arXiv subjects

Tomohiro Koana

Publications and source records attributed to Tomohiro Koana.

At least 19 recordsLinked to original sources

arXiv2026

High-Multiplicity Bin Packing is FPT

Bin packing asks whether a collection of items can be packed into at most a given number of bins of a given capacity. We consider the high-multiplicity setting with $d$ distinct item sizes, in which both the item sizes and the number of items of each size are encoded in binary. Goemans and Rothvos (JACM 2020) gave an XP algorithm parameterized by $d$. Whether this problem is fixed-parameter tractable (FPT) in $d$ has remained a central open problem. We resolve this question by giving a deterministic $O^*(2^{d^{O(d)}})$-time algorithm. We formulate bin packing as an integer linear program (ILP) with at most $(d+1)d^d$ variables. A bin configuration records the number of items of each type in one bin. We partition these configurations by their coordinate remainders modulo $d$. For each class, we use one variable for the bin count and $d$ variables for the total item counts. The convex hull of each class has the integer decomposition property, which guarantees that every feasible ILP solution corresponds to a packing.

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arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

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arXiv2026

Lawler-Moore Speedups via Additive Combinatorics

The Lawler-Moore dynamic programming framework is a classical tool in scheduling on parallel machines. It applies when the objective is regular, i.e. monotone in job completion times, and each machine follows a fixed priority order such as Smith's Rule or Jackson's Rule. For the basic objectives $Pm||\sum w_jC_j$, $Pm||L_{\max}$, and $Pm||\sum w_jU_j$, it gives running times $O(P^{m-1}n)$, $O(P^{m-1}n)$, and $O(P^mn)$, respectively, where $P$ is the total processing time. Recent SETH-based lower bounds indicate that the dependence on $P$ is essentially optimal, but they do not rule out improved dependence on the maximum processing time $p_{\max}$. We give the first major speedup of the Lawler-Moore recurrence. Our main ingredients are a new state-pruning method and a swapping argument based on an additive-combinatorial lemma. We prove that, whenever this swap does not increase the objective value, there exists an optimal schedule in which, for every prefix of jobs, the load difference between any two machines is at most $4p_{\max}^2$. This lets us prune redundant states throughout the dynamic program, replacing the dependence on $P$ by a dependence on $p_{\max}^2$. We show that the swap is non-increasing for all three objectives above. Hence $Pm||\sum w_jC_j$ and $Pm||L_{\max}$ admit algorithms with running time $O(p_{\max}^{2m-2}n)$, while $Pm||\sum w_jU_j$ can be solved in time $O(p_{\max}^{2m-2}Pn)\le O(p_{\max}^{2m-1}n^2)$. These bounds strictly improve the original Lawler-Moore runtimes whenever $p_{\max}=o(\sqrt{P})$. In particular, for $Pm||\sum w_jC_j$ and $Pm||L_{\max}$, we obtain the first near-linear-time algorithms when processing times are polylogarithmic in $n$.

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arXiv2026

Graph Coloring with Color Preferences

We study graph coloring with color preferences, in which each vertex ranks the available colors. In addition to assigning different colors to adjacent vertices, we require the coloring to be stable: no group of vertices can cyclically exchange their assigned colors so that each strictly prefers its new color to its original one. We define the stable chromatic number $χ_\mathrm{stable}(G)$ of a graph $G$ as the minimum integer $k$ such that every preference profile admits a stable $k$-coloring of $G$. We establish several upper and lower bounds. In particular, for any acyclic orientation of the edges of $G$, the largest number of vertices reachable from a vertex by directed paths, including the vertex itself, is an upper bound on $χ_\mathrm{stable}(G)$. This shows that $χ_\mathrm{stable}(G)$ is well-defined. We also show that $O(t \log (1+n/t))$ colors suffice for an $n$-vertex graph $G$ of treewidth $t$, and complement this with a lower bound in terms of the Grundy number. Turning to the problem of finding a minimum stable coloring for a given profile, we show that stable $2$-colorability is polynomial-time solvable, whereas stable $k$-colorability is NP-complete for every fixed $k\ge 3$. Using the treewidth bound, we give a fixed-parameter tractable algorithm parameterized by treewidth.

