1. The Subgraph Testing Model
Oded Goldreich
Weizmann Institute of Science
Joint work with Dana Ron.

2. Property Testing: informal definition
A relaxation of a decision problem:
For a fixed property P and any given object O,
determine whether O has property P
or is far from having property P (i.e., O is far from any other object having P).
Objects viewed as functions.
Inspecting = querying the function/oracle.
Focus: sub-linear time algorithms = performing the task by inspecting the object at few locations. ?
Objects viewed as functions, inspecting == querying the function/orcale

3. Property Testing: the standard (one-sided error) def’n
A property P = n Pn , where Pn is a set of functions with domain Dn.
The tester T gets explicit input n and ,
and oracle access to a function f with domain Dn.
If f  Pn then Prob[Tf(n,) accepts] = 1 (or > 2/3).
If f is -far from Pn then Prob[Tf(n,) rejects] > 2/3. (Distance is defined as fraction of disagreements.)
N.B.: special (but not exclusive) focus on testability within complexity independent of the domain size.
Focus: query complexity, q(n,) « |Dn|
Special focus: q(n,)=q(), independent of n.
Terminology:  is called the proximity parameter.

4. The new model: Testing properties of subgraphs
For a fixed base graph G=(V,E), let PG be a set of Boolean functions with domain E representing a set of subgraphs of G.
The tester T gets explicit input G=(V,E) and ,
and oracle access to a function f:E0,1.
If f  PG then Prob[Tf(G,) accepts] = 1 (or > 2/3).
If f is -far from PG then Prob[Tf(G,) rejects] > 2/3. (Distance is defined as fraction of disagreements.)
Terminology: Base graph. Tested objects are subgraphs of the base graph, which is fixed or given explicitly.
The orientation model [FLMNY] has the same syntax, but a different semantics.
Focus: query complexity, q(G,) « |E|
Special focus: q(G,)=q(), independent of G.
Terminology:  is called the proximity parameter.

5. Initial observations and actual focus
The tester T gets explicit (based graph) input G=(V,E) and , and oracle access to a function f:E0,1.
If f  PG then Prob[Tf(G,) accepts] = 1 (or > 2/3).
If f is -far from PG then Prob[Tf(G,) rejects] > 2/3. (Distance is defined as fraction of disagreements.)
The dense graph model is a special case (on input n, let G = n-vertex clique).
Testing Boolean functions is a special case (on input n, let G = n-vertex “augmented” path).
BDG = Bounded-Degree Graph.
In the DENSE graph mdel, graphs are represented by the adjacency predicate; that is G=(V,E) is represented by g:[n][n]{0,1} s.t. g(u,v)=1 iff {u,v}E.
Focus: The base graph G has bounded-degree.
Compare the complexity of testing properties of subgraphs to complexity in the BDG model.

6. Background: the bounded-degree graph (BDG) model
For a fixed d, we consider properties of graphs of max. degree d, represented by their incidence function. An n-vertex graph is represented by g:[n][d][n]0 such that g(v,i) is the ith neighbor of v (and g(v,i)=0 if v has less than I neighbors).
The tester T gets explicit input n and , and oracle access to a function g representing an n-vertex graph.
If f  Pn then Prob[Tf(n,) accepts] = 1 (or > 2/3).
If f is -far from Pn then Prob[Tf(n,) rejects] > 2/3. (Distance is defined as fraction of disagreements.)
Recalling the BDG model: Graphs (of degree bound d) are represented by their incidence function.

7. Results (sample): Downward Monotone Properties
Downwards monotone = Preserved under omission of edges.
Thm1: If a down.mono.  is testable within query complexity Qd(,n) in the BDG model, then, for every n-vertex graph G of degree d, testing whether a subgraph of G is in  can be done in query complexity Qd(/d,n).
Note: If G, then all subgraphs are in  (i.e., testing is trivial).
Thm2: For c2,3, testing c-colorability of subgraphs of some base graphs is as hard as testing c-colorability in the BDG model; that is, (n) for c=3, and (n1/2) for c=2 and =1/polylog(n).
BDG = Bounded-Degree Graph. Thm1 is proved by emulation (and the analysis uses the down.mono.).
Thm2 is proved by reductions: For c=2 we reduce by allowing to embed any bounded-degree graph in the base graph;
For c=3 we reduce 3SAT to 3COL (the formula is mapped to a base graph and the assignment to a subgraph), and use the hardness of testing assignments for 3CNF [BHR]. 
Take home msg (from Thms 1 & 2): Testing subgraphs in never harder than testing in BDG model, but it may be just as hard.

8. Results (sample): Non-Downward Monotone Properties
Thm3 (testing subgraphs may be harder than in the BDG): There exist (upwards mono.)  that is testable within complexity poly(1/) in the BDG model; but, for some bounded-degree n-vertex graph G, testing whether a subgraph of G is in  requires (loglog(n)) queries.
Thm4: For every bounded-degree graph, connectivity of subgraphs can be tested using poly(1/) queries.
BDG = Bounded-Degree Graph. In Thm4 the bound is quadratic. Thm3 is proved by reduction from the orientation model [FLMNY].
In Open1 we ask for *any* since for sure the answer is yes (via triviality) for some base graphs.
The partial answer uses the set of d-regular graphs that are 3-colorable. If the d-regular base graph is nort 3-colorable, then trivial. O.w., test degrees.
Note: The bound in Thm4 matches the bound in the BDG model. Open: What about 2-connectivity?

9. Results (more): Non-Downward Monotone Propoerties
Thm5: Let  be a locally characterizable property (i.e., it can be expressed as conjunction of constraints on O(1)-neighborhoods) and suppose that the base graph G is outerplanar. Then, we can test whether the subgraph of G is in  using O(-1log(n)) queries.
Note: Generalizes to base graphs with O(1)-size separators.
Open Problems:
Can this be improved to poly(1/)? How about testing regularity or even just 1-regularity (which means perfect-matching)?
What about testing regularity (or just 1-regularity) when the base graph G is a (two-dim) grid?
What about testing if the subgraph is Eulerian? Can do it in poly(1/) time when G is a grid, but what about any base graph?
BDG = Bounded-Degree Graph. Open 1 refer to outerplanar based graph. In Open 2 we consider a simple non-outerplanar base graph.

10. A kind of partial summary
Sometimes (e.g., for all Down. Mono. properties) testing in the subgraph model is not harder than in the BDG model, and sometimes they are not easier.
Sometimes (e.g., Thm3), testing in the subgraph model is harder than in the BDG model, and sometimes they are easier for some base graphs
(e.g., trivial cases).
BDG = Bounded-Degree Graph. In Thm4 the bound is quadratic. Thm3 is proved by reduction from the orientation model [FLMNY].
In Open1 we ask for *any* since for sure the answer is yes (via triviality) for some base graphs.
The partial answer uses the set of d-regular graphs that are 3-colorable. If the d-regular base graph is nort 3-colorable, then trivial. O.w., test degrees.
Open: Can testing subgraphs be easier for any base graph?
Yes, if the property is allowed to depend on the degree bound in the BDG model (i.e.,  contains only d-regular graphs).

11. We have introduced a new model, presented a few results, and posed many open problem.
END
Slides available at Paper available at
Summarize: We have introduced a new model, presented a few results, and suggest lots of open problems.

12. Property Testing (super-fast approximate decision): an illustration
Gothic cathedral ?
One Motivation: Real objects are far apart.
Other motivations: Approx. per se, or a preliminary step.
Compare to learning/deciding which cathedral this is…
Deciding by inspecting few locations in the object.