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 V 1 and V 2 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, D L  Σ *, of instances x In MAX CUT, D MAX CUT is the set of undirected, edge- weighted graphs –A polynomial p –For each x є D L, a set, F L (x)  Σ p(|x|), of feasible solutions s F MAX CUT (x) is the set of partitions of V into subsets V 1 and V 2. 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 F L (x), a set of neighbors, N L (s,x)  F L (x) –N MAX 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 F L (x), an integer measure, M L (s,x) –M MAX 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 D L  Σ *, feasible solutions F L (x)  Σ p(|x|), and neighbors N L (s,x)  F L (x) should all be polynomial-time recognizable –Polynomial-time algorithms must exist: Algorithm A L : on input x є D L, produces an initial feasible solution A L (x) є F L (x) Algorithm M L : for x є D L and s є F L (x), computes M L (s,x). Algorithm C L : for inputs x є D L and s є F L (x), either –Correctly reports that s is locally optimal –Produces a better solution s’ є N L (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) –D NAE kSAT : Formulas of “not-all-equal” clauses of at most k literals or constants, each having positive integer weights –F NAE kSAT (x): Assignments to the variables of x –M NAE kSAT (s,x): The sum of the weights of all clauses in x satisfied by the assignment s –N NAE 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 : D π  D L –There is a polynomial-time computable function, g : F L (f(x))  D π  F π (x) (e.g. taking (solution of f(x), x) pairs to solutions of x) –For all instances x є F π (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.

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 V 1 to truth, and all variables corresponding to vertices in V 2 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 |V 1 | = |V 2 |

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 w ij on the edges –F(G): assignments to the nodes x i of G from {+1,-1} –M(s,G): Σ i,j w ij x i x j –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. 22 nd 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.