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A Rendezvous of Complexity, Logic, Algebra, andCombinatorics
Hubie Chen
Universitat Pompeu FabraBarcelona, Spain
BCAM - September 2011
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3-Colorability
Given an undirected graph...
...is it 3-colorable?
Idea: divide a set of objects into three groups so that no twoincompatible objects are placed into same group
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Boolean satisfiability
Given a conjunction of propositional clauses...
(¬a ∨ ¬b ∨ c) ∧ (¬c ∨ d) ∧ (a ∨ ¬d)
...is there a satisfying assignment to the variables?
a → 1b → 0c → 0d → 1
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Digraph acyclicity
Given a directed graph, does it contain a cycle?
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Efficient algorithms
A broad goal of computer science:
Develop fast, usable algorithms for computational problems
A broad goal of theoretical computer science:
Classify problems according to whether or notthey have “efficient” algorithms
That is, classify problems according to their inherentcomplexity...
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Classifying problems
Theoretical computer science: place problems into complexityclasses...
P: polynomial time
Theoretical notion of efficiency/tractabilityProblems for which there’s an algorithm running in time O(nk)Ex: digraph acyclicity
NP: non-deterministic polynomial time
P ⊆ NP, believed that NP is “much bigger”Idea: problems for which a solution can be efficiently verified
Exs: all three problems seen
NP-hard: problems “as hard as”/that “can simulate”all problems in NP
Exs: 3-colorability, boolean sat.
Generally believed that
P = NP ⇔ no NP-hard problem is in P
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A dream
(Garey & Johnson ’79): Compendium of 100s of NP-hardproblems
Dream: to have an encyclopedia – a general classificationtheorem that, given a problem, would tell us if tractable ornot (if in P or if NP-hard)
This talk: discuss (my) research towards this dream...
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Classification
Classification projects
Identify a broad family of problemsTry to classify each problem as tractable/intractableOften, obtain generic conditions for tractability/intractability
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Exhibit #1: Constraint Satisfaction
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Examples
Given an undirected graph, is it 3-colorable?
Given a conjunction of propositional clauses, is there asatisfying assignment?
(¬a ∨ ¬b ∨ c) ∧ (¬c) ∧ (a)
Given a directed graph, does it contain a cycle?
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Animals
Are these problems inherently different animals?
...or can we place them into an organizational scheme?
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Constraint Satisfaction
All of these problems are constraint satisfaction problems!
The constraint satisfaction problem (CSP)
Given a primitive positive (pp) sentence
φ = ∃v1 . . . ∃vn (R(vi1 , . . . , vik ) ∧ . . .)
and a relational structure B, decide if B |= φ.
Idea: decide if there is an assignment to variables satisfying acollection of constraints
Examples:
graph coloring (vi = vertices, B = colors,one constraint E (vi , vj) for each edge)boolean satisfiability (vi = variables, B = 0, 1,one constraint for each clause)
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Constraint Satisfaction
For each structure B, we can define:
CSP(B)
Given a primitive positive (pp) sentence φ, decide if B |= φ.
Can capture many studied problems:
Graph 3-Coloring (B = K3), generalizationsBoolean sat. problems: 3-SAT, 2-SAT, Horn-SATDigraph acyclicity (B = (Q;<))Temporal Reasoning (e.g. B = (Q;≤, <, =))Systems of equations over various algebraic structures
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Constraint Satisfaction
Unified framework, captures many problems
Numerous application areas:
Assignment/scheduling problemsBoolean sat. problems & generalizationsGraph-theoretic problemsSpatial and temporal reasoningInterval reasoningAlgebraic equations problems
Research question:
Which problems CSP(B) are tractable (in poly-time), and why?
Known:
(Schaefer ’78): Classification for 2-element structures(Bulatov ’02): Classification for 3-element structures
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Constraint Satisfaction and Consistency
Consistency methods: simple, efficient, well-studied,used-in-practice inference methods that (in general) candetect unsatisfiability
What are the scope of various consistency methods?Which problems CSP(B) can they solve?
My work: study ofarc consistency and extensions(Bodirsky & Chen ’10), (Chen, Dalmau & Grussien ’11)local-to-global consistency(Bodirsky & Chen ’09), (Bodirsky, Chen & Wrona, ongoing)
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An Algebraic Approach
Algebraic approach: associate to each struct. B an algebra AB
An operation f : Bm → B is a polymorphism of a relationU ⊆ Bk if for any choice of m tuples from U, applying fcoordinatewise yields a tuple (s1, . . . , sk) also in U.
