We study the problem of computing a competitive equilibrium with approximately optimal bundles in Fisher markets with separable piecewise-linear concave (SPLC) utility functions, meaning that every buyer receives a (1−δ)-optimal bundle, instead of a perfectly optimal one. We establish the first intractability result for the problem by showing that it is PPAD-hard for some constant δ > 0, assuming the PCP-for-PPAD conjecture. This hardness result holds even if all buyers have identical budgets (competitive equilibrium with equal incomes), linear capped utilities, and even if we also allow ε-approximate clearing instead of perfect clearing, for any constant ε < 1/9. Importantly, we show that the PCP-for-PPAD conjecture is in fact required to show hardness for constant δ: showing PPAD-hardness for finding such approximate market equilibria in a broad class of markets encompassing those generated by our hardness result would prove the conjecture. This is the first natural problem where the conjecture is provably required to establish hardness for it.

It is well known that solving a (non-convex) quadratic program is NP -hard. We show that the problem remains hard even if we are only looking for a Karush–Kuhn–Tucker (KKT) point, instead of a global optimum. Namely, we prove that computing a KKT point of a quadratic polynomial over the domain [0,1] n is complete for the class CLS = PPAD ∩ PLS .

Abstract. In the [Formula: see text]-Consensus-Halving problem, we are given [Formula: see text] probability measures [Formula: see text] on the interval [Formula: see text], and the goal is to partition [Formula: see text] into two parts [Formula: see text] and [Formula: see text] using at most [Formula: see text] cuts, so that [Formula: see text] for all [Formula: see text]. This fundamental fair division problem was the first natural problem shown to be complete for the class PPA , and all subsequent PPA -completeness results for other natural problems have been obtained by reducing from it. We show that [Formula: see text]-Consensus-Halving is PPA -complete even when the parameter [Formula: see text] is a constant. In fact, we prove that this holds for any constant [Formula: see text]. As a result, we obtain constant inapproximability results for all known natural PPA -complete problems, including necklace splitting, the discrete ham sandwich problem, two variants of the pizza sharing problem, and for finding fair independent sets in cycles and paths.

We provide polynomial-time reductions between three search problems from three distinct areas: the P-matrix linear complementarity problem (P-LCP), finding the sink of a unique sink orientation (USO), and a variant of the $α$-Ham Sandwich problem. For all three settings, we show that "two choices are enough", meaning that the general non-binary version of the problem can be reduced in polynomial time to the binary version. This specifically means that generalized P-LCPs are equivalent to P-LCPs, and grid USOs are equivalent to cube USOs. These results are obtained by showing that both the P-LCP and our $α$-Ham Sandwich variant are equivalent to a new problem we introduce, P-Lin-Bellman. This problem can be seen as a new tool for formulating problems as P-LCPs.

We study the problem of computing approximate market equilibria in Fisher markets with separable piecewise-linear concave (SPLC) utility functions. In this setting, the problem was only known to be PPAD-complete for inverse-polynomial approximations. We strengthen this result by showing PPAD-hardness for constant approximations. This means that the problem does not admit a polynomial time approximation scheme (PTAS) unless PPAD = P. In fact, we prove that computing any approximation better than 1/11 is PPAD-complete. As a direct byproduct of our main result, we get the same inapproximability bound for Arrow-Debreu exchange markets with SPLC utility functions.

It is well known that solving a (non-convex) quadratic program is NP-hard. We show that the problem remains hard even if we are only looking for a Karush-Kuhn-Tucker (KKT) point, instead of a global optimum. Namely, we prove that computing a KKT point of a quadratic polynomial over the domain [0,1]n is complete for the class CLS = PPAD∩PLS.

The current state-of-the-art methods for showing inapproximability in PPAD arise from the ɛ-Generalized-Circuit (ɛ- GCircuit ) problem. Rubinstein (2018) showed that there exists a small unknown constant ɛ for which ɛ- GCircuit is PPAD -hard, and subsequent work has shown hardness results for other problems in PPAD by using ɛ- GCircuit as an intermediate problem. We introduce Pure-Circuit , a new intermediate problem for PPAD , which can be thought of as ɛ- GCircuit pushed to the limit as ɛ → 1, and we show that the problem is PPAD -complete. We then prove that ɛ- GCircuit is PPAD -hard for all ɛ < 1/10 by a reduction from Pure-Circuit , and thus strengthen all prior work that has used GCircuit as an intermediate problem from the existential-constant regime to the large-constant regime. We show that stronger inapproximability results can be derived by reducing directly from Pure-Circuit . In particular, we prove tight inapproximability results for computing approximate Nash equilibria and approximate well-supported Nash equilibria in graphical games, for finding approximate well-supported Nash equilibria in polymatrix games, and for finding approximate equilibria in threshold games.

