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Register a new userReducing CMSO Model Checking to Highly Connected Graphs
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Added by
Ramanujan M. S.
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Given a Counting Monadic Second Order (CMSO) sentence $psi$, the CMSO$[psi]$ problem is defined as follows. The input to CMSO$[psi]$ is a graph $G$, and the objective is to determine whether $Gmodels psi$. Our main theorem states that for every CMSO sentence $psi$, if CMSO$[psi]$ is solvable in polynomial time on globally highly connected graphs, then CMSO$[psi]$ is solvable in polynomial time (on general graphs). We demonstrate the utility of our theorem in the design of parameterized algorithms. Specifically we show that technical problem-specific ingredients of a powerful method for designing parameterized algorithms, recursive understanding, can be replaced by a black-box invocation of our main theorem. We also show that our theorem can be easily deployed to show fixed parameterized tractability of a wide range of problems, where the input is a graph $G$ and the task is to find a connected induced subgraph of $G$ such that few vertices in this subgraph have neighbors outside the subgraph, and additionally the subgraph has a CMSO-definable property.
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Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA 14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, $K_t$-free unit $d$-dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of $d$-contractions, witness that the twin-width is at most $d$. We show that FO model checking, that is deciding if a given first-order formula $phi$ evaluates to true for a given binary structure $G$ on a domain $D$, is FPT in $|phi|$ on classes of bounded twin-width, provided the witness is given. More precisely, being given a $d$-contraction sequence for $G$, our algorithm runs in time $f(d,|phi|) cdot |D|$ where $f$ is a computable but non-elementary function. We also prove that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarsky et al. [FOCS 15].
In the Survivable Network Design Problem (SNDP), the input is an edge-weighted (di)graph $G$ and an integer $r_{uv}$ for every pair of vertices $u,vin V(G)$. The objective is to construct a subgraph $H$ of minimum weight which contains $r_{uv}$ edge-disjoint (or node-disjoint) $u$-$v$ paths. This is a fundamental problem in combinatorial optimization that captures numerous well-studied problems in graph theory and graph algorithms. In this paper, we consider the version of the problem where we are given a $lambda$-edge connected (di)graph $G$ with a non-negative weight function $w$ on the edges and an integer $k$, and the objective is to find a minimum weight spanning subgraph $H$ that is also $lambda$-edge connected, and has at least $k$ fewer edges than $G$. In other words, we are asked to compute a maximum weight subset of edges, of cardinality up to $k$, which may be safely deleted from $G$. Motivated by this question, we investigate the connectivity properties of $lambda$-edge connected (di)graphs and obtain algorithmically significant structural results. We demonstrate the importance of our structural results by presenting an algorithm running in time $2^{O(k log k)} |V(G)|^{O(1)}$ for $lambda$-ECS, thus proving its fixed-parameter tractability. We follow up on this result and obtain the {em first polynomial compression} for $lambda$-ECS on unweighted graphs. As a consequence, we also obtain the first fixed parameter tractable algorithm, and a polynomial kernel for a parameterized version of the classic Mininum Equivalent Graph problem. We believe that our structural results are of independent interest and will play a crucial role in the design of algorithms for connectivity-constrained problems in general and the SNDP problem in particular.
MAX CLIQUE problem (MCP) is an NPO problem, which asks to find the largest complete sub-graph in a graph $G, G = (V, E)$ (directed or undirected). MCP is well known to be $NP-Hard$ to approximate in polynomial time with an approximation ratio of $1 + epsilon$, for every $epsilon > 0$ [9] (and even a polynomial time approximation algorithm with a ratio $n^{1 - epsilon}$ has been conjectured to be non-existent [2] for MCP). Up to this date, the best known approximation ratio for MCP of a polynomial time algorithm is $O(n(log_2(log_2(n)))^2 / (log_2(n))^3)$ given by Feige [1]. In this paper, we show that MCP can be approximated with a constant factor in polynomial time through approximation ratio preserving reductions from MCP to MAX DNF and from MAX DNF to MIN SAT. A 2-approximation algorithm for MIN SAT was presented in [6]. An approximation ratio preserving reduction from MIN SAT to min vertex cover improves the approximation ratio to $2 - Theta(1/ sqrt{n})$ [10]. Hence we prove false the infamous conjecture, which argues that there cannot be a polynomial time algorithm for MCP with an approximation ratio of any constant factor.
In the field of constraint satisfaction problems (CSP), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: strict and weak, and in the associated decision problem one must distinguish between being able to satisfy all the strict constraints versus not being able to satisfy all the weak constraints. The most commonly cited example of a promise CSP is the approximate graph coloring problem--which has recently seen exciting progress [BKO19, WZ20] benefiting from a systematic algebraic approach to promise CSPs based on polymorphisms, operations that map tuples in the strict form of each constraint to tuples in the corresponding weak form. In this work, we present a simple algorithm which in polynomial time solves the decision problem for all promise CSPs that admit infinitely many symmetric polymorphisms, which are invariant under arbitrary coordinate permutations. This generalizes previous work of the first two authors [BG19]. We also extend this algorithm to a more general class of block-symmetric polymorphisms. As a corollary, this single algorithm solves all polynomial-time tractable Boolean CSPs simultaneously. These results give a new perspective on Schaefers classic dichotomy theorem and shed further light on how symmetries of polymorphisms enable algorithms. Finally, we show that block symmetric polymorphisms are not only sufficient but also necessary for this algorithm to work, thus establishing its precise power
We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a k-gap interval graph if it has a multiple interval representation with at most n+k intervals in total. In order to scale up the nice algorithmic properties of interval graphs (where k=0), we parameterize graph problems by k, and find FPT algorithms for several problems, including Feedback Vertex Set, Dominating Set, Independent Set, Clique, Clique Cover, and Multiple Interval Transversal. The Coloring problem turns out to be W[1]-hard and we design an XP algorithm for the recognition problem.
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