This new version fixes the bug found by Somnath Sikdar in the proof of Claim 8. In the repaired version the modification of the Almost 2-SAT problem called 2-SLASAT is no longer needed and only the modification called 2-ASLASAT remains relevant. Hence the whole manuscript is updated so that the 2-SLASAT problem is not mentioned there anymore
We consider the following problem. Given a 2-CNF formula, is it possible to remove at most $k$ clauses so that the resulting 2-CNF formula is satisfiable? This problem is known to different research communities in Theoretical Computer Science under the names 'Almost 2-SAT', 'All-but-$k$ 2-SAT', '2-CNF deletion', '2-SAT deletion'. The status of fixed-parameter tractability of this problem is a long-standing open question in the area of Parameterized Complexity. We resolve this open question by proposing an algorithm which solves this problem in $O(15^k*k*m^3)$ and thus we show that this problem is fixed-parameter tractable.
We investigate exact crossing minimization for graphs that differ from trees by a small number of additional edges, for several variants of the crossing minimization problem. In particular, we provide fixed parameter tractable algorithms for the 1-page book crossing number, the 2-page book crossing number, and the minimum number of crossed edges in 1-page and 2-page book drawings.
An important question in the study of constraint satisfaction problems (CSP) is understanding how the graph or hypergraph describing the incidence structure of the constraints influences the complexity of the problem. For binary CSP instances (i.e., where each constraint involves only two variables), the situation is well understood: the complexity of the problem essentially depends on the treewidth of the graph of the constraints. However, this is not the correct answer if constraints with unbounded number of variables are allowed, and in particular, for CSP instances arising from query evalu…
There has been great interest in identifying tractable subclasses of NP complete problems and designing efficient algorithms for these tractable classes. Constraint satisfaction and Bayesian network inference are two examples of such problems that are of great importance in AI and algorithms. In this paper we study, under the frameworks of random constraint satisfaction problems and random Bayesian networks, a typical tractable subclass characterized by the treewidth of the problems. We show that the property of having a bounded treewidth for CSPs and Bayesian network inference problem has a p…
The optimal value computation for turned-based stochastic games with reachability objectives, also known as simple stochastic games, is one of the few problems in $NP \cap coNP$ which are not known to be in $P$. However, there are some cases where these games can be easily solved, as for instance when the underlying graph is acyclic. In this work, we try to extend this tractability to several classes of games that can be thought as "almost" acyclic. We give some fixed-parameter tractable or polynomial algorithms in terms of different parameters such as the number of cycles or the size of the m…
For several decades, much effort has been put into identifying classes of CNF formulas whose satisfiability can be decided in polynomial time. Classic results are the linear-time tractability of Horn formulas (Aspvall, Plass, and Tarjan, 1979) and Krom (i.e., 2CNF) formulas (Dowling and Gallier, 1984). Backdoors, introduced by Williams Gomes and Selman (2003), gradually extend such a tractable class to all formulas of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a formula and a tractable class. Backdoor depth, introduced by Mählmann,…
In the Max Lin-2 problem we are given a system $S$ of $m$ linear equations in $n$ variables over $\mathbb{F}_2$ in which Equation $j$ is assigned a positive integral weight $w_j$ for each $j$. We wish to find an assignment of values to the variables which maximizes the total weight of satisfied equations. This problem generalizes Max Cut. The expected weight of satisfied equations is $W/2$, where $W=w_1+... +w_m$; $W/2$ is a tight lower bound on the optimal solution of Max Lin-2. Mahajan et al. (J. Comput. Syst. Sci. 75, 2009) stated the following parameterized version of Max Lin-2: decide whe…
In MaxSat, we are given a CNF formula $F$ with $n$ variables and $m$ clauses and asked to find a truth assignment satisfying the maximum number of clauses. Let $r_1,..., r_m$ be the number of literals in the clauses of $F$. Then $asat(F)=\sum_{i=1}^m (1-2^{-r_i})$ is the expected number of clauses satisfied by a random truth assignment (the truth values to the variables are distributed uniformly and independently). It is well-known that, in polynomial time, one can find a truth assignment satisfying at least $asat(F)$ clauses. In the parameterized problem MaxSat-AA, we are to decide whether th…
