[Arith] Memoize IntervalSet variable relaxation to avoid exponential blowup

[jinhongyii]

Problem

Analyzer::Bind could hang indefinitely (>300s, ~200% CPU, no GPU work) while binding a small expression for one variable. The root cause is general and lives in src/arith/int_set.cc.

Diagnosis: 100% of the time is spent in arith::Analyzer::Bind → IntSetAnalyzer → IntervalSetEvaluator, evaluating a 5-node bound expression. A counter showed >2^20 VisitExpr calls at recursion depth 67 with no end in sight.

Root cause

IntervalSetEvaluator::VisitExpr_(VarNode) relaxes a variable's bounds by recursively evaluating both the min and max sub-expressions of its mapped interval. For diamond-shaped variable dependency chains (a → {b, c}, b → {d, e}, …) the shared sub-expressions are re-expanded along every path, so cost is O(2^depth) in the length of the dependency chain — bounded only by dom_map_.size() (~67 interdependent vars in the failing case).

Fix

Memoize the fully-relaxed interval per variable (relax_memo_) and break cyclic dependencies with an in-progress set (relax_in_progress_). A variable's relaxed interval is deterministic for a given evaluator instance (dom_map_/dom_constraints_ are fixed), so memoizing collapses the diamonds to linear cost. Short chains — the common case, which never reached the old recur_depth_ >= dom_map_.size() cutoff — are unaffected, so the change is behavior-preserving outside the pathological case.

Tests

New regression tests in tests/python/arith/test_arith_intset.py:

• test_relax_deep_variable_dependency_chain — a 64-deep diamond (O(2^64) without the fix; verified to hang on a clean build), also asserting the relaxed result is correct (x0 → [-n, 100+n]).
• test_relax_cyclic_variable_dependency — a cyclic x↔y dependency must terminate.

Verification

• tests/python/arith/test_arith_intset.py — 20 passed (the deep-chain test completes instantly).
• Full tests/python/arith/ — 933 passed (1 pre-existing flaky random-seed failure in test_arith_solve_linear_equations.py unrelated to this change, passes on rerun).

[gemini-code-assist]

Code Review

This pull request introduces memoization and cycle detection to the IntervalSetEvaluator in src/arith/int_set.cc to prevent exponential re-expansion and infinite recursion during variable relaxation. It also adds corresponding unit tests. The feedback suggests strengthening the cyclic dependency test by using IntSetChecker to verify the exact expected symbolic interval instead of just asserting that the result is not null.

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[jinhongyii]

cc: @tqchen