arXiv subjects

Eun Jung Kim

Publications and source records attributed to Eun Jung Kim.

At least 19 recordsLinked to original sources

arXiv2026

CMSO-transductions for tree-like graph decompositions

We give $\operatorname{CMSO}$-transductions that, given a graph $G$, output its modular decomposition, its split decomposition and its bi-join decomposition. This improves results by Courcelle [Logical Methods in Computer Science, 2006] who gave such transductions using order-invariant $\operatorname{MSO}$, a strictly more expressive logic than $\operatorname{CMSO}$. Our methods more generally yield $\operatorname{C}_2 \operatorname{MSO}$-transductions that output the canonical decompositions of weakly-partitive set systems and weakly-bipartitive systems of bipartitions.

cs.LO↗

arXiv2026

Multiway $f$-Cut is fixed-parameter tractable

A connectivity function on a finite set $E$ is a function $f\colon 2^E\to\mathbb Z$ that is submodular and symmetric, with $f(\varnothing)=0$. Given a connectivity function $f$ via a value oracle, terminals $t_1,\ldots,t_r\in E$, and an integer $k$, the Multiway $f$-Cut problem asks whether $E$ has a partition $(P_1,\ldots,P_r)$ with $t_i\in P_i$ for every $i$ and $\sum_{i=1}^r f(P_i)\le k$. We prove that Multiway $f$-Cut is fixed-parameter tractable parameterized by $k$. Cut functions of graphs are connectivity functions, so as a special case we recover the classical result that Edge Multiway Cut in graphs is fixed-parameter tractable. Our proof of correctness is completely elementary, and is arguably the simplest known proof of this fact.

cs.DM↗

arXiv2026

Fast Shortest Path in Graphs With Sparse Signed Tree Models and Applications

A signed tree model of a graph $G$ is a compact binary structure consisting of a rooted binary tree whose leaves are bijectively mapped to the vertices of $G$, together with 2-colored edges $xy$, called transversal pairs, interpreted as bicliques or anti-bicliques whose sides are the leaves of the subtrees rooted at $x$ and at $y$. We design an algorithm that, given such a representation of an $n$-vertex graph $G$ with $p$ transversal pairs and a source $v \in V(G)$, computes a shortest-path tree rooted at $v$ in $G$ in time $O(p \log n)$. A wide variety of graph classes are such that for all $n$, their $n$-vertex graphs admit signed tree models with $O(n)$ transversal pairs: for instance, those of bounded symmetric difference, more generally of bounded sd-degeneracy, as well as interval graphs. As applications of our Single-Source Shortest Path algorithm and new techniques, we - improve the runtime of the fixed-parameter algorithm for first-order model checking on graphs given with a witness of low merge-width from cubic [Dreier and Toruńczyk, STOC '25] to quadratic; - give an $O(n^2 \log n)$-time algorithm for All-Pairs Shortest Path (APSP) on graphs given with a witness of low merge-width, generalizing a result known on twin-width [Twin-Width III, SICOMP '24]; - extend and simplify an $O(n^2 \log n)$-time algorithm for multiplying two $n \times n$ matrices $A, B$ of bounded twin-width in [Twin-Width V, STACS '23]: now $A$ solely has to be an adjacency matrix of a graph of bounded twin-width and $B$ can be arbitrary; - give an $O(n^2 \log^2 n)$-time algorithm for APSP on graphs of bounded twin-width, bypassing the need for contraction sequences in [Twin-Width III, SICOMP '24; Bannach et al. STACS '24]; - give an $O(n^{7/3} \log^2 n)$-time algorithm for APSP on graphs of symmetric difference $O(n^{1/3})$.

cs.DS↗

arXiv2025

Approximating Maximum Cut on Interval Graphs and Split Graphs beyond Goemans-Williamson

We present a polynomial-time $(α_{GW} + \varepsilon)$-approximation algorithm for the Maximum Cut problem on interval graphs and split graphs, where $α_{GW} \approx 0.878$ is the approximation guarantee of the Goemans-Williamson algorithm and $\varepsilon > 10^{-34}$ is a fixed constant. To attain this, we give an improved analysis of a slight modification of the Goemans-Williamson algorithm for graphs in which triangles can be packed into a constant fraction of their edges. We then pair this analysis with structural results showing that both interval graphs and split graphs either have such a triangle packing or have maximum cut close to their number of edges. We also show that, subject to the Small Set Expansion Hypothesis, there exists a constant $c > 0$ such that there is no polyomial-time $(1 - c)$-approximation for Maximum Cut on split graphs.

cs.DS↗

arXiv2025

Bandwidth Parameterized by Cluster Vertex Deletion Number

Given a graph $G$ and an integer $b$, Bandwidth asks whether there exists a bijection $π$ from $V(G)$ to $\{1, \ldots, |V(G)|\}$ such that $\max_{\{u, v \} \in E(G)} | π(u) - π(v) | \leq b$. This is a classical NP-complete problem, known to remain NP-complete even on very restricted classes of graphs, such as trees of maximum degree 3 and caterpillars of hair length 3. In the realm of parameterized complexity, these results imply that the problem remains NP-hard on graphs of bounded pathwidth, while it is additionally known to be W[1]-hard when parameterized by the tree-depth of the input graph. In contrast, the problem does become FPT when parameterized by the vertex cover number. In this paper we make progress in understanding the parameterized (in)tractability of Bandwidth. We first show that it is FPT when parameterized by the cluster vertex deletion number cvd plus the clique number $ω$, thus significantly strengthening the previously mentioned result for vertex cover number. On the other hand, we show that Bandwidth is W[1]-hard when parameterized only by cvd. Our results develop and generalize some of the methods of argumentation of the previous results and narrow some of the complexity gaps.

cs.DS↗

arXiv2025

Flow-augmentation III: Complexity dichotomy for Boolean CSPs parameterized by the number of unsatisfied constraints

We study the parameterized problem of satisfying ``almost all'' constraints of a given formula $F$ over a fixed, finite Boolean constraint language $Γ$, with or without weights. More precisely, for each finite Boolean constraint language $Γ$, we consider the following two problems. In Min SAT$(Γ)$, the input is a formula $F$ over $Γ$ and an integer $k$, and the task is to find an assignment $α\colon V(F) \to \{0,1\}$ that satisfies all but at most $k$ constraints of $F$, or determine that no such assignment exists. In Weighted Min SAT$(Γ$), the input additionally contains a weight function $w \colon F \to \mathbb{Z}_+$ and an integer $W$, and the task is to find an assignment $α$ such that (1) $α$ satisfies all but at most $k$ constraints of $F$, and (2) the total weight of the violated constraints is at most $W$. We give a complete dichotomy for the fixed-parameter tractability of these problems: We show that for every Boolean constraint language $Γ$, either Weighted Min SAT$(Γ)$ is FPT; or Weighted Min SAT$(Γ)$ is W[1]-hard but Min SAT$(Γ)$ is FPT; or Min SAT$(Γ)$ is W[1]-hard. This generalizes recent work of Kim et al. (SODA 2021) which did not consider weighted problems, and only considered languages $Γ$ that cannot express implications $(u \to v)$ (as is used to, e.g., model digraph cut problems). Our result generalizes and subsumes multiple previous results, including the FPT algorithms for Weighted Almost 2-SAT, weighted and unweighted $\ell$-Chain SAT, and Coupled Min-Cut, as well as weighted and directed versions of the latter. The main tool used in our algorithms is the recently developed method of directed flow-augmentation (Kim et al., STOC 2022).

cs.CC↗

arXiv2025

A Tight Meta-theorem for LOCAL Certification of MSO$_2$ Properties within Bounded Treewidth Graphs

Distributed networks are prone to errors so verifying their output is critical. Hence, we develop LOCAL certification protocols for graph properties in which nodes are given certificates that allow them to check whether their network as a whole satisfies some fixed property while only communicating with their local network. Most known LOCAL certification protocols are specifically tailored to the problem they work on and cannot be translated more generally. Thus we target general protocols that can certify any property expressible within a certain logical framework. We consider Monadic Second Order Logic (MSO$_2$), a powerful framework that can express properties such as non-$k$-colorability, Hamiltonicity, and $H$-minor-freeness. Unfortunately, in general, there are MSO$_2$-expressible properties that cannot be certified without huge certificates. For instance, non-3-colorability requires certificates of size $Ω(n^2/\log n)$ on general $n$-vertex graphs (Göös, Suomela 2016). Hence, we impose additional structural restrictions on the graph. We provide a LOCAL certification protocol for certifying any MSO$_2$-expressible property on graphs of bounded treewidth and, consequently, a LOCAL certification protocol for certifying bounded treewidth. That is for each integer $k$ and each MSO$_2$-expressible property $Π$ we give a LOCAL Certification protocol to certify that a graph satisfies $Π$ and has treewidth at most $k$ using certificates of size $\mathcal{O}(\log n)$ (which is asymptotically optimal). Our LOCAL certification protocol requires only one round of distributed communication, hence it is also proof-labeling scheme. Our result improves upon work by Fraigniaud, Montealegre, Rapaport, and Todinca (Algorithmica 2024), Bousquet, Feuilloley, Pierron (PODC 2022), and the very recent work of Baterisna and Chang.

cs.DC↗

arXiv2025

A Constant-factor Approximation for Weighted Bond Cover

The Weighted $\mathcal{F}$-Vertex Deletion for a class ${\cal F}$ of graphs asks, weighted graph $G$, for a minimum weight vertex set $S$ such that $G-S\in{\cal F}.$ The case when ${\cal F}$ is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted $\mathcal{F}$-Vertex Deletion. Only three cases of minor-closed ${\cal F}$ are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. We study the problem for the class ${\cal F}$ of $θ_c$-minor-free graphs, under the equivalent setting of the Weighted $c$-Bond Cover problem, and present a constant-factor approximation algorithm using the primal-dual method. For this, we leverage a structure theorem implicit in [Joret, Paul, Sau, Saurabh, and Thomassé, SIDMA'14] which states the following: any graph $G$ containing a $θ_c$-minor-model either contains a large two-terminal protrusion, or contains a constant-size $θ_c$-minor-model, or a collection of pairwise disjoint constant-sized connected sets that can be contracted simultaneously to yield a dense graph. In the first case, we tame the graph by replacing the protrusion with a special-purpose weighted gadget. For the second and third case, we provide a weighting scheme which guarantees a local approximation ratio. Besides making an important step in the quest of (dis)proving a constant-factor approximation for Weighted $\mathcal{F}$-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families.

cs.DS↗

arXiv2024

Representing Matroids over the Reals is $\exists \mathbb R$-complete

A matroid $M$ is an ordered pair $(E,I)$, where $E$ is a finite set called the ground set and a collection $I\subset 2^{E}$ called the independent sets which satisfy the conditions: (i) $\emptyset \in I$, (ii) $I'\subset I \in I$ implies $I'\in I$, and (iii) $I_1,I_2 \in I$ and $|I_1| < |I_2|$ implies that there is an $e\in I_2$ such that $I_1\cup \{e\} \in I$. The rank $rank(M)$ of a matroid $M$ is the maximum size of an independent set. We say that a matroid $M=(E,I)$ is representable over the reals if there is a map $φ\colon E \rightarrow \mathbb{R}^{rank(M)}$ such that $I\in I$ if and only if $φ(I)$ forms a linearly independent set. We study the problem of matroid realizability over the reals. Given a matroid $M$, we ask whether there is a set of points in the Euclidean space representing $M$. We show that matroid realizability is $\exists \mathbb R$-complete, already for matroids of rank 3. The complexity class $\exists \mathbb R$ can be defined as the family of algorithmic problems that is polynomial-time is equivalent to determining if a multivariate polynomial with integers coefficients has a real root. Our methods are similar to previous methods from the literature. Yet, the result itself was never pointed out and there is no proof readily available in the language of computer science.

cs.CC↗

arXiv2023

Attack of the Knights: A Non Uniform Cache Side-Channel Attack

For a distributed last-level cache (LLC) in a large multicore chip, the access time to one LLC bank can significantly differ from that to another due to the difference in physical distance. In this paper, we successfully demonstrated a new distance-based side-channel attack by timing the AES decryption operation and extracting part of an AES secret key on an Intel Knights Landing CPU. We introduce several techniques to overcome the challenges of the attack, including the use of multiple attack threads to ensure LLC hits, to detect vulnerable memory locations, and to obtain fine-grained timing of the victim operations. While operating as a covert channel, this attack can reach a bandwidth of 205 kbps with an error rate of only 0.02%. We also observed that the side-channel attack can extract 4 bytes of an AES key with 100% accuracy with only 4000 trial rounds of encryption

cs.CR↗

arXiv2023

New Results on Directed Edge Dominating Set

We study a family of generalizations of Edge Dominating Set on directed graphs called Directed $(p,q)$-Edge Dominating Set. In this problem an arc $(u,v)$ is said to dominate itself, as well as all arcs which are at distance at most $q$ from $v$, or at distance at most $p$ to $u$. First, we give significantly improved FPT algorithms for the two most important cases of the problem, $(0,1)$-dEDS and $(1,1)$-dEDS (that correspond to versions of Dominating Set on line graphs), as well as polynomial kernels. We also improve the best-known approximation for these cases from logarithmic to constant. In addition, we show that $(p,q)$-dEDS is FPT parameterized by $p+q+tw$, but W-hard parameterized by $tw$ (even if the size of the optimal is added as a second parameter), where $tw$ is the treewidth of the underlying graph of the input. We then go on to focus on the complexity of the problem on tournaments. Here, we provide a complete classification for every possible fixed value of $p,q$, which shows that the problem exhibits a surprising behavior, including cases which are in P; cases which are solvable in quasi-polynomial time but not in P; and a single case $(p=q=1)$ which is NP-hard (under randomized reductions) and cannot be solved in sub-exponential time, under standard assumptions.