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arXiv2026

A Single-Exponential FPT Algorithm for 2-Vertex-Connectivity Augmentation

We study restricted-link augmentation to $2$-vertex-connectivity. An instance consists of a graph $G$, possibly disconnected, a set $L$ of admissible links on its vertices, integer link costs in $\{1,\dots,W\}$, and an integer $k$; the task is to add at most $k$ links of minimum total cost so that the resulting multigraph is $2$-vertex-connected. Recent work gives $O^*(k^{O(k)})$-time algorithms for unweighted $λ$-vertex-connectivity augmentation for every $λ\leq 4$ [Carmesin and Ramanujan, SODA 2026], and an $O^*((k+λ)^{O(k)})$-time algorithm for arbitrary $λ$ [Korhonen and Thorup, arXiv 2026]. We give a deterministic algorithm with running time $O^*(36^kW)$. Thus, for $λ=2$, the unweighted running time improves from $O^*(k^{O(k)})$ to $O^*(36^k)$, and the algorithm also handles link costs with pseudo-polynomial dependence on $W$. We reduce the problem to a boundary-pair variant of $2$-vertex-connected spanning subgraph, where each vertex is assigned a pair of incident edges with an associated pair cost. We solve this variant using a cancellation identity, inspired by Cut&Count [Cygan et al., TALG 2022], obtained by applying Möbius inversion to decompositions along cut vertices: the identity cancels every connected spanning graph with more than one block and keeps exactly the $2$-vertex-connected spanning graphs.

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arXiv2026

Complexity of induced subgraph isomorphism and maximum common induced subgraph parameterized by cluster vertex deletion number

We study the parameterized complexity of Induced Subgraph Isomorphism (ISI) and Maximum Common Induced Subgraph (MCIS) with respect to the cluster vertex deletion number $k$. For ISI, we give a randomized $O^*(k^{O(k)})$-time algorithm, showing that ISI is fixed-parameter tractable under this parameter and resolving an open question of Hanaka et al. [WALCOM 2026]. Our algorithm is optimal under the Exponential Time Hypothesis (ETH), and is based on a reduction to Exact Multicolored Matching solvable via algebraic techniques. For MCIS, we present a randomized $O^*(2^{O(k^2)})$-time algorithm via a reduction to a weighted variant of Exact Multicolored Matching, and we prove a matching ETH-based lower bound by showing that a $k$-by-$k$ binary matrix feasibility problem with list-constrained rows and columns admits no $O^*(2^{o(k^2)})$-time algorithm, which may be of independent interest. These results reveal that, in this setting, MCIS is strictly harder than ISI. Finally, for the three-graph variant 3-MCIS, we show that it becomes NP-hard already when each input graph has cluster vertex deletion number 2.

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arXiv2026

Kernelization for $H$-Packing Revisited

\textsc{$H$-Packing} asks whether a graph $G$ contains $k$ vertex-disjoint copies of a fixed pattern graph $H$. Via the standard reduction to \textsc{$d$-Set Packing}, one obtains generic kernels with $O(k^{|V(H)|-1})$ vertices and $O(k^{|V(H)|})$ edges. We revisit the question of beating these bounds for specific patterns $H$. Our main results concern subdivided stars. Let $S_{d_1,d_2}$ denote the subdivided star with $d_1$ branches of length $1$ and $d_2$ branches of length $2$. We obtain kernels with $O(k^2)$ vertices and $O(k^3)$ edges for $P_5=S_{0,2}$, for $S_{1,2}$, and for every $S_{d_1,1}$, kernels with $O(k^4)$ vertices and $O(k^6)$ edges for every fixed $S_{d_1,d_2}$ with $d_1\ge 1$, and a kernel with $O(k^2)$ vertices and $O(k^4)$ edges for the paw. Our proofs proceed in two steps. First, we reduce to instances in which all but a small part of the graph is independent, or in which the graph has a small vertex cover. Second, we reduce the independent side by keeping only a bounded number of witness vertices for each subset of the small part. On the negative side, we prove a lower bound for the line $S_{0,d}$. For every $d\ge 3$ and every $\varepsilon>0$, \textsc{$S_{0,d}$-Packing} does not admit a compression of size $O(k^{d-\varepsilon})$ unless $\NP\subseteq \coNP/\poly$. Thus, deleting a single vertex from the pattern may, surprisingly, make kernelization provably harder, showing that compressibility of \textsc{$H$-Packing} is not monotone under taking induced subgraphs.

