Lower Bounds for Property Testing

1
Lower Bounds for Property Testing

• Luca Trevisan
• U.C. Berkeley
• Joint work with Andrej Bogdanov and Kenji Obata

2
Sub-linear Time Algorithms

• Want to design algorithms that run in less than
  linear time (and so cannot read entire input).
• Must be probabilistic and approximate
• For optimization problems
• Compute numerical apx of optimum cost (and
  implicit representation of apx solution?)
• For decision problems
• What is approximation for decision problems?

3
(Graph) Property Testing

• Testing a property P with accuracy e in adjacency
  matrix representation
• Given graph G that has property P, accept with
  probability gt3/4
• Given graph G that is e-far from property P
  accept with probability lt1/4
• e-far must change efraction of adjacency
  matrix to get property P
  (add/remove gt en2 edges)

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Example GGR,AK

• Testing bipartiteness of a given graph G
• Pick (1/e)polylog(1/e) vertices, and check if
  they induce a bipartite graph if so accept
  otherwise reject
• If G is bipartite then alg accepts with prob 1
• If G is e-far from bipartite, then whp algorithm
  discovers an odd cycle (non-trivial to prove)
• Running time O ((1/e2)polylog(1/e))
• We will discuss matching lower bound if time
  allows

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Paleontologists approach
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Bounded Degree Graphs

• Testing a property P with accuracy e in adjacency
  lists representation
• Given graph G that has property P, accept with
  probability gt3/4
• Given graph G that is e-far from property P
  accept with probability lt1/4
• e-far must change efraction of adjacency
  lists entries to get property P
  (add/remove gt edn edges)

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Bipartiteness GR

• Testing bipartiteness
• Repeat polylog n times
• Start at random point, and pick sqrt(n) random
  walks of length polylog n, if two of them combine
  to form an odd cycle reject, otherwise accept
• Analysis
• in a graph where you need to remove constant
  fraction of edges to make it bipartite, algorithm
  finds odd cycle

8
Matching Lower Bound GR

• Define two distributions of graphs
• Gfar a random hamiltonian circuit, plus a random
  matching(whp 1/100-far from bipartite)
• Gbip a random hamiltonian circuit, plus a random
  matching conditioned on making the graph
  bipartite
• Gfar and Gbip are indistinguishable to algorithms
  of query complexity o(sqrt(n)).

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Sub-linear Time Approximation

• Minimum spanning tree
• given a connected weighted graph of degree d with
  weights in range 1,,w, can approximate MST
  weight within (1e) in time about
  O(dw/e2)Chazelle, Rubinfeld, T
• Max SAT
• Given a CNF where every variable occurs at most d
  times, can approximate Max SAT optimum within
  .618, presumably also 2/3, in O(d) timework in
  progress, hopefully will get 3/4-d

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Sublinear Time Approximation

• Problems restricted to dense instances
• Max CUT and other graph problems can be
  approximated within (1e) in graphs with at least
  an2 edges in time 2poly(1/ea) GGR
• Max 3SAT can be approximated within (1e) in
  instances with at least an3 clauses in time
  2poly(1/ea) and similar results for other
  satisfiability problemsAFKK

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General Goals

• When looking for polynomial-time algorithms
• Several algorithmic techniques of general
  applicability
• A general technique to prove impossibility
  (NP-completeness)
• For sublinear-time algorithms
• General algorithmic techniques?
• Impossibility results?

