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arxiv: 2605.21267 · v1 · pith:4NZ5DD3Snew · submitted 2026-05-20 · 💻 cs.CC

Towards Single Exponential Time for Temporal and Spatial Reasoning: A Study via Redundancy and Dynamic Programming

Victor Lagerkvist , Johanna Groven , Leif Eriksson This is my paper

Pith reviewed 2026-05-21 03:34 UTC · model grok-4.3

classification 💻 cs.CC
keywords interval algebraregion connection calculusdynamic programmingqualitative spatial reasoningqualitative temporal reasoningsingle exponential timenon-redundant constraintsNP-hard fragments
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The pith

Dynamic programming achieves 4^n time for a hard fragment of interval algebra and matches the o(n)^n bound for region connection calculus.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates paths toward single-exponential time for the NP-hard qualitative reasoning problems RCC and IA, which are currently solvable in 2^{O(n log n)} time. It first classifies the maximum number of non-redundant constraints to show that simple branching cannot improve on exhaustive search. The main algorithmic contributions are a dynamic programming procedure that solves a non-trivial NP-hard fragment of IA in 4^n time, covering the interval graph sandwich problem, and a refined DP method for RCC with its eight basic relations that reaches the o(n)^n bound already known for IA.

Core claim

We present a 4^n time algorithm for a non-trivial NP-hard fragment of IA, which includes the interval graph sandwich problem, and a more sophisticated dynamic programming approach for RCC with 8 basic relations that asymptotically matches the o(n)^n bound known for IA.

What carries the argument

Dynamic programming whose states are indexed by the classified sets of non-redundant constraints.

If this is right

  • A non-trivial NP-hard fragment of Allen's interval algebra admits single-exponential time.
  • RCC with its eight basic relations can be solved in o(n)^n time.
  • Branching on constraints is provably insufficient once the number of non-redundant constraints is taken into account.
  • These DP techniques narrow the gap between current upper bounds and the open question of true single-exponential time for the full problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the polynomial-time state update property holds more broadly, the same state representation might be adapted to other qualitative spatial or temporal calculi.
  • The redundancy classification step could be reused as a preprocessing routine to speed up existing solvers for IA and RCC fragments.
  • Testing the DP on the full IA would clarify whether the extra sophistication needed for RCC can be carried over without losing the single-exponential character.

Load-bearing premise

The chosen DP state representation based on non-redundant constraints can be updated in polynomial time per state without hidden exponential blow-up when extending partial solutions.

What would settle it

A concrete IA instance in the solvable fragment where maintaining the DP states over non-redundant constraints requires super-polynomial work per state, pushing the total time above 4^n.

Figures

Figures reproduced from arXiv: 2605.21267 by Johanna Groven, Leif Eriksson, Victor Lagerkvist.

Figure 1
Figure 1. Figure 1: Illustration of the basic relations of RCC- [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Graphic representation of the instance described in Example 5. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
read the original abstract

The region connection calculus ($RCC$) and Allen's interval algebra ($IA$) are two well-known NP-hard spatial-temporal qualitative reasoning problems. They are solvable in $2^{O(n \log n)}$ time, where $n$ is the number of variables, and $IA$ is additionally known to be solvable in $o(n)^n$ time. However, no improvement over exhaustive search is known for $RCC$, and if they are also solvable in single exponential time $2^{O(n)}$ is unknown. We investigate multiple avenues towards reaching such bounds. First, we show that branching is insufficient since there are too many non-redundant constraints. Concretely, we classify the maximum number of non-redundant constraints in $RCC$ and $IA$. Algorithmically, we make two significant contributions based on dynamic programming (DP). The first algorithm runs in $4^n$ time and is applicable to a non-trivial, NP-hard fragment of $IA$, which includes the well-known interval graph sandwich problem of Golumbic and Shamir (1993). For the richer $RCC$ problem with 8 basic relations we use a more sophisticated approach which asymptotically matches the $o(n)^n$ bound for $IA$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper classifies the maximum number of non-redundant constraints in RCC and IA to show branching is insufficient for single-exponential time. It then gives a 4^n DP algorithm for a non-trivial NP-hard fragment of IA (including the interval graph sandwich problem) and a more sophisticated DP for RCC with its 8 basic relations that asymptotically matches the known o(n)^n bound for IA.