(t11, t12, . . . t1k) ∈ U...
.... . .
...(tm1, tm2, . . . tmk) ∈ U
f ↓ f ↓ . . . f ↓
(s1, s2, . . . sk) ∈ U
An operation f is a polymorphism of a structure if it is apolymorphism of all relations (of the structure)
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An Algebraic Approach
For a struct. B, define the algebra of polymorphisms AB to bethe algebra
with universe B , andwhose operations are all polymorphisms of B
Key Theorem (Geiger/Bodcharnuk et al. ’60s, Jeavons et al. ’90s):
If finite structs. B,B have AB = AB , thenB,B can pp-define (∃,∧) the same relations, andCSP(B), CSP(B) have same complexity(up to polytime reduction)
Use properties of algebra AB to understand struct. B
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Exhibit #2: Quantified Constraint Satisfaction
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Quantified Constraint Satisfaction
The quantified constraint satisfaction problem (QCSP)
Given a conjunctive positive (cp) sentence
φ = Q1v1 . . .Qnvn (R(vi1 , . . . , vik ) ∧ . . .)
(with Qi ∈ ∀, ∃), and a relational structure B, decide if B |= φ.
Can model situations with uncertainty, two-party interaction,... that cannot be modelled with CSP(complexity is higher: generally PSPACE-complete)
Wide range of applications: verification, game playing,adversarial reasoning, planning, ...
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Quantified Constraint Satisfaction
QCSP(B)
Given a conjunctive positive sentence φ, decide if B |= φ.
Framework goes back to:
(Aspvall, Plass, Tarjan ’79) - Quantified 2-SAT
(Karpinski, Kleine Buning, Schmitt ’87) - Quantified HornSAT
Research issues:
Search for generic sources of tractability / hardness
Develop algebraic approach
Attempt to establish systematic classifications
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Polynomially Generated Powers (PGP)
(Chen ’08, ’11): study of polynomially generated powers(PGP) property – algebraic property
A combinatorial property of algebras concerning the size ofgenerating sets for An
(Wiegold ’70s-’80s), (Quick & Ruskuc ’10), (Awang, Garrido,McLeman, Quick & Ruskuc), ...
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Polynomially Generated Powers (PGP)
Def: An algebra A has polynomially generated powers if thereis a polynomial p ∈ N[X ] such that, for each n ≥ 1,the algebra An has a generating set of size ≤ p(n)
Example: let G be a finite group with identity elt. eConsider tuples where all entries [with one possible exception]are = e. In G 4:
(g1, e, e, e)(e, g2, e, e)(e, e, g3, e)(e, e, e, g4)↓ ↓ ↓ ↓
(g1, g2, g3, g4)
Have generating set for Gn of size ≤ |G |n: linear growth
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Polynomially Generated Powers (PGP)
(Chen ’11): Gave broad conditions for reducing QCSP to CSPbased on PGP
Observation: Suppose that algebra AB has the PGP, and agenerating set for An
B can be computed in poly time (in n)
Then, Π2-QCSPc(B) reduces to CSPc(B):
Consider an instance Φ = ∀y1 . . . ∀yn∃x1 . . . ∃xmψCompute a generating set t1, . . . , tk for An
BLet Φ[ti ] be the CSP instance where the vars. yj arereplaced with the values in tiΦ is equivalent to the CSP instance Φ[t1] ∧ · · · ∧ Φ[tk ]
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Polynomially Generated Powers (PGP)
Thm (Chen ’11): Let AB be a 3-elt algebra[that’s idempotent and where CSP(B) tractable].
Either:
AB is switchable, has the PGP, and QCSP(B) tractable; orAB has exponentially generated powers (EGP):generating sets require size Ω(cn), with c > 1
Dichotomy theorem: all studied algebras have PGP or EGP
Reduces the classification of tractable 3-elt QCSP(B) to aparticular, robust class of algebras
Open questions:
Full QCSP classification for 3-elt structuresClassification of PGP/EGP on all finite algebras
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Exhibit #3: Constraint Satisfaction,from the other side
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Constraint Satisfaction, the other side
Recall...
The Constraint Satisfaction Problem (CSP)
Given a primitive positive sentence φ = ∃v1 . . . ∃vn ∧ αj anda structure B, decide if B |= φ.
Before, we restricted the structure. Now we restrict the formulas.
