[Paper Review] Reducing CMSO Model Checking to Highly Connected Graphs

Daniel Lokshtanov, M. S. Ramanujan+2 more|arXiv (Cornell University)|Jan 1, 2018

Formal Methods in Verification29 references3 citations

TL;DR

This paper introduces a theorem reducing Counting Monadic Second-Order (CMSO) model checking on general graphs to model checking on highly connected (unbreakable) graphs. It proves that if a CMSO sentence ψ is solvable in O(nd) time on (s,c)-unbreakable graphs for d > 4, then it is also solvable in O(nd) time on all graphs. The result enables replacing complex recursive understanding techniques in parameterized algorithms with a black-box invocation of this theorem, simplifying FPT algorithm design for problems like Vertex Multiway Cut Uncut.

ABSTRACT

Given a Counting Monadic Second Order (CMSO) sentence $ψ$, the CMSO$[ψ]$ problem is defined as follows. The input to CMSO$[ψ]$ is a graph $G$, and the objective is to determine whether $G\models ψ$. Our main theorem states that for every CMSO sentence $ψ$, if CMSO$[ψ]$ is solvable in polynomial time on "globally highly connected graphs", then CMSO$[ψ]$ 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.

Motivation & Objective

• To establish a general reduction that transforms CMSO model checking on arbitrary graphs into model checking on highly connected graphs.
• To simplify the design of parameterized algorithms by replacing intricate recursive understanding techniques with a black-box theorem.
• To demonstrate the theorem's utility by showing fixed-parameter tractability for problems involving connected subgraphs with few external neighbors and CMSO-definable properties.
• To provide a unified framework for proving FPT results for problems such as Vertex Multiway Cut Uncut using CMSO logic and unbreakability.

Proposed method

• Define (s,c)-unbreakable graphs as those without small separators that split large vertex sets into two large components.
• Prove that if CMSO[ψ] is solvable in O(nd) time on (s,c)-unbreakable graphs for d > 4, then it is solvable in O(nd) time on all graphs.
• Use the theorem to replace technical recursive understanding arguments with a black-box reduction to unbreakable graphs.
• Formulate the Vertex Multiway Cut Uncut problem as a CMSO model checking problem using a formula ϕ with free variables R and S.
• Design an FPT algorithm for the reduced problem on (s(k),k)-unbreakable graphs by leveraging bounded component size and connectivity constraints.
• Apply Theorem 3 to lift the FPT result from unbreakable graphs to general graphs, completing the proof of fixed-parameter tractability.

Experimental results

Research questions

• RQ1 Can CMSO model checking on general graphs be reduced to model checking on highly connected graphs without loss of efficiency?
• RQ2 Is it possible to replace complex recursive understanding techniques in parameterized algorithms with a simpler black-box reduction?
• RQ3 Does the existence of an efficient algorithm for CMSO[ψ] on (s,c)-unbreakable graphs imply an efficient algorithm on all graphs for the same formula?
• RQ4 Can this reduction be used to prove fixed-parameter tractability for problems involving connected induced subgraphs with few external neighbors and CMSO-definable properties?
• RQ5 Can the theorem be effectively applied to known hard problems like Vertex Multiway Cut Uncut to yield FPT algorithms?

Key findings

• For every CMSO sentence ψ, if CMSO[ψ] is solvable in O(nd) time on (s,c)-unbreakable graphs for d > 4, then it is solvable in O(nd) time on all graphs.
• The main theorem allows replacing problem-specific recursive understanding techniques with a general black-box reduction to unbreakable graphs.
• The Vertex Multiway Cut Uncut problem is shown to be fixed-parameter tractable by formulating it as a CMSO model checking problem and applying the main theorem.
• An FPT algorithm for the related Vertex-Restricted Bounded-Cut-Union problem (V-RBCU) is constructed on (s(k),k)-unbreakable graphs, with running time O(n(n+m)).
• The correctness of the FPT algorithm for V-MWCU follows from combining the main theorem, a CMSO formula ϕ, and the bounded component size property in unbreakable graphs.
• The result demonstrates that the complexity of designing FPT algorithms can be significantly reduced by focusing on unbreakable instances and leveraging the main theorem.