12
Testing 3-Colorability

• Easy in adjacency matrix representation
• NP-hard in adjacency list representation
• Only for small enough e
• Can find 3-coloring good for 80 of the edges in
  a 3-colorable graph using SDP
• NP-hard to find 3-coloring good for 98 (?)
  fraction of edges
• Non-tight, and conditional lower bound for query
  complexity

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Other problems

• The query complexity of following problems is
  equivalent to query complexity of testing 3col
• Testing satisfiability of 3SAT instance
• Every variable occurs in O(1) clauses, adjacency
  list representation
• Approximating max cut, vertex cover, independent
  set, . . ., in bounded-degree graphs
• Approximating Max SAT, Max 2SAT, . . .
• Lower bound of sqrt(n) for all problems
• Nothing better except with complexity assumptions

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Our Results

• For one-sided error algorithms
• W(n) query complexity to distinguish 3-colorable
  graphs from graphs that are (1/3 d)-far
• Lower bound applies to testing problems that are
  solvable in polynomial time
• For two-sided error algorithms
• For some e, W(n) query complexity to distinguish
  3-colorable graphs from graphs that are e-far.

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Additional Results

• Unconditionally, algorithms running in time o(n)
  cannot
• Approximate Max 3SAT better than 7/8
• Approximate Max Cut in bounded-degree graphs
  better than 16/17
• . . .
• Hastad97 proved above problems are NP-hard

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The 3-Coloring Lower Bound

• Consider first one-sided error algorithms
• Its enough to find a graph G that is (1/3
  d)-far from 3-colorable, but every subgraph of
  size lt an is 3-colorable
• (for every d there is an a such that . . .)
• Then an algorithm of query complexity lt an either
  accepts G (which is wrong) or rejects some
  3-colorable graph (which means the algorithm has
  not one-sided error)

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The Graph

• Pick a graph of degree O(1/d2) at random (pick so
  many random matchings)
• Then it is (1/3 d)-far whp
• But, for some a, whp, every subgraph induced by k
  lt an vertices contains lt1.5k edges
• In a minimal non-3-colorable graph, every vertex
  has degree at least 3
• Every subgraph induced by lt an vertices is
  3-colorable
• Erdos

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Explicit Construction

• Can the previous construction be derandomized?
• For constants d, e, a, and for every suff large
  n, we can explicitly construct a graph on n
  vertices, max degree d, e-far from 3-colorable,
  and such that every subset of an vertices induces
  a 3-colorable subgraph.

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Explicit Construction

• We construct a 3SAT formula such that for
  constants k, e, a
• Every variable occurs k times
• No assignment satisfies more than 1-e fraction
  of clauses
• Every a fraction of clauses is satisfiable
• Then we use (slightly new) reduction from 3SAT to
  3Coloring

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The Formula

• Fix a degree-d expander graph G(V,E) such that
  for every cut (S,V-S) at least minS,V-S
  edges cross the cut(enough d14)
• Have two variables xuv and xvu for each egde
  (u,v)
• For every vertex v have the (3SAT equivalent of)
  the constraint
• Su xuv 1 Sw xvw

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Structure of the Analysis

• Impossible to satisfy more than a fraction
  1/(d1) of the constraints
• Can always satisfy half of the constraint
• define an auxiliary network
• show that the auxiliary network has no small cut
  because of expansion
• then there is a large flow
• use large flow to find assignment for subset of
  constraint

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Flow Argument

• Want to satisfy constraints corresponding to
  vertices in C, with C lt V/2

Construct flow network with new source s, sink t
obtained by collapsing V-C, and vertices in C
V-C
s
t
C
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Flow Argument
A edges
A
t

• Every cut has size at least C
• There is a 0/1 flow of cost at least C
• Interpreted as an assignment, satisfies all
  constraints in C

s
C-A edges
C-A
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Two-Sided Error Algorithms

• Need to define two distributions of graphs Gcol
  and Gfar such that
• Graphs in Gcol are (almost) always 3-colorable
• Graphs in Gfar are (almost) always far from
  3-colorable
• To an algorithm of bounded query complexity, Gcol
  and Gfar look (almost) the same

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Main Step

• Define two distributions Dsat and Dfar of
  instances of E3LIN-2(systems over GF(2) with 3
  variables per equation)
• Systems in Dsat are always satisfiable
• Systems in Dfar are (almost) always (1/2-d)-far
  from satisfiable
• To an algorithm of bounded query complexity, Dsat
  and Dfar look the same
• We get Gcol and Gfar using reduction
  fromapproximate E3LIN-2 to approximate
  3-coloring

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E3LIN-2

• X1 X3 X10 0 mod 2
• X2 X3 X4 1 mod 2
• X1 X2 X9 0 mod 2
• . . .

