The complexity of the parity argument with potential

Abstract

The parity argument principle states that every finite graph has an even number of odd degree vertices. We consider the problem whose totality is guaranteed by the parity argument on a graph with potential. In this paper, we show that the problem of finding an unknown odd-degree vertex or a local optimum vertex on a graph with potential is polynomially equivalent to EndOfPotentialLine if the maximum degree is at most three. However, even if the maximum degree is 4, such a problem is $\mathtt{PPA}\cap \mathtt{PLS}$-complete. To show the complexity of this problem, we provide new results on multiple-source variants of EndOfPotentialLine, which is the canonical problem for EOPL. This result extends the work by Goldberg and Hollender; they studied similar variants of EndOfLine.

Introduction

We are interested in the complexity class TFNP: a set of total search problems belonging to FNP [21]. Every problem in TFNP satisfies the following two properties: every instance always has a solution, and we can verify that a solution is correct in polynomial time. The class TFNP contains some of the most fundamental, elegant, and intriguing computational problems. The most famous examples are factoring, computing a Nash equilibrium, and finding a local optimum. These problems have no known polynomial-time algorithms, but it seems to be not NP-hard either. In fact, if TFNP has an NP-hard problem, then we have $\mathtt{NP}=\mathtt{coNP}$ [21].

Unfortunately, TFNP is a “semantic” class. Generally, such classes seem to have no complete problem like language classes RP, ZPP, and BPP [22]. Thus, we study “syntactic” subclasses of TFNP. The best-well known such subclasses are PLS, PPP, PPA, and PPAD [22], [16], [17]. These are classes of search problems whose totality is guaranteed by the corresponding mathematical lemma. The subclasses of TFNP have been studied productively and extensively, and have been shown to have many interesting search problems as complete problems. Section 2.3 gives details of some complexity classes contained in TFNP.

The above subclasses of TFNP are usually defined in terms of the corresponding search problem on an exponentially-large graph, which is represented by functions. Note that a graph in a problem is presented via a function that computes the neighbors of a given vertex on the graph,¹ not an adjacency matrix. Hence, we need exponential effort to get the structure of the entire graph.

For example, the class PPA is defined as a set of problems that reduce to Leaf: given a neighborhood function on an undirected graph on $2^n$ vertices whose degree at most two and a known leaf, find an unknown leaf. It is known that PPA has the following complete problems: ConsensusHalving, NecklaceSplitting, DiscreteHamSandwich, and OctahedralTucker [13], [14], [9], [1].

Similarly, the class PPAD is defined as a set of problems that reduce to EndOfLine: given a successor circuit and a predecessor circuit on a digraph on $2^n$ vertices whose in-degree/out-degree at most one and a standard source, find a sink or a non-standard source. PPAD includes a search problems that seems easier than PPA. The most famous PPAD-complete problem is the problem of finding a Nash equilibrium [6], [22].

Fearnley et al. [12] studied the search problems on digraphs with potential and produced the new computational complexity class EOPL. Although the natural EOPL-complete problem is still unknown, it contains some fascinating search problems, e.g., solving a Linear Complementarity Problem for P-matrices, solving parity games, and solving simple stochastic games [12].

A natural question worth considering is how easier the problem is by adding a potential condition to a problem known as a known PPA-complete problem. We consider the hardness of finding an unknown odd-degree vertex or a local optimum vertex when we are given a known odd-degree vertex on the graph with potential. To show the complexity of such a problem, we provide the new results on multiple-source variants of EndOfPotentialLine, which is a canonical problem for EOPL.

Recently, Goldberg and Hollender [15] have studied the robustness of the classification by EndOfLine. They considered combinatorial principles related to PPAD, leading to the following problems, on digraphs with degree at most two:

• given k sources and $l\ne k$ sinks, find another sink or source;

• given k sources and $l<k$ sinks, find $k-l$ other sinks; and

• given k sources, find k sinks or k other sources.

They proved that these above problems are also PPAD-complete. Moreover, they showed that the problem Imbalance, in which given a digraph and an unbalanced vertex, i.e., a vertex with in-degree ≠ out-degree, find another unbalanced vertex, is also PPAD-complete. These facts imply that the classification by EndOfLine is very robust.

Hollender and Goldberg [18] left an open question: Is a multiple-source variant of EndOfPotentialLine also EOPL-complete? We resolve this question in this paper, and we show that the classification of the class EOPL based on EndOfPotentialLine is robust.

In this paper, we extend Goldberg and Hollenders' [15] results to EndOfPotentialLine. In this problem, given a successor circuit, a predecessor circuit, a potential function, and one standard source, the objective is to find one of a sink, a non-standard source, and a non-increasing arc. We first consider combinatorial principles related to EOPL, leading to the following problems, on graphs with potential with degree at most two:

• given k sources, find another degree-1 vertex or a non-increasing arc; and

• given k sources, find k distinct vertices that are at least one of a sink, other source, and a non-increasing arc.

We show that these variants of EndOfPotentialLine can be classified in terms of the original problem, that is, these problems are also EOPL-complete. Furthermore, we consider the problem of generalizing EndOfPotentialLine to higher degree digraphs. In this problem, given a digraph with potential and one unbalanced vertex, the objective is to find another unbalanced vertex or a non-increasing arc. Naturally, this problem also belongs to EOPL.

