhanced version of the redistribution-reconfiguration (red-rec) algorithm, which offers superior operational and runtime performance. We detail its efficient implementation on a GPU using a parallel approach. Next, we present an optimized version of the assignment-reconfiguration-ordering (aro) algorithm, specifically tailored for unweighted grid graphs. Finally, we introduce the bird algorithm to solve reconfiguration problems on grids, achieving performance gains over both red-rec and aro. These algorithms can be used to prepare defect-free configurations of neutral atoms in arrays of optical traps, serve as subroutines in more complex algorithms, or cross-benchmark the operational and runtime performance of new algorithms. They are suitable for realizing quantum circuits incorporating displacement operations and are optimized for real-time operation on increasingly large system sizes.

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arXiv2024

Kernelization Complexity of Solution Discovery Problems

In the solution discovery variant of a vertex (edge) subset problem $\Pi$ on graphs, we are given an initial configuration of tokens on the vertices (edges) of an input graph $G$ together with a budget $b$. The question is whether we can transform this configuration into a feasible solution of $\Pi$ on $G$ with at most $b$ modification steps. We consider the token sliding variant of the solution discovery framework, where each modification step consists of sliding a token to an adjacent vertex (edge). The framework of solution discovery was recently introduced by Fellows et al. [Fellows et al., ECAI 2023] and for many solution discovery problems the classical as well as the parameterized complexity has been established. In this work, we study the kernelization complexity of the solution discovery variants of Vertex Cover, Independent Set, Dominating Set, Shortest Path, Matching, and Vertex Cut with respect to the parameters number of tokens $k$, discovery budget $b$, as well as structural parameters such as pathwidth.

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arXiv2024

Reconfiguration of Multisets with Applications to Bin Packing

We use the reconfiguration framework to analyze problems that involve the rearrangement of items among groups. In various applications, a group of items could correspond to the files or jobs assigned to a particular machine, and the goal of rearrangement could be improving efficiency or increasing locality. To cover problems arising in a wide range of application areas, we define the general Repacking problem as the rearrangement of multisets of multisets. We present hardness results for the general case and algorithms for various classes of instances that arise in real-life scenarios. By limiting the total size of items in each multiset, our results can be viewed as an offline approach to Bin Packing, in which each bin is represented as a multiset. In addition to providing the first results on reconfiguration of multisets, our contributions open up several research avenues: the interplay between reconfiguration and online algorithms and parallel algorithms; the use of the tools of linear programming in reconfiguration; and, in the longer term, a focus on extra resources in reconfiguration.

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arXiv2022

Efficient algorithms to solve atom reconfiguration problems. II. The assignment-rerouting-ordering (aro) algorithm

Programmable arrays of optical traps enable the assembly of configurations of single atoms to perform controlled experiments on quantum many-body systems. Finding the sequence of control operations to transform an arbitrary configuration of atoms into a predetermined one requires solving an atom reconfiguration problem quickly and efficiently. A typical approach to solve atom reconfiguration problems is to use an assignment algorithm to determine which atoms to move to which traps. This approach results in control protocols that exactly minimize the number of displacement operations; however, this approach does not optimize for the number of displaced atoms or the number of times each atom is displaced, resulting in unnecessary control operations that increase the execution time and failure rate of the control protocol. In this work, we propose the assignment-rerouting-ordering (aro) algorithm to improve the performance of assignment-based algorithms in solving atom reconfiguration problems. The aro algorithm uses an assignment subroutine to minimize the total distance traveled by all atoms, a rerouting subroutine to reduce the number of displaced atoms, and an ordering subroutine to guarantee that each atom is displaced at most once. The ordering subroutine relies on the existence of a partial ordering of moves that can be obtained using a polynomial-time algorithm that we introduce within the formal framework of graph theory. We numerically quantify the performance of the aro algorithm in the presence and in the absence of loss, and show that it outperforms the exact, approximation, and heuristic algorithms that we use as benchmarks. Our results are useful for assembling large configurations of atoms with high success probability and fast preparation time, as well as for designing and benchmarking novel atom reconfiguration algorithms.

