The Stable Circuit Problem

1
The Stable Circuit Problem
A Short Introduction Brendan Juba
2
The Stable Circuit Problem

• An instance of the Stable Circuit Problem is a
  circuit of MIN, MAX, and AVG gates, with input
  wires hardwired from 0,1
• Feedback is allowed
• All gates have fan-in two
• Solutions are settings of the internal wires
  where the output of every gate is set to the
  appropriate function of the inputs to the gate
• Solutions can be shown to always exist
• Originally believed to be in FP or FBPP after
  many flawed approaches, believed to be hard.

3
An Introduction to The Complexity Class PLS

• A Rapid Summary of Local Search
• Brendan Juba

4
Motivation

• PLS (Polynomial-time Local Search) is a
  complexity class intended to exemplify local
  search problems.
• Consider a local search version of MAX CUT
• Given an undirected graph G(V,E) with weighted
  edges
• The value of a partition of V into V1 and V2 is
  the sum of the weights on edges with endpoints in
  different sets.
• Start with an initial partition of V
• Move vertices between the sets until the value of
  the partition cannot be increased.

5
Definition Local Search

• A Local Search problem L consists of
• A set, DL ? S, of instances x
• In MAX CUT, DMAX CUT is the set of undirected,
  edge-weighted graphs
• A polynomial p
• For each x ? DL , a set, FL(x) ? Sp(x), of
  feasible solutions s
• FMAX CUT(x) is the set of partitions of V into
  subsets V1 and V2. Clearly, these partitions may
  be represented in a way polynomially bounded by
  the size of the graph.
• (continued)

6
Definition Local Search

• For each solution s ? FL(x), a set of neighbors,
  NL(s,x) ? FL(x)
• NMAX CUT(s,x) is the set of partitions of the
  vertices of x that differ from s in the placement
  of a single vertex
• For each solution s ? FL(x), an integer measure,
  ML(s,x)
• MMAX CUT(s,x) is the sum of the weights on the
  edges in x with endpoints crossing the partition
  s
• s is locally optimal for x when it has no
  strictly better neighbors
• An algorithm that finds local optima solves L

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Definition PLS

• For a local search problem L ? PLS,
• Instances DL ? S, feasible solutions FL (x) ?
  Sp(x), and neighbors NL(s,x) ? FL(x) should all
  be polynomial-time recognizable
• Polynomial-time algorithms must exist
• Algorithm AL on input x ? DL , produces an
  initial feasible solution AL(x) ? FL (x)
• Algorithm ML for x ? DL and s ? FL (x), computes
  ML(s,x).
• Algorithm CL for inputs x ? DL and s ? FL (x),
  either
• Correctly reports that s is locally optimal
• Produces a better solution s ? NL(s,x)
• We are usually interested in finding any local
  optimum for L.

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Examples Problems in PLS

• MAX CUT was in PLS
• NAE kSAT (a maximization problem)
• DNAE kSAT Formulas of not-all-equal clauses of
  at most k literals or constants, each having
  positive integer weights
• FNAE kSAT(x) Assignments to the variables of x
• MNAE kSAT(s,x) The sum of the weights of all
  clauses in x satisfied by the assignment s
• NNAE kSAT(s,x) Assignments that differ from s in
  the setting of any single variable
• POS NAE kSAT has no negated literals (hence, only
  positive literals)

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Definition PLS-reducibility

• A problem ? PLS-reduces to a problem L when
• ?, L ? PLS
• There is a polynomial-time computable function,
  f Dp ? DL
• There is a polynomial-time computable function, g
  FL(f(x)) ? Dp ? Fp(x) (e.g. taking (solution of
  f(x), x) pairs to solutions of x)
• For all instances x ? Fp(x), if s is a locally
  optimal solution of f(x), then g(s,x) is locally
  optimal for x.
• Notice that PLS-reductions are composable, and
  solve ? in polynomial time, using L.

