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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 ΠPLS-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 Πє 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, 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., 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. 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.