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arXiv2026

New Parameterized and Exact Exponential Time Algorithms for Strongly Connected Steiner Subgraph

The Strongly Connected Steiner Subgraph (SCSS) problem is a well-studied network design problem that asks for a minimum subgraph that strongly connects a given set of terminals. In this paper, we present several new algorithmic and complexity results for SCSS. As our main result, we show that SCSS can be solved in time $17^{\mathrm{tw}} n^{O(1)}$ on directed graphs with $n$ vertices when a tree decomposition of the underlying graph of width $\mathrm{tw}$ is provided. This improves over a natural $\mathrm{tw}^{O(\mathrm{tw})}n^{O(1)}$ time algorithm, and is the first algorithm with this kind of running time for a problem involving strong connectivity. Second, we give an exact exponential-time algorithm that solves SCSS in $2^n n^{O(1)}$ time, improving the known bounds for general directed graphs. Finally, we investigate kernelization with respect to vertex cover. We prove that SCSS does not admit a polynomial kernel when parameterized by the size of a vertex cover, unless the polynomial hierarchy collapses. In contrast, we show that the closely related Strongly Connected Spanning Subgraph problem does admit a polynomial kernel under the same parameterization.

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arXiv2025

Determinantal Sieving

We introduce determinantal sieving, a new, remarkably powerful tool in the toolbox of algebraic FPT algorithms. Given a polynomial $P(X)$ on a set of variables $X=\{x_1,\ldots,x_n\}$ and a linear matroid $M=(X,\mathcal{I})$ of rank $k$, both over a field $\mathbb{F}$ of characteristic 2, in $2^k$ evaluations we can sieve for those terms in the monomial expansion of $P$ which are multilinear and whose support is a basis for $M$. Alternatively, using $2^k$ evaluations of $P$ we can sieve for those monomials whose odd support spans $M$. Applying this framework, we improve on a range of algebraic FPT algorithms, such as: 1. Solving $q$-Matroid Intersection in time $O^*(2^{(q-2)k})$ and $q$-Matroid Parity in time $O^*(2^{qk})$, improving on $O^*(4^{qk})$ over general fields (Brand and Pratt, ICALP 2021) 2. Long $(s,t)$-Path in $O^*(1.66^k)$ time, improving on $O^*(2^k)$, and Rank $k$ $(S,T)$-Linkage in so-called frameworks in $O^*(2^k)$ time, improving on $O^*(2^{|S|+O(k^2 \log(k+|\mathbb{F}|))})$ over general fields (Fomin et al., SODA 2023). 3. Many instances of the Diverse X paradigm, finding a collection of $r$ solutions to a problem with a minimum mutual distance of $d$ in time $O^*(2^{r(r-1)d/2})$, improving solutions for $k$-Distinct Branchings from time $2^{O(k \log k)}$ to $O^*(2^k)$ (Bang-Jensen et al., ESA 2021), and for Diverse Perfect Matchings from $O^*(2^{2^{O(rd)}})$ to $O^*(2^{r^2d/2})$ (Fomin et al., STACS 2021). Here, all matroids are assumed to be represented over fields of characteristic 2. Over general fields, we achieve similar results at the cost of using exponential space by working over the exterior algebra. For a class of arithmetic circuits we call strongly monotone, this is even achieved without any loss of running time. However, the odd support sieving result appears to be specific to working over characteristic 2.

