Must each endpoint have the same supporting assignment?

No. Different endpoints can have different supporting tuples.

Does bounds consistency remove every impossible interior value?

No. Domain consistency is the stronger all-values check under comparable support definitions.

Can search still be necessary?

Yes. Locally consistent domains can leave unresolved choices or hide incompatibility across constraints.

AI Glossary / Graphs and symbolic AI / Intermediate

Bounds consistent

Definition

Bounds consistent describes a constraint whose remaining domain endpoints each have a supporting assignment under the chosen definition of bounds consistency. In a finite-domain interpretation, every variable's lowest and highest permitted values can participate in a tuple satisfying that constraint using permitted values for the other variables. It is a condition used to reduce search in constraint solving.

What it means

A variable's domain lists the values it may take. Checking every combination can be expensive, so a solver may focus on the extremes of each domain. If a smallest or largest value cannot possibly satisfy a constraint, it can be removed. The new endpoint is then examined as needed.
The phrase describes the support of endpoints, not a promise that every interior value works or that the complete model has a solution. Technical literature distinguishes variants according to whether supporting values must belong to the actual domains, integer intervals, or real intervals. This entry uses the finite-domain support interpretation and keeps those conventions explicit.

How it works

Consider one constraint at a time. For each variable, ask whether its current minimum has at least one compatible assignment of the other variables, and repeat for its maximum. Unsupported endpoints are removed without discarding any solution of that constraint. Changes may expose new unsupported endpoints or trigger neighboring constraints.
An assignment supporting one endpoint need not support another endpoint simultaneously. Likewise, the supporting assignments for different constraints need not be mutually compatible. This is why local consistency is useful pruning rather than a general substitute for search. A practical solver's achieved consistency also depends on the propagator selected for each constraint.

Three practical examples

In scheduling, a task's earliest start time is increased when it cannot respect a predecessor's completion constraint.
In allocation, the maximum number of units assigned to one location is reduced when no permitted allocation elsewhere can satisfy a total-capacity constraint.
In configuration, an extreme component count is removed when it has no supporting combination under a numerical compatibility rule.

Think of it this way

Think of narrowing the ends of a window until both edges have a valid reason to remain. This captures endpoint pruning. The analogy stops there: a domain can contain gaps, and keeping valid edges does not prove that every point between them is valid. Several individually acceptable windows can also fail to fit together across a complete model.

Worked example

Let integer x have domain {1, 2, 3, 4, 5}, y have domain {2, 3}, and require x + y = 6. For x = 1, y would need to be 5, which is unavailable. For x = 2, y would need to be 4, also unavailable. Remove those lower values. For x = 5, y would need to be 1, so remove that upper value too.
The remaining x domain is {3, 4}. Its endpoints have support: (3, 3) and (4, 2). Both endpoints of y also have support, so the constraint is bounds consistent under the stated interpretation. This small example happens to leave only fully supported values. That does not hold in general: with x = y, x domain {1, 2, 3}, and y domain {1, 3}, both endpoints have support although x = 2 does not. Bounds consistency alone need not remove that interior value.

Illustrative case study

At fictional Juniper Scheduling, a planner maintains earliest and latest start times for maintenance tasks. The team uses bounds reasoning to discard times that conflict with task precedence. It then performs further search to assign concrete start times that satisfy all shared-resource restrictions.
When a user sees a narrowed time window, the interface calls it a remaining possibility range rather than a confirmed appointment. This fictional case demonstrates a communication benefit: useful pruning can be exposed without presenting locally supported endpoints as a complete feasible schedule.

Common misconceptions and mistakes

Bounds consistent is not the same as fully solved. It also differs from domain consistency, which checks support for every domain value under its definition. Interior holes can matter even when endpoints look acceptable.
Do not confuse consistency of each constraint with consistency of the whole conjunction. Separate constraints can each support an endpoint using incompatible assignments. Finally, bounds consistency has several formal variants; an integer-domain definition and a real-interval relaxation may prune differently. Documentation should specify which notion a solver or paper uses rather than assuming identical behavior from the shared name.

Business use

Bounds consistency can reduce work in models with large numerical domains, such as quantities and time windows. It provides a useful balance between inexpensive reasoning and pruning strength, although the balance depends on the constraint. For procurement or scheduling software, ask what consistency is maintained and whether the displayed ranges are proven feasible choices or merely values not yet excluded. That distinction affects how confidently users can commit resources.

Check your understanding

For x = y, x has {1, 2, 3} and y has {1, 3}. Are supported endpoints enough to prove x = 2 is valid?

Answer: No. The endpoints are supported, but x = 2 has no matching y value.

Related terms

Constraint satisfaction
Domain
Bounds propagation
domain consistency
arc consistency
propagator
finite-domain solver

Learn next

Draw domains with and without interior gaps, then compare endpoint support with support for every value. Repeat with two constraints sharing a variable. The exercise shows why the exact consistency definition and the distinction between local and global feasibility both matter.

Sources

Sources: SICStus Prolog, CLPFD Glossary, official bounds and domain consistency definitions: https://sicstus.sics.se/sicstus/docs/4.10.0/html/sicstus/CLPFD-Glossary.html . John Hooker, original CP Summer School tutorial on integrating constraint programming and mathematical programming: https://johnhooker.tepper.cmu.edu/summerSchool2011.pdf .

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