Hardness of Continuous Local Search: Query Complexity and Cryptographic Lower Bounds

Abstract

Local search proved to be an extremely useful tool when facing hard optimization problems (e.g., via the simplex algorithm, simulated annealing, or genetic algorithms). Although powerful, it has its limitations: there are functions for which exponentially many queries are needed to find a local optimum. In many contexts the optimization problem is defined by a continuous function, which might offer an advantage when performing the local search. This leads us to study the following natural question: How hard is continuous local search? The computational complexity of such search problems is captured by the complexity class CLS (Daskalakis and Papadimitriou SODA'11) which is contained in the intersection of PLS and PPAD, two important subclasses of TFNP (the class of NP search problems with a guaranteed solution).In this work, we show the first hardness results for CLS (the smallest non-trivial class among the currently defined subclasses of TFNP). Our hardness results are in terms of black-box (where only oracle access to the function is given) and white-box (where the function is represented succinctly by a circuit). In the black-box case, we show instances for which any (computationally unbounded) randomized algorithm must perform exponentially many queries in order to find a local optimum. In the white-box case, we show hardness for computationally bounded algorithms under cryptographic assumptions. Our results demonstrate a strong conceptual barrier precluding design of efficient algorithms for solving local search problems even over continuous domains.As our main technical contribution we introduce a new total search problem which we call End-of-Metered-Line. The special structure of End-of-Metered-Line enables us to: (1) show that it is contained in CLS, and (2) prove hardness for it both in the black-box and the white-box setting.

The complexity of splitting necklaces and bisecting ham sandwiches

“…Furthermore, they show that two well-known problems in CLS, the P-Matrix Linear Complementarity Problem (P-LCP), and finding a fixpoint of a piecewise-linear contraction map (PL-Contraction) belong to the class. The End of Potential Line problem of [27] is closely related to the End of Metered Line of [39].…”

Communication complexity of approximate Nash equilibria

“…See Section 3 for details. We remark that related constructions were known for two dimensions [Vav93,HY17,BM20,CDHS20].…”

“…6 For our construction, we essentially use the End-of-Metered-Line [HY17,FGMS19] variant of the problem, which also encodes a potential function on the line.…”

“…Query complexity Let L P = (l P (0), l P (1), ..., l P (T )) be a line of length T in the Pyramid graph with starting point 0 2 ; i.e., l P (t) ∈ V P y , L P (0) = (0, 0), and (l P (t), l P (t + 1)) ∈ E P . Note that any line over the Pyramid graph is, in fact, a metered line (see [HY17]). Namely, if a line goes through a given vertex (x, y) we know that the line went exactly x + y steps so far, and exactly T − x − y steps remained.…”

PPP-Completeness with Connections to Cryptography

“…This was generalized in [Jer16] by proving that the problem of factoring integers is in PPA ∩ PPP under randomized reductions. Recently, strong cryptographic assumptions were used to prove the average-case hardness of PPAD and CLS [BPR15, GPS16,HY17]. In [RSS17] it was shown that average-case PPAD hardness does not imply one-way function under black-box reductions, whereas in [HNY17] it was shown that any hard on average problem in NP implies the average case hardness of TFNP.…”