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arXiv2022

Efficient algorithms to solve atom reconfiguration problems. I. The redistribution-reconfiguration (red-rec) algorithm

We propose the redistribution-reconfiguration (red-rec) algorithm to prepare large configurations of atoms using arrays of dynamic optical traps. Red-rec exploits simple heuristics and exact subroutines to solve atom reconfiguration problems on grids. It admits a fast and efficient implementation suitable for real-time operation. We numerically quantify its performance using realistic physical parameters and operational constraints, both in the absence and presence of loss. Red-rec enables assembling large configurations of atoms with high mean success probability. Fast preparation times are achieved by harnessing parallel control operations that actuate multiple traps simultaneously. Faster preparation times are achieved by rejecting configurations of atoms containing fewer atoms than a given threshold. However, the number of traps required to prepare a compact-centered configuration of atoms on a grid with a probability of 0.5 scales as the 3/2 power of the number of desired atoms. This finding highlights some of the challenges associated with scaling up configurations of atoms beyond tens of thousands of atoms.

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arXiv2022

A survey on the parameterized complexity of the independent set and (connected) dominating set reconfiguration problems

A graph vertex-subset problem defines which subsets of the vertices of an input graph are feasible solutions. We view a feasible solution as a set of tokens placed on the vertices of the graph. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions of size $k$, whether it is possible to transform one into the other by a sequence of token slides (along edges of the graph) or token jumps (between arbitrary vertices of the graph) such that each intermediate set remains a feasible solution of size $k$. Many algorithmic questions present themselves in the form of reconfiguration problems: Given the description of an initial system state and the description of a target state, is it possible to transform the system from its initial state into the target one while preserving certain properties of the system in the process? Such questions have received a substantial amount of attention under the so-called combinatorial reconfiguration framework. We consider reconfiguration variants of three fundamental underlying graph vertex-subset problems, namely Independent Set, Dominating Set, and Connected Dominating Set. We survey both older and more recent work on the parameterized complexity of all three problems when parameterized by the number of tokens $k$. The emphasis will be on positive results and the most common techniques for the design of fixed-parameter tractable algorithms.

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arXiv2021

Parameterized complexity of reconfiguration of atoms

Our work is motivated by the challenges presented in preparing arrays of atoms for use in quantum simulation. The recently-developed process of loading atoms into traps results in approximately half of the traps being filled. To consolidate the atoms so that they form a dense and regular arrangement, such as all locations in a grid, atoms are rearranged using moving optical tweezers. Time is of the essence, as the longer that the process takes and the more that atoms are moved, the higher the chance that atoms will be lost in the process. Viewed as a problem on graphs, we wish to solve the problem of reconfiguring one arrangement of tokens (representing atoms) to another using as few moves as possible. Because the problem is NP-complete on general graphs as well as on grids, we focus on the parameterized complexity for various parameters, considering both undirected and directed graphs, and tokens with and without labels. For unlabelled tokens, the problem is in FPT when parameterizing by the number of tokens, the number of moves, or the number of moves plus the number of vertices without tokens in either the source or target configuration, but intractable when parameterizing by the difference between the number of moves and the number of differences in the placement of tokens in the source and target configurations. When labels are added to tokens, however, most of the tractability results are replaced by hardness results.

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arXiv2020

Vertex Cover Reconfiguration and Beyond

In the Vertex Cover Reconfiguration (VCR) problem, given a graph $G$, positive integers $k$ and $\ell$ and two vertex covers $S$ and $T$ of $G$ of size at most $k$, we determine whether $S$ can be transformed into $T$ by a sequence of at most $\ell$ vertex additions or removals such that every operation results in a vertex cover of size at most $k$. Motivated by results establishing the W[1]-hardness of VCR when parameterized by $\ell$, we delineate the complexity of the problem restricted to various graph classes. In particular, we show that VCR remains W[1]-hard on bipartite graphs, is NP-hard, but fixed-parameter tractable on (regular) graphs of bounded degree and more generally on nowhere dense graphs and is solvable in polynomial time on trees and (with some additional restrictions) on cactus graphs.

