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Extended Version of: On the Structural Hardness of Answer Set Programming: Can Structure Efficiently Confine the Power of Disjunctions?

Markus Hecher, Rafael Kiesel

TL;DR

The results provide an in-depth hardness study, relying on a novel reduction from normal to disjunctive programs, trading the increase of complexity for an exponential parameter compression.

Abstract

Answer Set Programming (ASP) is a generic problem modeling and solving framework with a strong focus on knowledge representation and a rapid growth of industrial applications. So far, the study of complexity resulted in characterizing hardness and determining their sources, fine-grained insights in the form of dichotomy-style results, as well as detailed parameterized complexity landscapes. Unfortunately, for the well-known parameter treewidth disjunctive programs require double-exponential runtime under reasonable complexity assumptions. This quickly becomes out of reach. We deal with the classification of structural parameters for disjunctive ASP on the program's rule structure (incidence graph). First, we provide a polynomial kernel to obtain single-exponential runtime in terms of vertex cover size, despite subset-minimization being not represented in the program's structure. Then we turn our attention to strictly better structural parameters between vertex cover size and treewidth. Here, we provide double-exponential lower bounds for the most prominent parameters in that range: treedepth, feedback vertex size, and cliquewidth. Based on this, we argue that unfortunately our options beyond vertex cover size are limited. Our results provide an in-depth hardness study, relying on a novel reduction from normal to disjunctive programs, trading the increase of complexity for an exponential parameter compression.

Extended Version of: On the Structural Hardness of Answer Set Programming: Can Structure Efficiently Confine the Power of Disjunctions?

TL;DR

The results provide an in-depth hardness study, relying on a novel reduction from normal to disjunctive programs, trading the increase of complexity for an exponential parameter compression.

Abstract

Answer Set Programming (ASP) is a generic problem modeling and solving framework with a strong focus on knowledge representation and a rapid growth of industrial applications. So far, the study of complexity resulted in characterizing hardness and determining their sources, fine-grained insights in the form of dichotomy-style results, as well as detailed parameterized complexity landscapes. Unfortunately, for the well-known parameter treewidth disjunctive programs require double-exponential runtime under reasonable complexity assumptions. This quickly becomes out of reach. We deal with the classification of structural parameters for disjunctive ASP on the program's rule structure (incidence graph). First, we provide a polynomial kernel to obtain single-exponential runtime in terms of vertex cover size, despite subset-minimization being not represented in the program's structure. Then we turn our attention to strictly better structural parameters between vertex cover size and treewidth. Here, we provide double-exponential lower bounds for the most prominent parameters in that range: treedepth, feedback vertex size, and cliquewidth. Based on this, we argue that unfortunately our options beyond vertex cover size are limited. Our results provide an in-depth hardness study, relying on a novel reduction from normal to disjunctive programs, trading the increase of complexity for an exponential parameter compression.
Paper Structure (10 sections, 31 theorems, 2 equations, 5 figures)

This paper contains 10 sections, 31 theorems, 2 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vertex Cover Limits the Cost of Disjunction
  4. Decrease Cyclicity and Treedepth by the Power of Disjunction
  5. Hardness for Bandwidth and Cutwidth
  6. Discussion and Conclusion
  7. Acknowledgments

Key Result

Theorem 1

Let $\Pi$ be a program such that for every rule $r \in \Pi$ we have $|H_r| + |B_r^-| + |B_r^+| \leq c$ for $c \in \mathbb{N}$. Further, let $S \subseteq \text{\normalfont at}(\Pi)$ be a vertex cover of $\mathcal{G}_{\Pi}$. Then there is a program $\Pi'$ where we have (i) $|\text{\normalfont at}(\Pi'

Figures (5)

  • Figure 1: (Left): The primal graph $\mathcal{G}_{\Pi_1}$ of program $\Pi_1$ of Example \ref{['ex:running1']}. (Right): The incidence graph $\mathcal{I}_{\Pi_1}$ of $\Pi_1$.
  • Figure 2: (Left): A TD (PD) $\mathcal{T}$ of $\mathcal{G}_{\Pi_1}$ of Figure \ref{['fig:graphs']}. (Right): A Trémaux tree of $\mathcal{G}_{\Pi_1}$.
  • Figure 3: The reduction $\mathcal{R}_{fvs}$ that takes a normalized fully tight program $\Pi$ and a sparse feedback vertex set $S$ of $\mathcal{G}_{\Pi}$.
  • Figure 4: Visualization of the structure of $\mathcal{R}_{fvs}$ for some normalized fully tight program $\Pi$. (Left): Primal graph $\mathcal{G}_{\Pi}$ together with a sparse FVS $S$ that connects $\mathcal{G}_{\Pi}$. (Right): $\mathcal{G}_{\Pi'}$ (simplified) and a sparse FVS $S'$ of $\Pi'$, obtained by $\mathcal{R}_{fvs}$.
  • Figure 5: The reduction $\mathcal{R}_{td}$ that takes a normalized fully tight program $\Pi$ and an annotated nice TD $\mathcal{T}=(T,\chi,\varphi)$ of $\mathcal{G}_{\Pi}$.

Theorems & Definitions (53)

  • Example 1
  • Example 2
  • Theorem 1: Polynomial VC-Kernel
  • proof : Proof (Sketch)
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Example 3
  • Lemma 1: Runtime & Correctness
  • Theorem 2
  • ...and 43 more