cs.CC↗

arXiv2023

Flow-augmentation II: Undirected graphs

We present an undirected version of the recently introduced flow-augmentation technique: Given an undirected multigraph $G$ with distinguished vertices $s,t \in V(G)$ and an integer $k$, one can in randomized $k^{O(1)} \cdot (|V(G)| + |E(G)|)$ time sample a set $A \subseteq \binom{V(G)}{2}$ such that the following holds: for every inclusion-wise minimal $st$-cut $Z$ in $G$ of cardinality at most $k$, $Z$ becomes a minimum-cardinality cut between $s$ and $t$ in $G+A$ (i.e., in the multigraph $G$ with all edges of $A$ added) with probability $2^{-O(k \log k)}$. Compared to the version for directed graphs [STOC 2022], the version presented here has improved success probability ($2^{-O(k \log k)}$ instead of $2^{-O(k^4 \log k)}$), linear dependency on the graph size in the running time bound, and an arguably simpler proof. An immediate corollary is that the Bi-objective $st$-Cut problem can be solved in randomized FPT time $2^{O(k \log k)} (|V(G)|+|E(G)|)$ on undirected graphs.

cs.DS↗

arXiv2023

Flow-augmentation I: Directed graphs

We show a flow-augmentation algorithm in directed graphs: There exists a randomized polynomial-time algorithm that, given a directed graph $G$, two vertices $s,t \in V(G)$, and an integer $k$, adds (randomly) to $G$ a number of arcs such that for every minimal $st$-cut $Z$ in $G$ of size at most $k$, with probability $2^{-\mathrm{poly}(k)}$ the set $Z$ becomes a minimum $st$-cut in the resulting graph. We also provide a deterministic counterpart of this procedure. The directed flow-augmentation tool allows us to prove fixed-parameter tractability of a number of problems parameterized by the cardinality of the deletion set, whose parameterized complexity status was repeatedly posed as open problems: (1) Chain SAT, defined by Chitnis, Egri, and Marx [ESA'13, Algorithmica'17], (2) a number of weighted variants of classic directed cut problems, such as Weighted $st$-Cut} or Weighted Directed Feedback Vertex Set. By proving that Chain SAT is FPT, we confirm a conjecture of Chitnis, Egri, and Marx that, for any graph $H$, if the List $H$-Coloring problem is polynomial-time solvable, then the corresponding vertex-deletion problem is fixed-parameter tractable.

cs.DS↗

arXiv2022

Obstructions for matroids of path-width at most k and graphs of linear rank-width at most k

Every minor-closed class of matroids of bounded branch-width can be characterized by a list of excluded minors, but unlike graphs, this list may need to be infinite in general. However, for each fixed finite field $\mathbb F$, the list needs to contain only finitely many $\mathbb F$-representable matroids, due to the well-quasi-ordering of $\mathbb F$-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these $\mathbb F$-representable excluded minors in general. We consider the class of matroids of path-width at most $k$ for fixed $k$. We prove that for a finite field $\mathbb F$, every $\mathbb F$-representable excluded minor for the class of matroids of path-width at most $k$ has at most $2^{|\mathbb{F}|^{O(k^2)}}$ elements. We can therefore compute, for any integer $k$ and a fixed finite field $\mathbb F$, the set of $\mathbb F$-representable excluded minors for the class of matroids of path-width $k$, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an $\mathbb F$-represented matroid is at most $k$. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most $k$ has at most $2^{2^{O(k^2)}}$ vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.

math.CO↗

arXiv2022

On weighted graph separation problems and flow-augmentation

One of the first application of the recently introduced technique of \emph{flow-augmentation} [Kim et al., STOC 2022] is a fixed-parameter algorithm for the weighted version of \textsc{Directed Feedback Vertex Set}, a landmark problem in parameterized complexity. In this note we explore applicability of flow-augmentation to other weighted graph separation problems parameterized by the size of the cutset. We show the following. -- In weighted undirected graphs \textsc{Multicut} is FPT, both in the edge- and vertex-deletion version. -- The weighted version of \textsc{Group Feedback Vertex Set} is FPT, even with an oracle access to group operations. -- The weighted version of \textsc{Directed Subset Feedback Vertex Set} is FPT. Our study reveals \textsc{Directed Symmetric Multicut} as the next important graph separation problem whose parameterized complexity remains unknown, even in the unweighted setting.