Finally, we consider a problem: the goal is to seek an unknown odd-degree vertex or a local optimum vertex on a given graph with potential on $2^n$ vertices with one known odd-degree vertex. We prove that if the maximum degree of a given graph is at most three, this problem is EOPL-complete. However, this problem is not always EOPL-complete, even if the maximum degree of a given graph is 4.

An overview of our results and the known relationship of complexity classes is depicted in Fig. 1. In this figure, each arrow $\alpha \to \beta$ denotes that there is a polynomial-time reduction from α to β.

In Section 2.4, we introduce the notion called the normalization to simplify arguments in this paper. Some search problems require that every instance has several properties. The best-known example of such a problem is Brouwer. In this problem, the function given by the instance is required Lipschitz continuous. Generally, it seems hard to decide whether a given function is Lipschitz continuous. However, given a witness, it is easy to check it. We often add every witness as a solution called a violation. Informally speaking, our normalization is to transform from an instance which has violations to another instance satisfying the required conditions. The formal definition is given in Section 2.4.

Directed graphs with potential. In Section 3, we extend the elegant results by Goldberg and Hollender [15] to EndOfPotentialLine, and we show the robustness of EOPL. We introduce the new variant of EndOfPotentialLine. We call this problem Multiple-Source EndOfPotentialLine. We prove that this problem is also EOPL-complete. Furthermore, we introduce the new problem generalizing EndOfPotentialLine to higher degree digraphs, which is called PotentialImbalance. We also prove that this problem is EOPL-complete.

Undirected graphs with potential. In Section 4, to study the hardness of the parity argument with potential, we introduce the complexity class $\mathtt{PPA}\cap \mathtt{PLS}$. This class consists of all search problems belonging to both PPA and PLS (see Section 2.3.6). We define PotentialOdd; this problem is a generalization of PotentialImbalance to an undirected graph with potential. We show that PotentialOdd is, generally, $\mathtt{PPA}\cap \mathtt{PLS}$-complete. Specifically, if the maximum degree on a given graph is at most three, then PotentialOdd belongs to EOPL, that is, this problem is EOPL-complete. However, even if the maximum degree on a given degree is 4, then PotentialOdd is $\mathtt{PPA}\cap \mathtt{PLS}$-complete.

Daskalakis and Papadimitriou [7] introduced the new complexity class CLS. This class is defined as a set of all search problems that reduce to ContinuousLocalOPT ² in polynomial time. They showed that this class includes some interesting search problems: finding an approximate fixed point of a contraction map, finding a stationary point of a polynomial, and finding a mixed Nash equilibrium on congestion games [7]. In particular, the problem of finding a pure Nash equilibrium on congestion games is PLS-complete [10]. Daskalakis et al. [8] proved that the problem of finding an approximate fixed point of a metric contraction map is CLS-complete. This is the first natural CLS-complete problem. Most recently, Fearnley et al. [11] showed that $\mathtt{CLS}=\mathtt{PPAD}\cap \mathtt{PLS}$. They proved that finding a KKT point is CLS-complete. Babichenko and Rubinstein [2] showed that finding a mixed Nash equilibrium on explicit congestion games is $\mathtt{PPAD}\cap \mathtt{PLS}$-complete; this implies that $\mathtt{CCLS}=\mathtt{PPAD}\cap \mathtt{PLS}$, where the class CCLS is another subclass of $\mathtt{PPAD}\cap \mathtt{PLS}$ introduced by Daskalakis and Papadimitriou [7]. Therefore, three classes $\mathtt{PPAD}\cap \mathtt{PLS}$, CLS, and CCLS are equivalent.

Hubáček and Yogev [19] introduced the new search problem called EndOfMeteredLine, and showed that this problem belongs to CLS. Fearnley et al. [12] defined the class EOPL by using the problem EndOfPotentialLine. Furthermore, they showed that EndOfPotentialLine is equivalent to EndOfMeteredLine in polynomial time. That is, EOPL is a subclass of CLS.

Beame et al. [3] defined two types of search problems, which are a directed analog of Odd. One is called Excess, and the other is called Imbalance. Specifically, Excess is a generalization of a Sink³ to higher degree digraphs. The problem Excess is defined as: given a digraph and a vertex satisfying that in-degree < out-degree, find a vertex satisfying that in-degree > out-degree on the given graph. We can easily see that this problem is PPADS-complete. On the other hand, Imbalance is a generalization of EndOfLine to higher degree digraphs. Obviously, this problem is PPAD-hard; however, the PPAD-completeness of Imbalance has been open for two decades. Goldberg and Hollender [15] proved that Imbalance is PPAD-complete. Furthermore, they applied the PPAD-completeness of Multiple-Source EndOfLine to show that k-D-HairyBall is PPAD-complete for all even $k\ge 2$.

Daskalakis et al. [6] proved the PPAD-completeness for finding a Nash equilibrium on graphical games, which is called polymatrix games, and three-players forms games. Daskalakis et al. [4] showed that we can compute a Nash equilibrium on the zero-sum polymatrix game by using a liner programming, that is, we can find it in polynomial time. Chen et al. [5] showed that finding a Nash equilibrium in a two-players game is PPAD-hard.