Randomized contractions meet lean decompositions

We show an algorithm that, given an $n$-vertex graph $G$ and a parameter $k$, in time $2^{O(k \log k)} n^{O(1)}$ finds a tree decomposition of $G$ with the following properties: * every adhesion of the tree decomposition is of size at most $k$, and * every bag of the tree decomposition is $(i,i)$-unbreakable in $G$ for every $1 \leq i \leq k$. Here, a set $X \subseteq V(G)$ is $(a,b)$-unbreakable in $G$ if for every separation $(A,B)$ of order at most $b$ in $G$, we have $|A \cap X| \leq a$ or $|B \cap X| \leq a$. The resulting tree decomposition has arguably best possible adhesion size boundsand unbreakability guarantees. Furthermore, the parametric factor in the running time bound is significantly smaller than in previous similar constructions. These improvements allow us to present parameterized algorithms for Minimum Bisection, Steiner Cut, and Steiner Multicut with improved parameteric factor in the running time bound. The main technical insight is to adapt the notion of lean decompositions of Thomas and the subsequent construction algorithm of Bellenbaum and Diestel to the parameterized setting.Comment: v2: New co-author (Magnus) and improved results on vertex unbreakability of bags, v3: final changes, including new abstrac

Parameterized Complexity of Fair Bisection: FPT-Approximation meets Unbreakability

In the Minimum Bisection problem, input is a graph $G$ and the goal is to partition the vertex set into two parts $A$ and $B$, such that $||A|-|B|| \le 1$ and the number $k$ of edges between $A$ and $B$ is minimized. This problem can be viewed as a clustering problem where edges represent similarity, and the task is to partition the vertices into two equally sized clusters, while minimizing the number of pairs of similar objects that end up in different clusters. In this paper, we initiate the study of a fair version of Minimum Bisection. In this problem, the vertices of the graph are colored using one of $c \ge 1$ colors. The goal is to find a bisection $(A, B)$ with at most $k$ edges between the parts, such that for each color $i\in [c]$, $A$ has exactly $r_i$ vertices of color $i$. We first show that Fair Bisection is $W$[1]-hard parameterized by $c$ even when $k = 0$. On the other hand, our main technical contribution shows that is that this hardness result is simply a consequence of the very strict requirement that each color class $i$ has {\em exactly} $r_i$ vertices in $A$. In particular, we give an $f(k,c,\epsilon)n^{O(1)}$ time algorithm that finds a balanced partition $(A, B)$ with at most $k$ edges between them, such that for each color $i\in [c]$, there are at most $(1\pm \epsilon)r_i$ vertices of color $i$ in $A$. Our approximation algorithm is best viewed as a proof of concept that the technique introduced by [Lampis, ICALP '18] for obtaining FPT-approximation algorithms for problems of bounded tree-width or clique-width can be efficiently exploited even on graphs of unbounded width. The key insight is that the technique of Lampis is applicable on tree decompositions with unbreakable bags (as introduced in [Cygan et al., SIAM Journal on Computing '14]). Along the way, we also derive a combinatorial result regarding tree decompositions of graphs.Comment: Full version of ESA 2023 paper. Abstract shortened to meet the character limi

Solving hard cut problems via flow-augmentation

We present a new technique for designing FPT algorithms for graph cut problems in undirected graphs, which we call flow augmentation. Our technique is applicable to problems that can be phrased as a search for an (edge) $(s,t)$-cut of cardinality at most $k$ in an undirected graph $G$ with designated terminals $s$ and $t$. More precisely, we consider problems where an (unknown) solution is a set $Z \subseteq E(G)$ of size at most $k$ such that (1) in $G-Z$, $s$ and $t$ are in distinct connected components, (2) every edge of $Z$ connects two distinct connected components of $G-Z$, and (3) if we define the set $Z_{s,t} \subseteq Z$ as these edges $e \in Z$ for which there exists an $(s,t)$-path $P_e$ with $E(P_e) \cap Z = \{e\}$, then $Z_{s,t}$ separates $s$ from $t$. We prove that in this scenario one can in randomized time $k^{O(1)} (|V(G)|+|E(G)|)$ add a number of edges to the graph so that with $2^{-O(k \log k)}$ probability no added edge connects two components of $G-Z$ and $Z_{s,t}$ becomes a minimum cut between $s$ and $t$. We apply our method to obtain a randomized FPT algorithm for a notorious "hard nut" graph cut problem we call Coupled Min-Cut. This problem emerges out of the study of FPT algorithms for Min CSP problems, and was unamenable to other techniques for parameterized algorithms in graph cut problems, such as Randomized Contractions, Treewidth Reduction or Shadow Removal. To demonstrate the power of the approach, we consider more generally Min SAT($\Gamma$), parameterized by the solution cost. We show that every problem Min SAT($\Gamma$) is either (1) FPT, (2) W[1]-hard, or (3) able to express the soft constraint $(u \to v)$, and thereby also the min-cut problem in directed graphs. All the W[1]-hard cases were known or immediate, and the main new result is an FPT algorithm for a generalization of Coupled Min-Cut

Exploiting Dense Structures in Parameterized Complexity

Over the past few decades, the study of dense structures from the perspective of approximation algorithms has become a wide area of research. However, from the viewpoint of parameterized algorithm, this area is largely unexplored. In particular, properties of random samples have been successfully deployed to design approximation schemes for a number of fundamental problems on dense structures [Arora et al. FOCS 1995, Goldreich et al. FOCS 1996, Giotis and Guruswami SODA 2006, Karpinksi and Schudy STOC 2009]. In this paper, we fill this gap, and harness the power of random samples as well as structure theory to design kernelization as well as parameterized algorithms on dense structures. In particular, we obtain linear vertex kernels for Edge-Disjoint Paths, Edge Odd Cycle Transversal, Minimum Bisection, d-Way Cut, Multiway Cut and Multicut on everywhere dense graphs. In fact, these kernels are obtained by designing a polynomial-time algorithm when the corresponding parameter is at most ?(n). Additionally, we obtain a cubic kernel for Vertex-Disjoint Paths on everywhere dense graphs. In addition to kernelization results, we obtain randomized subexponential-time parameterized algorithms for Edge Odd Cycle Transversal, Minimum Bisection, and d-Way Cut. Finally, we show how all of our results (as well as EPASes for these problems) can be de-randomized

Open Problems in (Hyper)Graph Decomposition

Large networks are useful in a wide range of applications. Sometimes problem instances are composed of billions of entities. Decomposing and analyzing these structures helps us gain new insights about our surroundings. Even if the final application concerns a different problem (such as traversal, finding paths, trees, and flows), decomposing large graphs is often an important subproblem for complexity reduction or parallelization. This report is a summary of discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph Decomposition" and presents currently open problems and future directions in the area of (hyper)graph decomposition

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 algorithmfor the weighted version of \textsc{Directed Feedback Vertex Set}, a landmarkproblem in parameterized complexity. In this note we explore applicability offlow-augmentation to other weighted graph separation problems parameterized bythe size of the cutset. We show the following. -- In weighted undirected graphs\textsc{Multicut} is FPT, both in the edge- and vertex-deletion version. -- Theweighted version of \textsc{Group Feedback Vertex Set} is FPT, even with anoracle access to group operations. -- The weighted version of \textsc{DirectedSubset Feedback Vertex Set} is FPT. Our study reveals \textsc{DirectedSymmetric Multicut} as the next important graph separation problem whoseparameterized complexity remains unknown, even in the unweighted setting.<br