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arXiv2025

Graph Coloring Below Guarantees via Co-Triangle Packing

In the $\ell$-Coloring Problem, we are given a graph on $n$ nodes, and tasked with determining if its vertices can be properly colored using $\ell$ colors. In this paper we study below-guarantee graph coloring, which tests whether an $n$-vertex graph can be properly colored using $g-k$ colors, where $g$ is a trivial upper bound such as $n$. We introduce an algorithmic framework that builds on a packing of co-triangles $\overline{K_3}$ (independent sets of three vertices): the algorithm greedily finds co-triangles and employs a win-win analysis. If many are found, we immediately return YES; otherwise these co-triangles form a small co-triangle modulator, whose deletion makes the graph co-triangle-free. Extending the work of [Gutin et al., SIDMA 2021], who solved $\ell$-Coloring (for any $\ell$) in randomized $O^*(2^{k})$ time when given a $\overline{K_2}$-free modulator of size $k$, we show that this problem can likewise be solved in randomized $O^*(2^{k})$ time when given a $\overline{K_3}$-free modulator of size~$k$. This result in turn yields a randomized $O^{*}(2^{3k/2})$ algorithm for $(n-k)$-Coloring (also known as Dual Coloring), improving the previous $O^{*}(4^{k})$ bound. We then introduce a smaller parameterization, $(ω+\overlineμ-k)$-Coloring, where $ω$ is the clique number and $\overlineμ$ is the size of a maximum matching in the complement graph; since $ω+\overlineμ\le n$ for any graph, this problem is strictly harder. Using the same co-triangle-packing argument, we obtain a randomized $O^{*}(2^{6k})$ algorithm, establishing its fixed-parameter tractability for a smaller parameter. Complementing this finding, we show that no fixed-parameter tractable algorithm exists for $(ω-k)$-Coloring or $(\overlineμ-k)$-Coloring under standard complexity assumptions.

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arXiv2025

Kronecker scaling of tensors with applications to arithmetic circuits and algorithms

We show that sufficiently low tensor rank for the balanced tripartitioning tensor $P_d(x,y,z)=\sum_{A,B,C\in\binom{[3d]}{d}:A\cup B\cup C=[3d]}x_Ay_Bz_C$ for a large enough constant $d$ implies uniform arithmetic circuits for the matrix permanent that are exponentially smaller than circuits obtainable from Ryser's formula. We show that the same low-rank assumption implies exponential time improvements over the state of the art for a wide variety of other related counting and decision problems. As our main methodological contribution, we show that the tensors $P_n$ have a desirable Kronecker scaling property: They can be decomposed efficiently into a small sum of restrictions of Kronecker powers of $P_d$ for constant $d$. We prove this with a new technique relying on Steinitz's lemma, which we hence call Steinitz balancing. As a consequence of our methods, we show that the mentioned low rank assumption (and hence the improved algorithms) is implied by Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)], a bold conjecture that has recently seen intriguing progress.

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arXiv2025

Faster Minimization of Total Weighted Completion Time on Parallel Machines

We study the classical problem of minimizing the total weighted completion time on a fixed set of $m$ identical machines working in parallel, the $Pm||\sum w_jC_j$ problem in the standard three field notation for scheduling problems. This problem is well known to be NP-hard, but only in the ordinary sense, and appears as one of the fundamental problems in any scheduling textbook. In particular, the problem served as a proof of concept for applying pseudo-polynomial time algorithms and approximation schemes to scheduling problems. The fastest known pseudo-polynomial time algorithm for $Pm||\sum w_jC_j$ is the famous Lawler and Moore algorithm from the late 1960's which runs in $\tilde{O}(P^{m-1}n)$ time, where $P$ is the total processing time of all jobs in the input. After more than 50 years, we are the first to present an algorithm, alternative to that of Lawler and Moore, which is faster for certain range of the problem parameters (e.g., when their values are all $O(1)$).

