Parameterized Complexity of MinCSP over the Point Algebra

TL;DR

We study the parameterized complexity of MinCSP over the Point Algebra with constraint relations $<,\le,=,\neq$ under a cost budget $k$. The main result is a complete classification: if the constraint language $\Gamma$ contains both $\le$ and $\neq$, MinCSP$(\Gamma)$ is $W[1]$-hard; otherwise it is fixed-parameter tractable. For the tractable cases, we obtain an $FPT$ algorithm for MinCSP$(<,=,\neq)$ by compress-branch-cut, reducing to a Boolean MinCSP that is solved via flow augmentation; this also generalizes known results for related problems like Directed Feedback Arc Set and Edge Multicut. Additionally, we establish $W[1]$-hardness for Directed Symmetric Multicut, solving an open problem and clarifying the boundary between directed and undirected transversal problems. Collectively, the results connect CSP constraint languages with classical graph separation problems and advance parameterized techniques for temporal reasoning formalisms.

Abstract

The input in the Minimum-Cost Constraint Satisfaction Problem (MinCSP) over the Point Algebra contains a set of variables, a collection of constraints of the form $x < y$, $x = y$, $x \leq y$ and $x \neq y$, and a budget $k$. The goal is to check whether it is possible to assign rational values to the variables while breaking constraints of total cost at most $k$. This problem generalizes several prominent graph separation and transversal problems: MinCSP$(<)$ is equivalent to Directed Feedback Arc Set, MinCSP$(<,\leq)$ is equivalent to Directed Subset Feedback Arc Set, MinCSP$(=,\neq)$ is equivalent to Edge Multicut, and MinCSP$(\leq,\neq)$ is equivalent to Directed Symmetric Multicut. Apart from trivial cases, MinCSP$(Γ)$ for $Γ\subseteq \{<,=,\leq,\neq\}$ is NP-hard even to approximate within any constant factor under the Unique Games Conjecture. Hence, we study parameterized complexity of this problem under a natural parameterization by the solution cost $k$. We obtain a complete classification: if $Γ\subseteq \{<,=,\leq,\neq\}$ contains both $\leq$ and $\neq$, then MinCSP$(Γ)$ is W[1]-hard, otherwise it is fixed-parameter tractable. For the positive cases, we solve MinCSP$(<,=,\neq)$, generalizing the FPT results for Directed Feedback Arc Set and Edge Multicut as well as their weighted versions. Our algorithm works by reducing the problem into a Boolean MinCSP, which is in turn solved by flow augmentation. For the lower bounds, we prove that Directed Symmetric Multicut is W[1]-hard, solving an open problem.

Figures (3)

• Figure 1: These are seven subsets of Point Algebra. Arrow represent polynomial-time cost-preserving reductions between corresponding MinCSPs that either follow by inclusion or using $(x = y) \equiv (x \leq y) \land (y \leq x)$ and $(x < y) \equiv (x \leq y) \land (x \neq y)$. MinCSPs for all of them are NP-hard, and in FPT for all except the red language, for which it is W[1]-hard. There are eight more non-empty subsets, out of which four give rise to polynomial-time solvable MinCSPs ($\neq$ and $\{\leq,=\}$ with subsets), subset $\{<,\neq\}$, which reduced to $\{<\}$ because all $\neq$-constraints can be safely disregarded, and three more subsets that contain both $\leq$ and $\neq$.
• Figure 2: A diamond digraph on the left and a sequence of three joined diamonds on the right. Undeletable arcs are drawn with thick lines, and deletable arcs are drawn with a thin line.
• Figure 3: Snippet of the construction of the gadget (bottom) for the graph $G$ of non-edges (top) with $n = 4$ and $k = 2$. Vertices $x_0,\dots,x_{16}$ and $y_0,\dots,y_{16}$ are junction vertices of the necklaces corresponding to the left hand side and the right hand side of $G$, respectively. Colorful arrows in the gadget represent paths of crossing arcs. Solid delimiters at $x_0$, $x_{12}$ and $y_0$, $y_{12}$ indicate starts of new strings.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Proposition 3
• Theorem 4: Theorem 1.2 in kim2023flowIIIsoda
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof