Scarf's lemma is one of the fundamental results in combinatorics, originally introduced to study the core of an N-person game. Over the last four decades, the usefulness of Scarf's lemma has been demonstrated in several important combinatorial problems seeking "stable" solutions. However, the complexity of the computational version of Scarf's lemma (SCARF) remained open. In this paper, we prove that SCARF is complete for the complexity class PPAD. This proves that SCARF is as hard as the computational versions of Brouwer's fixed point theorem and Sperner's lemma. Hence, there is no polynomial-time algorithm for SCARF unless PPAD P. We also show that fractional stable paths problem and finding strong fractional kernels in digraphs are PPAD-hard.
The complexity class TFNP was introduced by Megiddo and Papadimitriou @cite_9 . The class PLS (for polynomial local search) was introduced by Johnson, Papadimitriou and Yannakakis @cite_16 . In @cite_10 Papadimitriou defined the complexity classes PPA (polynomial parity argument), PPAD (polynomial parity argument in directed graphs), PPP (polynomial pigeon-hole principle) and PPM (polynomial probabilistic method). He proved that computational versions of Brouwer's fixed point theorem and Sperner's lemma are PPAD-complete.
@cite_9 @cite_16 @cite_10
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Papadimitriou introduced several classes of NP search problems based on combinatorial principles which guarantee the existence of solutions to the problems. Many interesting search problems not known to be solvable in polynomial time are contained in these classes, and a number of them are complete problems. We consider the question of the relative complexity of these search problem classes. We prove several separations which show that in a generic relativized world the search classes are distinct and there is a standard search problem in each of them that is not computationally equivalent to any decision problem. (Naturally, absolute separations would imply thatP?NP.) Our separation proofs have interesting combinatorial content and go to the heart of the combinatorial principles on which the classes are based. We derive one result via new lower bounds on the degrees of poly- nomials asserted to exist by Hilbert's nullstellensatz over finite fields. We define several new complexity classes of search problems, ''between'' the classes FP and FNP. These new classes are contained, along with factoring, and the class PLS, in the class TFNP of search problems in FNP that always have a witness. A problem in each of these new classes is defined in terms of an implicitly given, exponentially large graph. The existence of the solution sought is established via a simple graph-theoretic argument with an inefficiently constructive proof; for example, PLS can be thought of as corresponding to the lemma ''every dag has a sink.'' The new classes, are based on lemmata such as ''every graph has an even number of odd-degree nodes.'' They contain several important problems for which no polynomial time algorithm is presently known, including the computational versions of Sperner's lemma, Brouwer's fixpoint theorem, Chevalley's theorem, and the Borsuk-Ulam theorem, the linear complementarity problem for P-matrices, finding a mixed equilibrium in a non-zero sum game, finding a second Hamilton circuit in a Hamiltonian cubic graph, a second Hamiltonian decomposition in a quartic graph, and others. Some of these problems are shown to be complete. Abstract Nondeterministic multivalued functions with values that are polynomially verifiable and guaranteed to exist form an interesting complexity class between P and NP. We show that this class, which we call TFNP, contains a host of important problems, whose membership in P is currently not known. These include, besides factoring, local optimization, Brouwer's fixed points, a computational version of Sperner's Lemma, bimatrix equilibria in games, and linear complementarity for P -matrices. For certain subclasses of NP, @math P, or #P characterized by local constraints, it is known that if there exist any problems within that subclass that are not polynomial time computable, then all the problems in the subclass are NP-complete, @math P-complete, or #P-complete. Such dichotomy results have been proved for characterizations such as constraint satisfaction problems and directed and undirected graph homomorphism problems, often with additional restrictions. Here we give a dichotomy result for the more expressive framework of Holant problems. For example, these additionally allow for the expression of matching problems, which have had pivotal roles in the development of complexity theory. As our main result we prove the dichotomy theorem that, for the class @math P, every set of symmetric Holant signatures of any arities that is not polynomial time computable is @math P-complete. The result exploits some special properties of the class @math P and characterizes four distinct tractable ...