Significance. The explicit classification of non-redundant constraints is a useful technical contribution that grounds the state space for the DPs. If the transition costs are polynomial per state, the RCC result would be the first improvement over 2^{O(n log n)} for that problem and would close the gap with the best IA bound; this would be a meaningful step toward single-exponential algorithms for qualitative spatial-temporal reasoning.

major comments (2)
  1. [§5] §5 (RCC DP): the states are indexed by the classified non-redundant constraints on the first k variables, yet the manuscript supplies neither the explicit recurrence for the transition that adds variable k+1 nor a proof that this update (including consistency checking and redundancy re-classification) runs in time polynomial in the current state size. Without this, the o(n)^n claim cannot be verified and may fail if the per-state cost grows with k.
  2. [§4] §4 (IA fragment DP): the 4^n bound for the fragment containing the interval graph sandwich problem is stated, but the proof that the DP transitions on the classified non-redundant constraints can be performed without hidden exponential blow-up when extending partial solutions is not detailed; this is load-bearing for the claimed running time.
minor comments (2)
  1. [Abstract] The abstract refers to 'a more sophisticated dynamic programming approach' for RCC but does not restate the precise asymptotic bound achieved; a single sentence clarifying that it matches the o(n)^n bound for IA would improve readability.
  2. [§3] Notation for the classified constraint sets is introduced without a compact summary table; adding one would help readers track how the state space size is bounded.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments on our manuscript. We address each major comment below and commit to revisions that will make the dynamic programming arguments fully rigorous and verifiable.

read point-by-point responses

  1. Referee: [§5] §5 (RCC DP): the states are indexed by the classified non-redundant constraints on the first k variables, yet the manuscript supplies neither the explicit recurrence for the transition that adds variable k+1 nor a proof that this update (including consistency checking and redundancy re-classification) runs in time polynomial in the current state size. Without this, the o(n)^n claim cannot be verified and may fail if the per-state cost grows with k.

    Authors: We agree that the current presentation of the RCC dynamic program is high-level and that an explicit recurrence together with a polynomial-time transition proof is necessary to substantiate the o(n)^n bound. The manuscript relies on the prior classification of non-redundant constraints to bound the state space, but does not spell out the precise update rule when the (k+1)th variable is introduced or prove that consistency checking and redundancy re-classification can be performed in time polynomial in the current state size. In the revised version we will insert a dedicated subsection that (i) states the recurrence explicitly, (ii) describes how the new constraints involving variable k+1 are generated and checked for consistency against the existing non-redundant set, and (iii) shows that each such operation runs in time polynomial in the number of non-redundant constraints (which is O(n) by our classification). This will confirm that the per-state cost does not grow with k and that the overall running time remains o(n)^n. revision: yes

  2. Referee: [§4] §4 (IA fragment DP): the 4^n bound for the fragment containing the interval graph sandwich problem is stated, but the proof that the DP transitions on the classified non-redundant constraints can be performed without hidden exponential blow-up when extending partial solutions is not detailed; this is load-bearing for the claimed running time.

    Authors: We acknowledge that the proof of polynomial-time transitions for the 4^n DP on the IA fragment is only sketched and that a fully detailed argument is required to rule out hidden exponential factors. The manuscript describes the state space via the classified non-redundant constraints and asserts that extending a partial solution by one variable can be done efficiently, yet it does not supply an explicit transition function or a complexity analysis showing that consistency and redundancy updates remain polynomial. In the revision we will expand §4 with (i) the precise recurrence, (ii) a description of how the new interval constraints are added and checked against the non-redundant set, and (iii) a proof that each extension step runs in time polynomial in the state size (which is bounded by 4^n overall). This will establish that the claimed 4^n bound holds without exponential blow-up. revision: yes

Circularity Check

0 steps flagged

No significant circularity; time bounds follow from explicit DP state enumeration

full rationale

The paper derives its 4^n bound for the IA fragment and the o(n)^n-matching bound for RCC directly from counting states indexed by classified non-redundant constraints and stating polynomial-time transitions per state. These counts and costs are presented as consequences of the algorithmic construction itself rather than fitted to the target bound or reduced to a self-citation. No self-definitional loop, fitted-input prediction, or load-bearing self-citation appears in the abstract or described approach; the derivation remains self-contained against the stated DP representation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard complexity-theoretic assumptions and the structural properties of the constraint languages; no free parameters or invented entities are introduced.

axioms (1)
  • standard math NP-hardness of the considered fragments follows from known polynomial-time reductions.
    Invoked to justify that the problems remain hard even after restricting to the fragments studied.

pith-pipeline@v0.9.0 · 5758 in / 1037 out tokens · 52114 ms · 2026-05-21T03:34:49.296449+00:00 · methodology

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Reference graph

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