cs.DS↗

arXiv2022

Algorithmic Applications of Tree-Cut Width

The recently introduced graph parameter tree-cut width plays a similar role with respect to immersions as the graph parameter treewidth plays with respect to minors. In this paper, we provide the first algorithmic applications of tree-cut width to hard combinatorial problems. Tree-cut width is known to be lower-bounded by a function of treewidth, but it can be much larger and hence has the potential to facilitate the efficient solution of problems that are not known to be fixed-parameter tractable (FPT) when parameterized by treewidth. We introduce the notion of nice tree-cut decompositions and provide FPT algorithms for the showcase problems Capacitated Vertex Cover, Capacitated Dominating Set, and Imbalance parameterized by the tree-cut width of an input graph. On the other hand, we show that List Coloring, Precoloring Extension, and Boolean CSP (the latter parameterized by the tree-cut width of the incidence graph) are W[1]-hard and hence unlikely to be fixed-parameter tractable when parameterized by tree-cut width.

cs.DS↗

arXiv2022

Twin-width VI: the lens of contraction sequences

A contraction sequence of a graph consists of iteratively merging two of its vertices until only one vertex remains. The recently introduced twin-width graph invariant is based on contraction sequences. More precisely, if one puts red edges between two vertices representing non-homogeneous subsets, the twin-width is the minimum integer $d$ such that a contraction sequence keeps red degree at most $d$. By changing the condition imposed on the trigraphs (i.e., graphs with some edges being red) and possibly slightly tweaking the notion of contractions, we show how to characterize the well-established bounded rank-width, tree-width, linear rank-width, path-width, and proper minor-closed classes by means of contraction sequences. As an application we give a transparent alternative proof of the celebrated Courcelle's theorem (actually of its generalization by Courcelle, Makowsky, and Rotics), that MSO$_2$ (resp. MSO$_1$) model checking on graphs with bounded tree-width (resp. bounded rank-width) is fixed-parameter tractable in the size of the input sentence. We then explore new avenues along the general theme of contraction sequences both in order to refine the landscape between bounded tree-width and bounded twin-width (via spanning twin-width) and to capture more general classes than bounded twin-width. To this end, we define an oriented version of twin-width, where appearing red edges are oriented away from the newly contracted vertex, and the mere red out-degree should remain bounded. Surprisingly, classes of bounded oriented twin-width coincide with those of bounded twin-width. Finally we examine, from an algorithmic standpoint, the concept of partial contraction sequences, where, instead of terminating on a single-vertex graph, the sequence ends when reaching a particular target class.

cs.DS↗

arXiv2022

Grundy Distinguishes Treewidth from Pathwidth

Structural graph parameters, such as treewidth, pathwidth, and clique-width, are a central topic of study in parameterized complexity. A main aim of research in this area is to understand the "price of generality" of these widths: as we transition from more restrictive to more general notions, which are the problems that see their complexity status deteriorate from fixed-parameter tractable to intractable? This type of question is by now very well-studied, but, somewhat strikingly, the algorithmic frontier between the two (arguably) most central width notions, treewidth and pathwidth, is still not understood: currently, no natural graph problem is known to be W-hard for one but FPT for the other. Indeed, a surprising development of the last few years has been the observation that for many of the most paradigmatic problems, their complexities for the two parameters actually coincide exactly, despite the fact that treewidth is a much more general parameter. It would thus appear that the extra generality of treewidth over pathwidth often comes "for free". Our main contribution in this paper is to uncover the first natural example where this generality comes with a high price. We consider Grundy Coloring, a variation of coloring where one seeks to calculate the worst possible coloring that could be assigned to a graph by a greedy First-Fit algorithm. We show that this well-studied problem is FPT parameterized by pathwidth; however, it becomes significantly harder (W[1]-hard) when parameterized by treewidth. Furthermore, we show that Grundy Coloring makes a second complexity jump for more general widths, as it becomes para-NP-hard for clique-width. Hence, Grundy Coloring nicely captures the complexity trade-offs between the three most well-studied parameters. Completing the picture, we show that Grundy Coloring is FPT parameterized by modular-width.

cs.DS↗