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arXiv2018

Incremental Optimization of Independent Sets under Reachability Constraints

We introduce a new framework for reconfiguration problems, and apply it to independent sets as the first example. Suppose that we are given an independent set $I_0$ of a graph $G$, and an integer $l \ge 0$ which represents a lower bound on the size of any independent set of $G$. Then, we are asked to find an independent set of $G$ having the maximum size among independent sets that are reachable from $I_0$ by either adding or removing a single vertex at a time such that all intermediate independent sets are of size at least $l$. We show that this problem is PSPACE-hard even for bounded pathwidth graphs, and remains NP-hard for planar graphs. On the other hand, we give a linear-time algorithm to solve the problem for chordal graphs. We also study the fixed-parameter (in)tractability of the problem with respect to the following three parameters: the degeneracy $d$ of an input graph, a lower bound $l$ on the size of the independent sets, and a lower bound $s$ on the solution size. We show that the problem is fixed-parameter intractable when only one of $d$, $l$, and $s$ is taken as a parameter. On the other hand, we give a fixed-parameter algorithm when parameterized by $s+d$; this result implies that the problem parameterized only by $s$ is fixed-parameter tractable for planar graphs, and for bounded treewidth graphs.

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arXiv2018

Reconfiguration of graph minors

Under the reconfiguration framework, we consider the various ways that a target graph $H$ is a {\em minor} of a host graph $G$, where a subgraph of $G$ can be transformed into $H$ by means of {\em edge contraction} (replacement of both endpoints of an edge by a new vertex adjacent to any vertex adjacent to either endpoint). Equivalently, an {\em $H$-model} of $G$ is a labeling of the vertices of $G$ with the vertices of $H$, where the contraction of all edges between identically-labeled vertices results in a graph containing representations of all edges in $H$. We explore the properties of $G$ and $H$ that result in a connected {\em reconfiguration graph}, in which nodes represent $H$-models and two nodes are adjacent if their corresponding $H$-models differ by the label of a single vertex of $G$. Various operations on $G$ or $H$ are shown to preserve connectivity. In addition, we demonstrate properties of graphs $G$ that result in connectivity for the target graphs $K_2$, $K_3$, and $K_4$, including a full characterization of graphs $G$ that result in connectivity for $K_2$-models, as well as the relationship between connectivity of $G$ and other $H$-models.

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arXiv2018

Reconfiguring spanning and induced subgraphs

Subgraph reconfiguration is a family of problems focusing on the reachability of the solution space in which feasible solutions are subgraphs, represented either as sets of vertices or sets of edges, satisfying a prescribed graph structure property. Although there has been previous work that can be categorized as subgraph reconfiguration, most of the related results appear under the name of the property under consideration; for example, independent set, clique, and matching. In this paper, we systematically clarify the complexity status of subgraph reconfiguration with respect to graph structure properties.

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arXiv2015

The complexity of dominating set reconfiguration

Suppose that we are given two dominating sets $D_s$ and $D_t$ of a graph $G$ whose cardinalities are at most a given threshold $k$. Then, we are asked whether there exists a sequence of dominating sets of $G$ between $D_s$ and $D_t$ such that each dominating set in the sequence is of cardinality at most $k$ and can be obtained from the previous one by either adding or deleting exactly one vertex. This problem is known to be PSPACE-complete in general. In this paper, we study the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem remains PSPACE-complete even for planar graphs, bounded bandwidth graphs, split graphs, and bipartite graphs. We then give a general scheme to construct linear-time algorithms and show that the problem can be solved in linear time for cographs, trees, and interval graphs. Furthermore, for these tractable cases, we can obtain a desired sequence such that the number of additions and deletions is bounded by $O(n)$, where $n$ is the number of vertices in the input graph.