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arXiv2025

FPT algorithms over linear delta-matroids with applications

Matroids, particularly linear ones, have been a powerful tool in parameterized complexity for algorithms and kernelization. They have sped up or replaced dynamic programming. Delta-matroids generalize matroids by encapsulating structures such as non-maximum matchings in general graphs and various path-packing and topological configurations. Linear delta-matroids (represented by skew-symmetric matrices) offer significant expressive power and enable powerful algorithms. We investigate parameterized complexity aspects of problems defined over linear delta-matroids or with delta-matroid constraints. Our analysis of basic intersection and packing problems reveals a different complexity landscape compared to the familiar matroid case. In particular, there is a stark contrast between the cardinality parameter $k$ and the rank parameter $r$. For example, finding an intersection of size $k$ of three linear delta-matroids is W[1]-hard when parameterized by $k$, while more general problems (e.g., finding a set packing of size $k$ feasible in a linear delta-matroid) are FPT when parameterized by $r$. We extend the recent determinantal sieving procedure of Eiben, Koana and Wahlström (SODA 2024) to sieve a polynomial for a monomial whose support is feasible in a linear delta-matroid by $r$. Second, we investigate a class of problems that remains FPT when parameterized by $k$, even on delta-matroids of unbounded rank. We begin with Delta-matroid Triangle Cover - finding a feasible set of size $k$ that can be covered by a vertex-disjoint packing of triangles (sets of size 3) from a given collection. This approach allows us to find a packing of $K_3$'s and $K_2$'s in a graph with a maximum number of edges, parameterized above the matching number. As applications, we settle questions on the FPT status of Cluster Subgraph and Strong Triadic Closure parameterized above the matching number.

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arXiv2025

Faster Edge Coloring by Partition Sieving

In the Edge Coloring problem, we are given an undirected graph $G$ with $n$ vertices and $m$ edges, and are tasked with finding the smallest positive integer $k$ so that the edges of $G$ can be assigned $k$ colors in such a way that no two edges incident to the same vertex are assigned the same color. Edge Coloring is a classic NP-hard problem, and so significant research has gone into designing fast exponential-time algorithms for solving Edge Coloring and its variants exactly. Prior work showed that Edge Coloring can be solved in $2^m\text{poly}(n)$ time and polynomial space, and in graphs with average degree $d$ in $2^{(1-\varepsilon_d)m}\text{poly}(n)$ time and exponential space, where $\varepsilon_d = (1/d)^{Θ(d^3)}$. We present an algorithm that solves Edge Coloring in $2^{m-3n/5}\text{poly}(n)$ time and polynomial space. Our result is the first algorithm for this problem which simultaneously runs in faster than $2^m\text{poly}(m)$ time and uses only polynomial space. In graphs of average degree $d$, our algorithm runs in $2^{(1-6/(5d))m}\text{poly}(n)$ time, which has far better dependence in $d$ than previous results. We also generalize our algorithm to solve a problem known as List Edge Coloring, where each edge $e$ in the input graph comes with a list $L_e\subseteq\left\{1, \dots, k\right\}$ of colors, and we must determine whether we can assign each edge a color from its list so that no two edges incident to the same vertex receive the same color. We solve this problem in $2^{(1-6/(5k))m}\text{poly}(n)$ time and polynomial space. The previous best algorithm for List Edge Coloring took $2^m\text{poly}(n)$ time and space.

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arXiv2024

A Polynomial Time Algorithm for Steiner Tree when Terminals Avoid a $K_4$-Minor

We study a special case of the Steiner Tree problem in which the input graph does not have a minor model of a complete graph on 4 vertices for which all branch sets contain a terminal. We show that this problem can be solved in $O(n^4)$ time, where $n$ denotes the number of vertices in the input graph. This generalizes a seminal paper by Erickson et al. [Math. Oper. Res., 1987] that solves Steiner tree on planar graphs with all terminals on one face in polynomial time.

