Tractable Symmetry Breaking for CSPs with Interchangeable Values

1. Tractable Symmetry Breaking for CSPs with Interchangeable Values P. Van Hentenryck, P. Flener, J. Pearson and M. Agren IJCAI 2003 Presented by: ShantKarakashian

2. Outline • Purpose of the Paper • Definitions • CSP • Partial Assignment • Completion • Proposition: Nogood • Value-Interchangeable CSPs • Nogood Closure • Existential Nogood • Eliminating Existential Nogoods • Abstract Nogood • Basic Intuition with Example • Compact Variable Decomposition Tree • Other CSP Classes • Experimental Results • Conclusion

3. Purpose of the Paper • Important symmetry breaking schems: SBDS & SBDD • These schemes may require exponential resources: • Exponential space for storing the generated nogoods • Exponential time to check if assignment is symmetric to another assignment • Identify classes of CSPs: • That are practically relevant • Symmetry breaking is tractable • The symmetries of the problem are known • Exploit the knowledge to eliminate symmetries efficiently • In contrast, value interchangeability forms of Freuder discover symmetries inside CSPs

4. Definition: CSP • A CSP is a triplet (V, D, C) • V set of variables • D possible values for the variables • C: (V → D) → Boolean is a function that specifies if the assignment is a solution • A solution to the CSP is a function: σ: V → D; C(σ)=true

5. Definition: Partial Assignment • Given a CSP: P = (V, D, C) • A partial assignment for P is a function α: W ⊆ V → D • The domain of α: dom(α) is W • The image of α: image(α) is {α(v) | v ∈ dom(α)} • For each d ∈ image(α), α-1(d) = {v | v ∈ dom(α) ∧ α(v) = d} • Example: α : v1 = 1 ∧ v2 = 2 ∧ v3 = 3 dom(α) = {v1, v2, v3} image(α) = {1, 2, 3}

6. Definition: Completion • Given a CSP: P = (V, D, C) • A completion of a partial assignment α for P is an extension α’: V → D of α on P • The set of all completions of αfor P is: Comp(α,P) • The set of all solutions to P is Sol(P) • A nogoodα for P is a partial assignment such that Comp(α,P) ∩ Sol(P) = ∅

7. Proposition: Nogood • Given a CSP: P = (V, D, C) • D = {d1,…,dm} • Let α be a partial assignment for P • dom(α) = {vi1,…,vik} • For every i(1 ≤ i ≤ m), αi= α ∧ (vik+1= di) is a nogood for P • Then α is a nogood for P

8. Value-Interchangeable CSPs • Value-Interchangeable CSPs are problems where the exact values taken by the variables are not important • Freuder calls this notion of symmetry full interchangeability for all variables and all values • The symmetry breaking for these problems is tractable • Given a CSP: P = (V, D, C) • P is value interchangeable if: • For each solution σ ∈ Sol(P) • For each bijectionb : D → D • (b o σ) ∈ Sol(P)

9. Nogood Closure • Given a ICSP: P = (V, D, C) • Let α be a nogood for P • Let b : D → D be a bijection • (b o α)is a nogood • The closure of α for P is: Closure(α,P) = {b o α | b : D → D is a bijection}

10. Existential Nogood • Given a ICSP: P = (V, D, C) • Let α be a nogood for P • image(α) = {d1,…,dk} • The existential nogood of α for P is Enogood(α,P): • Set of functions γ : dom(α) → D such that • ∃ e1,…,ek ∈ D: • ∀ i ∈ 1..k :∀ vj∈ α-1(di) : γ(vj) = ei ∧ aldiff(ei,…,ek) • Enogood(α,P) = Closure(α,P)

