QCSP on partially reflexive forests Barnaby Martin Algorithms and - - PowerPoint PPT Presentation

QCSP on partially reflexive forests

Barnaby Martin

Algorithms and Complexity in Durham, Durham University, U.K.

CP 2011, Perugia. 14th August 2011.

Model Checking problem

We are interested in the parameterisation of the model checking problem by the model. Fix a logic L and fix D. The problem “L (D)” has

◮ Input: a sentence Φ of L . ◮ Question: does D |

= Φ? We consider syntactic fragments L of FO and structures D that are relational and finite.

Complexity of Model Checking

Fragment Dual Classification? {∃, ∨} {∀, ∧} L {∃, ∨, =} {∀, ∧, =} {∃, ∨, =} {∀, ∧, =} {∃, ∧, ∨} {∀, ∧, ∨} L if there is some element a such that all relations are {∃, ∧, ∨, =} {∀, ∧, ∨, =} a-valid, and NP-complete otherwise {∃, ∧, ∨, =} {∀, ∧, ∨, =} {∃, ∧} {∀, ∨} CSP dichotomy conjecture: P or NP-complete {∃, ∧, =} {∀, ∨, =} {∃, ∧, =} {∀, ∨, =} NP-complete for |D| ≥ 3, reduces to Schaefer classes other- wise. {∃, ∀, ∧} {∃, ∀, ∨} QCSP trichotomy??: P, NP-complete, or Pspace-complete {∃, ∀, ∧, =} {∃, ∀, ∨, =} {∃, ∀, ∧, =} {∃, ∀, ∨, =} Pspace-complete for |D| ≥ 3, reduces to Schaefer classes for Quantified Sat otherwise. {∀, ∃, ∧, ∨} Positive equality free: recently classified tetrachotomy {∀, ∃, ∧, ∨, =} {∀, ∃, ∧, ∨, =} L when |D| ≤ 1, Pspace-complete otherwise {¬, ∃, ∀, ∧, ∨, =} {¬, ∃, ∀, ∧, ∨} L when D contains only empty or full relations, Pspace- complete otherwise

The QCSP

The quantifed constraint satisfaction problem QCSP(D) is the problem with

◮ Input: a sentence

Φ := ∀x1∃x2 . . . ∀x2m−1∃x2m φ(x1, x1, . . . , x1, x1), where φ is a conjunction of positive atoms.

◮ Question: does D |

= Φ? QCSP(D) is always in Pspace.

Extant classifications

QCSP classifications are harder than CSP classifications.

◮ Boolean structures. Dichotomy P, Pspace-complete.

(Creignou et al. 2001/ Dalmau 1997.)

◮ Graphs of permutations. Trichotomy P, NP-complete,

Pspace-complete. (B¨

• rner et al. 2002.)

◮ Graphs with at most one cycle. Trichotomy L, NP-complete,

Pspace-complete. (M.-Madelaine 2006.)

◮ Structures with 2-semilattice polymorphism. Dichotomy P,

coNP-hard. (Chen 2004.) The algebraic approach is weaker for QCSPs and the combinatorial method has fewer constructs. Separating NP-hard into NP-complete and Pspace-complete is especially difficult.

Foresting renaissance

Recently a major classification appeared for the retraction problem Ret(H).

Theorem (Feder, Hell, Jonsson, Krokhin and Nordh 2010)

Let H be a partially reflexive pseudoforest. If some pseudotree T ⊆ H has disconnected self-loops, or any cycle of length ≥ 5, or a reflexive 4-cycle, or an irreflexive 3-cycle, then Ret(H) is NP-complete. Otherwise, Ret(H) is in NL. Following this, something similar but weaker came for the surjective homomorphism problem.

Theorem (Golovach, Paulusma and Song 2011)

Let H be a partially reflexive forest. If some tree T ⊆ H has disconnected self-loops, then SurHom(H) is NP-conmplete. Otherwise, SurHom(H) is in L.

This last result is our principle motivation. The L cases for trees are H that are loop-connected. All other trees are NP-complete. let λH be the longest distance from a vertex in H to a loop in H. If either 1.) there exists no looped vertex or 2.) there exists a single reflexive connected component T0 ⊆ H, such that there is a λH-walk from any vertex of H to T0, then H is quasi-loop-connected.

Theorem (CP 2011)

Let H be a forest. If H is disconnected (not a tree) then QCSP(H) is in NL. If H is quasi-loop-connected then QCSP(H) is in NL. Otherwise, QCSP(H) is NP-hard.

Theorem (CP 2011)

Let H be a path. If H is quasi-loop-connected then QCSP(H) is in

• NL. Otherwise, QCSP(H) is Pspace-complete.
• EASY

HARD EASY

Method – easy cases

Lemma (CP 2011)

If H is a loop-connected tree then H admits a majority polymorphism. It follows from Chen that QCSP(H) is in NL. What about the quasi-loop-connected case? We apply the following.

Theorem (Chen-Madelaine-M. 2008)

QCSP(A) =QCSP(B) iff exists a, b such that Aa − → → B and Bb − → → A.

Each of the following has the same QCSP

• Lemma (CP 2011)

For each quasi-loop-connected A there exists a loop-connected B and a b such that A− → → B and Bb − → → A.

• 1

2 −2 −1

Figure: P100

2 and its . . .

−0 −0 1 −1 1 −0 2 −2

Figure: . . . homomorphism to P00100

Method – hard cases

QCSP(P101): reduction from quantified not-all-equal 3-sat, borrowing heavily from Patrignani and Pizzonia 2001 proof that matching cut is NP-complete.

• ⊤

⊥ l1 l2 l3 v1 v2 ∀

Figure: Variable and clause gadgets in reduction to QCSP(P101).

Call a path weakly balanced if, starting from the centre and walking left (also right) one encounters at some point a non-loop followed by a loop.

◮ Weakly balanced 0-centred paths are very like P101.

Pspace-complete.

◮ Weakly balanced 1-centred paths are harder to classify,

especially P10101. Pspace-complete.

• v1

⊤ ⊥ v ′

1

v ′

2

v2 ∀

Figure: Variable gadgets in reduction to QCSP(P10101).

• ⊤

⊥ l1 l2 l3 l′

3

l′

1

l′

2

Figure: Clause gadget in reduction to QCSP(P10101).

Call a path weakly balanced if, starting from the centre and walking left (also right) one encounters at some point a non-loop followed by a loop.

◮ Weakly balanced 0-centred paths are very like P101.

Pspace-complete.

◮ Weakly balanced 1-centred paths are harder to classify,

especially P10101. Pspace-complete.

◮ Remaining not quasi-loop-connected paths are easy to classify.

Pspace-complete.

Pseudoforests

Conjecture

Let H be a pseudoforest. Then QCSP(H) is either in NL, NP-complete or Pspace-complete. This is settled for irreflexive pseudoforests and partially reflexive paths.