Combinatorial search techniques

Jo Devriendt †
† KTH Royal Institute of Technology, Sweden

jhmde@kth.se

TODO: look for Laurent Perron

On the intersection between AI and OR

Combinatorial problems

Vehicle routing

Nurse rostering

Planning

Graph coloring

Pigeon allocation

Max flow

Combinatorial search

Pick a satisfying solution from a set of candidate solutions

Combinatorial search in
Artifical Intelligence

• More focused on decision problems
• Lots of approaches
  • SAT, CP, SMT, ASP, ...
• Are converging towards "learning"
  • Fundamental algorithm:
    Conflict-Driven Clause Learning (CDCL)
  • Mathematical foundation through Proof Complexity
• Young technique: 1996 [1]
• Biggest successes: chip verification, planning, scheduling

[1] "GRASP - A New Search Algorithm for Satisfiability" Marques-Silva & Sakallah 1996

Conflict-driven clause-learning search

Propositional description

Proof Theory

(resolution)

Derived formulas

Inconsistency

Initial formulas

Proof Theory

(cutting planes)

Derived formulas

Inconsistency

Initial formulas

• Proof length implies lower bound on running time for inconsistent instances
• Resolution has exponential length for pigeonhole principle
• Cutting planes has polynomial length for pigeonhole principle

Combinatorial search in Operations Research

• More focused on optimization problems
• Two large classes of approaches
  • Local search
    • Incomplete heuristics
  • Mathematical programming
    • Solve equations over numerical variables
    • Fundamental algorithm: Simplex
• Old technique: 1947 [2]
• Biggest successes: traveling salesman, resource allocation

[2] "Maximization of a linear function of variables subject to linear inequalities" Dantzig 1947

Branch-and-bound and Simplex

Description in linear algebra

Depth-first search

Result

Simplex: solve a system of linear inequalities over real variables

Motivation for a hybrid approach

• Some hard problems for CDCL have no rational solutions
  • E.g., pigeonhole principle
  • CDCL cannot "count" or "add"
• Some easy problems for CDCL are hard for mathematical programming
  • No "tight" linearization
  • Simplex cannot "split"
• State-of-the-art hybrid: SCIP [3]
  • Clause learning and constraint propagation during branch-and-bound MIP solving
• Research goal: plug Simplex into CDCL
  • "Inverse SCIP" approach

[3] "The SCIP Optimization Suite 6.0" Gleixner et al. 2018

A rational solver for Pseudo-Boolean (PB) problems

• Linear constraints over Boolean variables:
  
  
   

• Characteristics of clauses and mathematical programs

  • CDCL solvers exist, with non-clausal learning

• End-goal: call rational solver (Simplex) during CDCL search

  • derive rational infeasability
  • learn new PB constraint

A rational solver for Pseudo-Boolean (PB) problems

• Solve PB problems for rational solutions

  • 10s timeout

  • 7070 instances

[4] "A fast linear-arithmetic solver for DPLL(T)" Dutertre & De Moura 2006

A rational solver for Pseudo-Boolean (PB) problems

Challenges:

• Timeout for many instances

• Incremental search

• Exact vs floating point calculations

• "Primal" vs "Dual"

Thanks for your attention!

Questions?