In the bounded-degree cut problem, we are given a multigraph $G=(V,E)$, two disjoint vertex subsets $A,B\subseteq V$, two functions $\mathrm{u}_A, \mathrm{u}_B:V\to \{0,1,\ldots,|E|\}$ on $V$, and an integer $k\geq 0$. The task is to determine whether there is a minimal $(A,B)$-cut $(V_A,V_B)$ of size at most $k$ such that the degree of each vertex $v\in V_A$ in the induced subgraph $G[V_A]$ is at most $\mathrm{u}_A(v)$ and the degree of each vertex $v\in V_B$ in the induced subgraph $G[V_B]$ is at most $\mathrm{u}_B(v)$. In this paper, we show that the bounded-degree cut problem is fixed-para…
University of Udine（乌迪内大学）；University of Newcastle（纽卡斯尔大学）
We study the complexity of the problem of searching for a set of patterns that separate two given sets of strings. This problem has applications in a wide variety of areas, most notably in data mining, computational biology, and in understanding the complexity of genetic algorithms. We show that the basic problem of finding a small set of patterns that match one set of strings but do not match any string in a second set is difficult (NP-complete, W[2]-hard when parameterized by the size of the pattern set, and APX-hard). We then perform a detailed parameterized analysis of the problem, separat…
We show that the Satisfiability (SAT) problem for CNF formulas with β-acyclic hypergraphs can be solved in polynomial time by using a special type of Davis-Putnam resolution in which each resolvent is a subset of a parent clause. We extend this class to CNF formulas for which this type of Davis-Putnam resolution still applies and show that testing membership in this class is NP-complete. We compare the class of β-acyclic formulas and this superclass with a number of known polynomial formula classes. We then study the parameterized complexity of SAT for "almost" β-acyclic instances, using as pa…
We study the question of when we can provide direct access to the k-th answer to a Conjunctive Query (CQ) according to a specified order over the answers in time logarithmic in the size of the database, following a preprocessing step that constructs a data structure in time quasilinear in database size. Specifically, we embark on the challenge of identifying the tractable answer orderings, that is, those orders that allow for such complexity guarantees. To better understand the computational challenge at hand, we also investigate the more modest task of providing access to only a single answer…
Bonnet et al. (FOCS 2020) introduced the graph invariant twin-width and showed that many NP-hard problems are tractable for graphs of bounded twin-width, generalizing similar results for other width measures, including treewidth and clique-width. In this paper, we investigate the use of twin-width for solving the propositional satisfiability problem (SAT) and propositional model counting. We particularly focus on Bounded-ones Weighted Model Counting (BWMC), which takes as input a CNF formula $F$ along with a bound $k$ and asks for the weighted sum of all models with at most $k$ positive litera…
We study the following general disjoint paths problem: given a supply graph $G$, a set $T\subseteq V(G)$ of terminals, a demand graph $H$ on the vertices $T$, and an integer $k$, the task is to find a set of $k$ pairwise vertex-disjoint valid paths, where we say that a path of the supply graph $G$ is valid if its endpoints are in $T$ and adjacent in the demand graph $H$. For a class $\mathcal{H}$ of graphs, we denote by $\mathcal{H}$-Maximum Disjoint Paths the restriction of this problem when the demand graph $H$ is assumed to be a member of $\mathcal{H}$. We study the fixed-parameter tractabi…
The three-way table problem is to decide if there exists an l x m x n table satisfying given line sums, and find a table if there is one. It is NP-complete already for l=3 and every bounded integer program can be isomorphically represented in polynomial time for some m and n as some 3 x m x n table problem. Recently, the problem was shown to be fixed-parameter tractable with parameters l,m. Here we extend this and show that the huge version of the problem, where the variable side n is a huge number encoded in binary, is also fixed-parameter tractable with parameters l,m. We also conclude that …
Institute of Information Systems, Vienna University of Technology（维也纳工业大学信息系统研究所）
A backdoor set is a set of variables of a propositional formula such that fixing the truth values of the variables in the backdoor set moves the formula into some polynomial-time decidable class. If we know a small backdoor set we can reduce the question of whether the given formula is satisfiable to the same question for one or several easy formulas that belong to the tractable class under consideration. In this survey we review parameterized complexity results for problems that arise in the context of backdoor sets, such as the problem of finding a backdoor set of size at most k, parameteriz…