?
f(x)
g
L
x
f
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Example PLS reduction

• We reduce MAX CUT to POS NAE 3SAT
• f given an encoding of a graph G(V,E),
  construct a formula x such that
• For each v ? V, there is a corresponding variable
  v of x
• For each edge (u,v) ? E, there is a corresponding
  NAE clause in x, NAE(u,v). The weight of this
  clause is the same as the weight of (u,v) in G.
• g assign the variables corresponding to vertices
  in V1 to truth, and all variables corresponding
  to vertices in V2 to falsehood.
• It is clear that the measures and neighborhoods
  are preserved hence, local optimality is
  preserved.
• This is an atypically clean reduction!

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PLS-completeness

• L ? PLS is PLS-complete when
• For every ? ? PLS, ? PLS-reduces to L
• Observe that we may show a problem to be
  PLS-complete by reducing a problem known to be
  PLS-complete to it
• Example since MAX CUT is known to be
  PLS-complete, we have just shown that POS NAE
  3SAT is PLS-complete as well

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PLS-complete problems (a partial list)

• CIRCUIT FLIP
• D combinational circuits without feedback
• F(x) binary settings for the inputs
• M(s,x) the output viewed as a binary integer
• N(s,x) settings differing in the setting of a
  single input
• 2SAT FLIP
• SWAP (graph partitioning)
• MAX CUT, with the restriction V1 V2

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PLS-complete problems (continued)

• GRAPH PARTITION (e.g. SWAP)
• Kernighan-Lin neighborhood
• Greedy swaps of pairs of vertices
• Fiduccia-Mattheyses neighborhood
• Greedy swaps of individual vertices
• TRAVELLING SALESMAN
• Lin-Kernighan neighborhood
• k-OPT neighboring tours differ by k edges (for a
  fixed but large k)

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PLS-complete problems (continued)

• STABLE NET (maximization)
• D graphs with integer (postive or negative)
  weights wij on the edges
• F(G) assignments to the nodes xi of G from
  1,-1
• M(s,G) Si,j wij xi xj
• N(s,G) assignments that differ in the setting of
  a single variable

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Hardness

• Stable circuit is now believed to be hard
• It is doubtful that Stable Circuit is NP-hard, as
  it is in TFNP (and hence, in NP?co-NP).
• The problem has other similarities with many of
  the PLS-complete problems
• Exponential worst-case behavior for known
  neighborhood structures
• An unweighted version of the problem (circuits
  with no AVG gates) is P-complete
• Thus, we conjecture that the problem is
  PLS-complete, and intend to show this.

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References PLS-completeness

• D.S. Johnson, C.H. Papadimitriou, and M.
  Yannakakis. How Easy Is Local Search? Journal of
  Computer and System Sciences, 3779-100, 1988.
• M.W. Krentel. Structure in Locally Optimal
  Solutions, Proc. of IEEE FOCS, pp216-221, 1989.
• C.H. Papadimitriou, A.A. Schäffer, and
  M.Yannakakis. On the Complexity of Local
  Optimality, Proc. 22nd Annual ACM STOC, Baltimore
  MD, 1990, pp84-94.
• A.A. Schäffer, and M.Yannakakis. Simple Local
  Search Problems That Are Hard to Solve, SIAM J.
  Comput., 2056-87, 1991.

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References Simple Stochastic Games

• Anne Condon, The Complexity of Stochastic Games,
  Information and Computation, vol. 96, No. 2,
  February 1992, pp. 203-224.
• Anne Condon, On Algorithms for Simple Stochastic
  Games, DIMACS Series in Discrete Mathematics and
  Theoretical Computer Science, Volume 13, edited
  by Jin-Yi Cai, American Mathematical Society,
  1993, pp. 51-73.
• Manuel Blum, Rachel Rue, and Ke Yang, On the
  Complexity of MAX/MIN/AVRG Circuits, Technical
  Report CMU-CS-02-110, Department of Computer
  Science, Carnegie Mellon University, 2002.