11. Eliminating Existential Nogoods • Testing Membership to an existential nogood involves existential quantificatons • It is easy to eliminate the existential elements by selecting a representative for each set α-1(di) • Example: • ∃e1, e2, e3 ∈ D : θ(v1) = e1 ∧ θ(v2) = e2 ∧ θ(v3) = e3 ∧ θ(v4) = e1 ∧ θ(v5) = e2 ∧ alldiff (e1, e2, e3) • The elements e1, e2 and e3 can be eliminated to produce: • θ(v1) = θ(v4) ∧ θ(v2) = θ(v5) ∧ alldiff (θ(v1), θ(v2), θ(v3))

12. Abstract Nogood • Given a ICSP: P = (V, D, C) • Let α be a nogood for P • image(α) = {d1,…,dk} • Let vri ∈ α-1(di)(1≤ i≤ k) • The abstract nogood of α wrtP, Anogood(α,P) is the set of functions γ : dom(α) → D satisfying: • ∀ i ∈ 1..k :∀ vj∈ α-1(di) : γ(vj) = γ(vvri) ∧ aldiff(γ(vvri),…, γ(vvrk)) • Enogood(α,P) = Anogood(α,P)

13. Basic Intuition with Example • Given a ICSP: P = (V, D, C), Consider the partial assignment: v1 = 1 ∧ v2 = 2 ∧ v3 = 3 ∧ v4 = 1 ∧ v5 = 2 • Assume that depth-first search tries to assign variable v6 from the set of possible values 1..10 • The failure of v6 = 1 produces the abstract nogood θ which holds if: allequal(θ(v1), θ(v4), θ(v6)) ∧ allequal(θ(v2), θ(v5)) ∧ alldiff(θ(v1), θ(v2), θ(v3)) • The assignments v6 = 2 and v6 = 3 produce similar abstract nogoods

14. Basic Intuition with Example (cont.) • When v6 is assigned a value i ∈ 4..10 say 6 • The abstract nogood θ is now defined as: allequal(θ(v1), θ(v4)) ∧ allequal(θ(v2), θ(v5)) ∧ alldiff(θ(v1), θ(v2), θ(v3), θ(v6)) • Which can be partially evaluated to: alldiff(1,2,3,θ(v6)) • The disjuction of all the nogoods can be partially evaluated to: θ(v6) = 1 ∨ θ(v6) = 2 ∨ θ(v6) = 3 ∨ alldiff(1,2,3,θ(v6)) • It follows that v6 must only be assigned to the previously assigned values 1,2,3 or to exactly one new value in 4..10 • Only some of the children need to be explored in the search tree: a value in dom(α) & exactly one other child

15. Compact Variable Decomposition Tree • Given a ICSP: P = (V, D, C) D = {d1,…,dm} • Compact variable decomposition treeT is a search tree where nodes represent partial assignments for P and nodes are decomposed as follows: • For a given node n: • represents the partial assignemntα • dom(α) = {di,…,dik} • is for some variable vik+1 ∈ V \ dom(α) • Children of n represent the partial assignment: • αi= α ∧ (vik+1 = d0) for all d0 ∈ image(α) • And exactly one child represents: • αi= α ∧ (vik+1 = dn) for some dn∈ (D \ image(α)) (if not empty) • The partial assignment of a node in T never matches any nogood generated during the exploration of T (except the one possibly it generates)

16. Other CSP Classes • Piecewise-Interchangeable CSPs • Values of the variables are chosen from disjoints sets • Values are interchangeable in each set • Wreath-Value Interchangeable CSPs • Each variable v is assigned a pair of values (d1,d2) • (d1,d2) ∈ D1 × D2 • Values in D1 are fully interchangeable • For a fixed value in D1 the values in D2 are fully interchangeable

17. Experimental Results • Results on Graph Coloring • Table shows time in seconds • Default labeling • Interchangeable CSP • Partitioned graph coloring • Colors divided into 4 groups and are fully interchangeable in each groups

18. Conclusion • Presented three classes of CSPs • The three classes feature various forms of value interchangeability • Symmetry breaking is performed in tractable time and space • Experimental results showed the the benefits of the symmetry breaking on these instances

19. Thank You! • Questions?