Remark: for a fixed sentence φ, the CSP on φ is in poly time
Question: on which sets of formulas is CSP tractable?What structural restrictions can we place on formulas, toensure tractability?
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Graph-based approach
Graph-based approach: given a CSP instance, look at graph wherevertices are variables, two variables linked if in a commonconstraint
Example:∃a∃b∃c∃d∃e∃f ∃g (R(a, b, c) ∧ S(b, c , d) ∧ T (d , e, f )
∧U(a, g) ∧W (f , g))
↓
a
b
c
d
e
f
g
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Treewidth
Thm (Freuder ’90):When the graphs have bounded treewidth,the CSP is tractable.
Tree, tw=1 Cycle, tw=2 k-Clique,tw=(k-1)
Thm (Grohe ’07): This theorem is optimal–unbounded treewidth implies CSP intractability.
In fact, bounded treewidth essentially determines tractabilityon formulas of bounded arity.
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Hypergraph-based approach
Hypergraph-based approach: given a CSP instance, look athypergraph where vertices are variables, have a hyperedge for eachconstraint
Example:
∃a∃b∃c∃d∃e∃f ∃g (R(a, b, c) ∧ S(b, c , d) ∧ T (d , e, f )∧U(a, g) ∧W (f , g))
↓
a
b
c
d
e
f
g
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Hypergraph-based approach
Basic question: classify sets of hypergraphs as tractable /intractable
(Gottlob et al., ’90s): Introduction of hypergraph complexitymeasures – hypertree width, generalized hypertree width
Showed that CSP is tractable when hypertree width is bounded
Thm (Chen & Dalmau ’05): CSPs are tractable when thegeneralized hypertree width is bounded.(Resolved primary open question of Gottlob et al. work)
Still open: which hypergraphs can be solvable in poly time?
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Exhibit #4: Quantified Constraint Satisfaction,from the other side
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Quantified Constraint Satisfaction, the other side
Recall...
The Quantified Constraint Satisfaction Problem (QCSP)
Given a conjunctive positive sentence φ = Q1v1 . . .Qnvn ∧ αj anda structure B, decide if B |= φ.
Before, we restricted the structure. Now we restrict the formulas.
Question: on which sets of formulas is QCSP tractable?What structural restrictions can we place on formulas, toensure tractability?
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Graph-based approach
Graph-based approach: given a QCSP instance, look atprefixed graph containing quantifier prefix & graph
Example:∀a∃b∃c∀d∃e∀f ∃g (R(a, b, c) ∧ S(b, c , d) ∧ T (d , e, f )
∧U(a, g) ∧W (f , g))
↓
(∀a∃b∃c∀d∃e∀f ∃g ,
a
b
c
d
e
f
g)
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Prefixed Graphs
Prefixed graph perspective:now have graph PLUS quantifier prefix
Examples:
X1
X2
X3
Y
∃X1∃X2∃X3 . . . ∃Xn∀YTractable!
Y1
Y2
Y3
X
∀Y 1∀Y 2∀Y 3 . . . ∃Yn∃XIntractable!
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Prefixed Graphs
(Chen & Dalmau ’05): Notion of treewidth for prefixedgraphs, proof that bounded treewidth ⇒ tractability
Pulina and Tacchella (’10) study this treewidth measure,showing that an approximation
“is a marker of empirical hardness”, and “it is theonly parameter that succeeds consistently in beingso among several other syntactic parameters whichare plausible candidates”
Close connection between
treewidth notions ↔ variable elimination algorithms(used in practice)
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Prefixed Graphs
(Chen & Dalmau, recent):
New & refined notion of width for prefixed graphs(assigns a natural number w ∈ N to each prefixed graph)
Generalization of treewidth
Complete classification of all prefixed graphs:A set of prefixed graphs is tractable iff it has bounded width
Open: Classification for formulas of bounded arity.
Perhaps the following will help:
Theorem (Chen, Madelaine & Martin ’08):Entailment and equivalence are decidable (computable) on QCSP(∀, ∃,∧) sentences.
(New decidability result for entailment!Optimal in that entailment undecidable in positive (∀, ∃,∧,∨).)
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Closing
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Themes
Study the computational complexity of fundamentalcomputational problems
Dream: to have an encyclopedic “master” classificationtheoremLook at broad families of problems,attempt to obtain comprehensive classification theoremsOften obtain general conditions for tractability/intractability
Use ideas from / interface with many areas: algorithms,constraint satisfaction, universal algebra, graph theory,combinatorics, logic ...
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