Directed Multicut is W[1]-hard, Even for Four Terminal Pairs

We prove that Multicut in directed graphs, parameterized by the size of the cutset, is W[1]-hard and hence unlikely to be fixed-parameter tractable even if restricted to instances with only four terminal pairs. This negative result almost completely resolves one of the central open problems in the area of parameterized complexity of graph separation problems, posted originally by Marx and Razgon [SIAM J. Comput. 43(2):355-388 (2014)], leaving only the case of three terminal pairs open. Our gadget methodology allows us also to prove W[1]-hardness of the Steiner Orientation problem parameterized by the number of terminal pairs, resolving an open problem of Cygan, Kortsarz, and Nutov [SIAM J. Discrete Math. 27(3):1503-1513 (2013)].Comment: v2: Added almost tight ETH lower bound

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 $\Gamma$, with or without weights. More precisely, for each finite Boolean constraint language $\Gamma$, we consider the following two problems. In Min SAT$(\Gamma)$, the input is a formula $F$ over $\Gamma$ and an integer $k$, and the task is to find an assignment $\alpha \colon V(F) \to \{0,1\}$ that satisfies all but at most $k$ constraints of $F$, or determine that no such assignment exists. In Weighted Min SAT$(\Gamma$), 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 $\alpha$ such that (1) $\alpha$ 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 $\Gamma$, either Weighted Min SAT$(\Gamma)$ is FPT; or Weighted Min SAT$(\Gamma)$ is W[1]-hard but Min SAT$(\Gamma)$ is FPT; or Min SAT$(\Gamma)$ is W[1]-hard. This generalizes recent work of Kim et al. (SODA 2021) which did not consider weighted problems, and only considered languages $\Gamma$ 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)

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

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