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  Σ*, of instances x
In MAX CUT, DMAX CUT is the set of undirected, edge-weighted graphs
A polynomial p
For each x є DL , a set, FL(x)  Σp(|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 є 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 є 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

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

8. 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)

9. Definition: PLS-reducibility
A problem Π PLS-reduces to a problem L when
Π, L є PLS
There is a polynomial-time computable function, f : Dπ  DL
There is a polynomial-time computable function, g : FL(f(x))  Dπ  Fπ(x) (e.g. taking (solution of f(x), x) pairs to solutions of x)
For all instances x є Fπ(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 Π in polynomial time, using L. Π
f(x) g x L f

10. 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 є V, there is a corresponding variable v of x
For each edge (u,v) є 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!

11. PLS-completeness L є PLS is PLS-complete when
For every Π є PLS, Π 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

12. 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|

13. 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)

14. 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): Σi,j wij xi xj
N(s,G): assignments that differ in the setting of a single variable

15. 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.

16. References: PLS-completeness
D.S. Johnson, C.H. Papadimitriou, and M. Yannakakis. How Easy Is Local Search? Journal of Computer and System Sciences, 37:79-100, 1988.
M.W. Krentel. Structure in Locally Optimal Solutions, Proc. of IEEE FOCS, pp , 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., 20:56-87, 1991.

17. References: Simple Stochastic Games
Anne Condon, The Complexity of Stochastic Games, Information and Computation, vol. 96, No. 2, February 1992, pp
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
Manuel Blum, Rachel Rue, and Ke Yang, On the Complexity of MAX/MIN/AVRG Circuits, Technical Report CMU-CS , Department of Computer Science, Carnegie Mellon University, 2002.