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arXiv2014

Reconfiguration over tree decompositions

A vertex-subset graph problem $Q$ defines which subsets of the vertices of an input graph are feasible solutions. The reconfiguration version of a vertex-subset problem $Q$ asks whether it is possible to transform one feasible solution for $Q$ into another in at most $\ell$ steps, where each step is a vertex addition or deletion, and each intermediate set is also a feasible solution for $Q$ of size bounded by $k$. Motivated by recent results establishing W[1]-hardness of the reconfiguration versions of most vertex-subset problems parameterized by $\ell$, we investigate the complexity of such problems restricted to graphs of bounded treewidth. We show that the reconfiguration versions of most vertex-subset problems remain PSPACE-complete on graphs of treewidth at most $t$ but are fixed-parameter tractable parameterized by $\ell + t$ for all vertex-subset problems definable in monadic second-order logic (MSOL). To prove the latter result, we introduce a technique which allows us to circumvent cardinality constraints and define reconfiguration problems in MSOL.

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arXiv2014

Shortest reconfiguration paths in the solution space of Boolean formulas

Given a Boolean formula and a satisfying assignment, a flip is an operation that changes the value of a variable in the assignment so that the resulting assignment remains satisfying. We study the problem of computing the shortest sequence of flips (if one exists) that transforms a given satisfying assignment $s$ to another satisfying assignment $t$ of a Boolean formula. Earlier work characterized the complexity of finding any (not necessarily the shortest) sequence of flips from one satisfying assignment to another using Schaefer's framework for classification of Boolean formulas. We build on it to provide a trichotomy for the complexity of finding the shortest sequence of flips and show that it is either in P, NP-complete, or PSPACE-complete. Our result adds to the small set of complexity results known for shortest reconfiguration sequence problems by providing an example where the shortest sequence can be found in polynomial time even though its length is not equal to the symmetric difference of the values of the variables in $s$ and $t$. This is in contrast to all reconfiguration problems studied so far, where polynomial time algorithms for computing the shortest path were known only for cases where the path modified the symmetric difference only.

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arXiv2014

Reconfiguration of Dominating Sets

We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph $G$ is a set $S$ of vertices such that each vertex is either in $S$ or has a neighbour in $S$. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions $s$ and $t$ such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of $k$, we consider properties of $D_k(G)$, the graph consisting of a vertex for each dominating set of size at most $k$ and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that $D_{Γ(G)+1}(G)$ is not necessarily connected, for $Γ(G)$ the maximum cardinality of a minimal dominating set in $G$. The result holds even when graphs are constrained to be planar, of bounded tree-width, or $b$-partite for $b \ge 3$. Moreover, we construct an infinite family of graphs such that $D_{γ(G)+1}(G)$ has exponential diameter, for $γ(G)$ the minimum size of a dominating set. On the positive side, we show that $D_{n-m}(G)$ is connected and of linear diameter for any graph $G$ on $n$ vertices having at least $m+1$ independent edges.

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arXiv2013

An Easy-to-use Scalable Framework for Parallel Recursive Backtracking

Supercomputers are equipped with an increasingly large number of cores to use computational power as a way of solving problems that are otherwise intractable. Unfortunately, getting serial algorithms to run in parallel to take advantage of these computational resources remains a challenge for several application domains. Many parallel algorithms can scale to only hundreds of cores. The limiting factors of such algorithms are usually communication overhead and poor load balancing. Solving NP-hard graph problems to optimality using exact algorithms is an example of an area in which there has so far been limited success in obtaining large scale parallelism. Many of these algorithms use recursive backtracking as their core solution paradigm. In this paper, we propose a lightweight, easy-to-use, scalable framework for transforming almost any recursive backtracking algorithm into a parallel one. Our framework incurs minimal communication overhead and guarantees a load-balancing strategy that is implicit, i.e., does not require any problem-specific knowledge. The key idea behind this framework is the use of an indexed search tree approach that is oblivious to the problem being solved. We test our framework with parallel implementations of algorithms for the well-known Vertex Cover and Dominating Set problems. On sufficiently hard instances, experimental results show linear speedups for thousands of cores, reducing running times from days to just a few minutes.

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