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arXiv2024

Subexponential Algorithms for Clique Cover on Unit Disk and Unit Ball Graphs

In Clique Cover, given a graph $G$ and an integer $k$, the task is to partition the vertices of $G$ into $k$ cliques. Clique Cover on unit ball graphs has a natural interpretation as a clustering problem, where the objective function is the maximum diameter of a cluster. Many classical NP-hard problems are known to admit $2^{O(n^{(1 - 1/d)})}$-time algorithms on unit ball graphs in $\mathbb{R}^d$ [de Berg et al., SIAM J. Comp 2018]. A notable exception is the Maximum Clique problem, which admits a polynomial-time algorithm on unit disk graphs and a subexponential algorithm on unit ball graphs in $\mathbb{R}^3$, but no subexponential algorithm on unit ball graphs in dimensions 4 or larger, assuming the ETH [Bonamy et al., JACM 2021]. In this work, we show that Clique Cover also suffers from a "curse of dimensionality", albeit in a significantly different way compared to Maximum Clique. We present a $2^{O(\sqrt{n})}$-time algorithm for unit disk graphs and argue that it is tight under the ETH. On the other hand, we show that Clique Cover does not admit a $2^{o(n)}$-time algorithm on unit ball graphs in dimension $5$, unless the ETH fails.

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arXiv2024

Structural Parameterizations of the Biclique-Free Vertex Deletion Problem

In this work, we study the Biclique-Free Vertex Deletion problem: Given a graph $G$ and integers $k$ and $i \le j$, find a set of at most $k$ vertices that intersects every (not necessarily induced) biclique $K_{i, j}$ in $G$. This is a natural generalization of the Bounded-Degree Deletion problem, wherein one asks whether there is a set of at most $k$ vertices whose deletion results in a graph of a given maximum degree $r$. The two problems coincide when $i = 1$ and $j = r + 1$. We show that Biclique-Free Vertex Deletion is fixed-parameter tractable with respect to $k + d$ for the degeneracy $d$ by developing a $2^{O(d k^2)} \cdot n^{O(1)}$-time algorithm. We also show that it can be solved in $2^{O(f k)} \cdot n^{O(1)}$ time for the feedback vertex number $f$ when $i \ge 2$. In contrast, we find that it is W[1]-hard for the treedepth for any integer $i \ge 1$. Finally, we show that Biclique-Free Vertex Deletion has a polynomial kernel for every $i \ge 1$ when parameterized by the feedback edge number. Previously, for this parameter, its fixed-parameter tractability for $i = 1$ was known (Betzler et al., 2012) but the existence of polynomial kernel was open.

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arXiv2024

Fully Polynomial-time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication

We study the computational complexity of several polynomial-time-solvable graph problems parameterized by vertex integrity, a measure of a graph's vulnerability to vertex removal in terms of connectivity. Vertex integrity is the smallest number $ι$ such that there is a set $S$ of $ι' \le ι$ vertices such that every connected component of $G-S$ contains at most $ι-ι'$ vertices. It is known that the vertex integrity lies between the well-studied parameters vertex cover number and tree-depth. Alon and Yuster [ESA 2007] designed algorithms for graphs with small vertex cover number using fast matrix multiplications. We demonstrate that fast matrix multiplication can also be effectively used when parameterizing by vertex integrity $ι$ by developing efficient algorithms for problems including an $O(ι^{ω-1}n)$-time algorithm for computing the girth of a graph, randomized $O(ι^{ω- 1}n)$-time algorithms for Maximum Matching and for finding any induced four-vertex subgraph except for a clique or an independent set, and an $O(ι^{(ω-1)/2}n^2) \subseteq O(ι^{0.687} n^2)$-time algorithm for All-Pairs Shortest Paths. These algorithms can be faster than previous algorithms parameterized by tree-depth, for which fast matrix multiplication is not known to be effective.

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