Lower Bounds for Property Testing

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Lower Bounds for Property Testing

• Luca Trevisan
• U C Berkeley

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Sub-linear Time Algorithms

• Want to design algorithms that run in less than
  linear time
• 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?

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Graph Property Testing GGR

• Testing a property P with accuracy e
• 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 representation
  of G to get property P
• Intuition input (not output) is approximate

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Different Representations

• G is represented as adjacency matrix
• e-far must add/remove en2 edges
• G has max degree d and is represented using
  adjacency lists
• e-far must add/remove edn edges
• (Some extra subtleties in bounded-degree case)

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Purpose of This Talk

• Discuss algorithms and lower bounds for
• Sub-linear time property testing for some basic
  graph properties
• Sub-linear time approximation algorithms for some
  basic optimization problems
• (well mostly discuss lower bounds)

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Motivations

• Large data sets
• web, wall-mart, amazon, phone calls, . . .
• linear time can still be infeasible
• Fine print most research on property testing
  focuses on problems having no connection to
  applications with large data sets
• Goal for theory research
• Develop general algorithmic techniques(like
  dynamic programming, local search, for P)
• Develop general techniques for impossibility
  results(like NP-completeness)

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Property Testing and Approximation inAdjacency
Matrix Representation
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Bipartiteness Algorithm 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))

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Lower Bounds BT

• W(1/e1.5) for adaptive algorithms
• W(1/e2) for non-adaptive algorithms
• The bounds apply to the query complexity of the
  algorithm(and to running time for a stronger
  reason)

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Proof for one-sided error case

• Pick a random graph with edge-probability 3e
• whp it is e-far from bipartite
• Consider view of (possibly adaptive) algorithm
  that makes q queries and finds odd cycle w.h.p.
• sees Q(eq) edges and O(e2q2) pairs of connected
  vertices
• a cycle can be discovered only by querying two
  vertices in same connected component
• it takes W(1/e) such attempts
• q W (1/e1.5 )

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One-sided error non-adaptive

• Pick a random graph with edge-probability 3e
• Consider view of non-adaptive algorithm that
  makes q queries
• Same as
• Start with q-edges graph
• Independently delete each edge with prob 1-e
• If qo(1/e2) then view is a forest w.p. 1-o(1)
• Proof There are at most O(qt/2) cycles of length
  t

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Two-Sided Error

• Two distributions
• Gfar random graph with edge probability 3e
• Gbip first random partition, then each edge
  crossing partition exists with prob 6e
• Distributions indistinguishable by
• Non-adaptive algorithms of query complexity
  o(1/e2)
• Adaptive algorithms of query complexity o(1/e1.5)
• Both tight for these distributions

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Generality/Lessons

• Possible lesson try random graph as a possible
  distribution of hard instances far from having
  the properties
• Not good for Triangle freeness property whose
  complexity is possibly most interesting open
  question in the adjacency matrix model.

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Triangle-free Graphs

• Want to distinguish triangle-free graphs from
  graphs where need to remove en2 edges to break
  all triangles
• Solvable in time super-exponential in 1/e
• Polynomial in 1/e is impossible Alon
• 2poly(1/e) possible?
• Simplest special case of more general (and
  important) question

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Sublinear Time Approximation

• 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 problems AFKK
• Lower bounds?

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Property Testing and Approximation in Adjacency
List Representation
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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

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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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Approximation Algorithms

• 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)
  timeHopefully will get 3/4-d

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Testing 3-Colorability

• 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
• Gives non-tight, and conditional lower bound for
  query complexity

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Other Problems

• 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
• Reduction from bipartiteness

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Tight Lower Bound BOT

• 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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Using Reductions. . .

• 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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Derandomization

• 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,
• such that every subset of an vertices induces a
  3-colorable subgraph.

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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?
• Need strong unique-neighbor expanders

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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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Generality/Lessons

• Reductions are useful and extend results to
  several problems
• In adjacency matrix (dense graph) setting,
  several and general algorithms. Few and ad-hoc
  lower bounds
• In adjacency list (sparse graph) setting, vice
  versa.

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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