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Main Building Block

• We show that for every c there is a such that
  there exists a left-hand side with
• n variables, cn equations, 3 variables per
  equations, every variable occurs in 3c equations
• every an equations are linearly independent
• Pick the left-hand side at random
• repeat 3c times pick at random a set of n/3
  disjoint triples of variables
• Explicit construction?

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Distributions

• The left-hand side is always as before
• In Dsat, we pick a random assignment to the
  variables, and set right-hand side consistently
• always satisfiable
• In Dfar, we pick the right-hand side uniformly at
  random
• With high probability, (1/2 O(1/sqrt c))-far

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Indistinguishability

• Two distributions differ only in right-hand side
• In Dfar uniformly distributed
• In Dsat, an-wise independent
• Linear independence implies statistical
  independence
• Look the same to algorithm that sees less than an
  equations

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Conclusion of the Argument

• No algorithm of query complexity o(n) can
  distinguish satisfiable instances of E3LIN-2 from
  instances that are (1/2-d)-far from satisfiable
• For some e, no algorithm of query complexity o(n)
  can distinguish 3-colorable graphs from graphs
  that efar from 3-col.
• No algorithm of query complexity o(n) can
  approximate Max 3SAT better than 7/8 . . .

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Open Questions

• Show that distinguishing 3-colorable graphs from
  (1/3-d)-far graphs requires query complexity W(n)
• we can only prove it for one-sided error
• Show that approximating Max SAT better than ¾ and
  Max CUT bettter than ½ requires query complexity
  W(n)
• we only know W(sqrt(n)) implicit in GR
• would explain why we need SDP

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Back to Dense Graphs

• Recall Alon-Krivelevich bipartiteness test for
  the adjacency matrix representation
• pick (1/e)polylog(1/e) vertices and look at
  induced subgraph
• if see odd cycle reject, otherwise accept
• Running time (1/e2)polylog(1/e)
• We prove
• W(1/e2) for non-adaptive algorithms
• W(1/e1.5) for adaptive algorithms

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Two Distributions

• Gfar every edge exists with probability e
• whp it is e/3-far from bipartite
• Gbip pick a random partition, then every edge
  that crosses the partition exists with
  probability 2e
• Thm1 look the same to non-adaptive algorithms
  making o(1/e2) queries
• Thm2 look the same to adaptive algorithms making
  o(1/e1.5) queries

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Proof of a Weaker Statement

• Thm1 (weaker) a non-adaptive algorithm making
  qo(1/e2) queries in Gfar is unlikely to see an
  odd cycle
• Proof
• a non-adaptive algorithm asks about some subgraph
  with q edges.
• There are at most about qt/2 cycles of length t,
  and each one exists with probability etqt/2,
  exponentially small in t.
• Summing over all t, its still unlikely that
  there is a cycle

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Proof of a Weaker Statement

• Thm2 (weaker) an adaptive algorithm making
  qo(1/e1.5) queries in Gfar is unlikely to see an
  odd cycle
• Proof
• the algorithm sees an edge only once in 1/e
  queries
• the algorithm sees a cycle only after querying a
  pair that it already sees as connects
• It takes 1/e.5 edges to have 1/e pairs of
  connected vertices
• It takes 1/e1.5 queries to have so many edges

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Some more open questions

• In adjacency matrix representation, most
  interesting problems solvable in constant (in e)
  time
• For some problems (eg testing triangle-freeness)
  analysis uses Szemeredys regularity lemma, and
  constant is hyper-exponential in e
• Lower bound (1/e)log 1/ e and only and for
  one-sided error
• Alternative analysis / stronger lower bounds?