Structure of Solutions to Random Constraint Satisfaction Problems
Structure of Solutions to Random Constraint Satisfaction Problems Dimitris Achlioptas UC Santa Cruz & RACTI
Constraint Satisfaction Problems • n variables with the same, small, discrete domains binding O(1) variables . . . m competing constraints each c 1 c 2 v 1 v 2. . . • Variables Constraints
Graph k-coloring • . . . Each edge is a “not-equal” constraint on two variables e 1 e 2 v 1 v 2. . . • Each vertex is a variable with domain {1, 2, …, k} Vertices Edges
k-SAT • • Each constraint (k-clause) binds k variables Forbids 1 out of the possible joint values c 1 c 2 x 1 x 2. . . k 2 Variables . . . • n Boolean variables Clauses • Example ( ):
Randomness as Complexity Each constraint forbids a constant fraction of all assignments e. g. , one k-clause forbids assignments P vs. NP: “Can crude constraints create fine sets? ”
Our Setting: Random CSPs • Locally tree-like and [approximately] regular e 1 e 2 v 1 v 2. . . • Lack of geometry or structure. Sparse instances, i. e. , Variables . . . • Each constraint binds a random subset of variables Constraints
Random Graph k-coloring • • G(n, m) random graph: the two vertices are chosen randomly • Random r-regular: each vertex is chosen r times e 1 e 2 v 1 v 2. . . • Vertices . . . Each vertex is a variable with domain {1, 2, …, k} Each edge is a “not-equal” constraint on two variables Edges
Random k-SAT • n Boolean variables • Among all possible . . . uniformly and independently. c 1 c 2 x 1 x 2. . . k-clauses, select m=rn Variables Clauses • Similarly: NAE k-SAT, Hypergraph 2 -coloring, Vertex Cover…
The chromatic number of a random graph [DA, Naor ’ 05]
Examples 5 or 6
k-coloring G(n, m=rn) Repeat if some vertex has 1 color left, color it else color a random vertex a random available color Works w. h. p when All known poly-time algorithms fail at [Bollobás, Thomasson 84] [Mc. Diarmid 84]
The satisfiability threshold conjecture
Bounds for the k-SAT threshold Since the mid-80 s: [Chao & Franco 86] [Chvatal, Reed ’ 92] [Frieze, Suen ’ 96] [Franco, Paull ’ 83]
Bounds for the k-SAT threshold Recently: ln k [Coja-Oghlan, ’ 09] [Franco, Paull ’ 83]
Bounds for the k-SAT threshold [DA, Peres ‘ 04]
A General Phenomenon • Easy to get an upper bound • Some naïve algorithm gives a (far away) lower bound • Poly-time algorithms hit a barrier • Non-algorithmic lower bounds [DA, Naor, Peres, Nature ‘ 05] 0 Algorithms Naïve Density No Solutions
A Second Moment Primer
A Second Moment Primer
Key point + Examples
Random Graph k-Coloring
The Next Question Why Algorithms Fail?
Evolution of Solution-Space Geometry 0 Algorithmic Barrier Density No Solutions
Definitions
Definition of Shattering
A Rough Energy Landscape
Theorems on Shattering
Freezing
Rigor[ous] Mortis [DA, Ricci-Tershenghi ‘ 06]
Proof Sketch
The “Big Picture” All random graphs with n vertices and m edges 1 1 1 1 1 1 1 1 1 1 1 1 Row balance Random (approximate): solution: column from then symmetry row probability ★Graphs ‘rich’ in are seen with higher Lemma. In ka-colorings balanced matrix both processes pick a uniformly random 1 from entire matrix Column balance (approximate): from concentration and are thus exactly equivalent Planted solution: row then column
One-way Functions Assume that for a random instance I of a CSP: Let K consist of (log n)2 independent planted instances:
One-way Functions Assume that for a random instance I of a CSP: Let K consist of (log n)2 independent planted instances:
One-way Functions Assume that for a random instance I of a CSP: Example on random k-CNF: Let K consist of (log n)2 independent keyed instances:
Future Work • Lower bounds for random walk type algorithms. • Prove universality of shattering for random CSPs and characterize which CSPs are like XOR. • Do we really need randomness?
Thank You
Transfer Theorems Recipe [Friedgut ’ 99]
Rigor[ous] Mortis [DA, Ricci-Tershenghi ‘ 06]
Proof Sketch
- Slides: 38