We provide a complete characterization for the computational complexity of finding approximate equilibria in two-action graphical games. We consider the two most well-studied approximation notions: ε-Nash equilibria (ε-NE) and ε-well-supported Nash equilibria (ε-WSNE), where ε is in [0,1]. We prove that computing an ε-NE is PPAD-complete for any constant ε smaller than 1/2, while a very simple algorithm (namely, letting all players mix uniformly between their two actions) yields a 1/2-NE. On the other hand, we show that computing an ε-WSNE is PPAD-complete for any constant ε smaller than 1, while a 1-WSNE is trivial to achieve, because any strategy profile is a 1-WSNE. All of our lower bounds immediately also apply to graphical games with more than two actions per player.

We study search problems that can be solved by performing Gradient Descent on a bounded convex polytopal domain and show that this class is equal to the intersection of two well-known classes: PPAD and PLS. As our main underlying technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a continuously differentiable function over the domain [0,1] 2 is PPAD ∩ PLS-complete. This is the first non-artificial problem to be shown complete for this class. Our results also imply that the class CLS (Continuous Local Search) – which was defined by Daskalakis and Papadimitriou as a more “natural” counterpart to PPAD ∩ PLS and contains many interesting problems – is itself equal to PPAD ∩ PLS.

We study the computational complexity of computing solutions for the straight-cut and square-cut pizza sharing problems. We show that finding an approximate solution is PPA-hard for the straight-cut problem, and PPA-complete for the square-cut problem, while finding an exact solution for the square-cut problem is FIXP-hard and in BU. Our PPA-hardness results apply even when all mass distributions are unions of non-overlapping squares, and our FIXP-hardness result applies even when all mass distributions are unions of weighted squares and right-angled triangles. We also prove that decision variants of the square-cut problem are hard: the approximate problem is NP-complete, and the exact problem is ETR-complete.

The current state-of-the-art methods for showing inapproximability in PPAD arise from the $\varepsilon$-Generalized-Circuit ($\varepsilon$-GCIRCUIT) problem. Rubinstein (2018) showed that there exists a small unknown constant $\varepsilon$ for which $\varepsilon$-GCIRCUIT is PPAD-hard, and subsequent work has shown hardness results for other problems in PPAD by using $\varepsilon$-GCIRCUIT as an intermediate problem.We introduce PURE-CIRCUIT, a new intermediate problem for PPAD, which can be thought of as $\varepsilon$-GCIRCUIT pushed to the limit as $\varepsilon\rightarrow 1$, and we show that the problem is PPAD-complete. We then prove that $\varepsilon$-GCIRCUIT is PPAD-hard for all $\varepsilon \lt 0.1$ by a reduction from PURE-CIRCUIT, and thus strengthen all prior work that has used GCIRCUIT as an intermediate problem from the existential-constant regime to the large-constant regime. We show that stronger inapproximability results can be derived by a direct reduction from PURE-CIRCUIT. In particular, we prove that finding an $\varepsilon$-well-supported Nash equilibrium in a polymatrix game is PPAD-hard for all $\varepsilon \lt 1/3$, and that this result is tight for two-action games.

In the ε-Consensus-Halving problem, we are given n probability measures v1, …, vn on the interval R = [0,1], and the goal is to partition R into two parts R+ and R− using at most n cuts, so that |vi(R+) − vi(R−)| ≤ ε for all i. This fundamental fair division problem was the first natural problem shown to be complete for the class PPA, and all subsequent PPA-completeness results for other natural problems have been obtained by reducing from it.

Dang et al. have given an algorithm that can find a Tarski fixed point in a k -dimensional lattice of width n using O (log k n ) queries [ 2 ]. Multiple authors have conjectured that this algorithm is optimal [ 2 , 7 ], and indeed this has been proven for two-dimensional instances [ 7 ]. We show that these conjectures are false in dimension three or higher by giving an O (log 2 n ) query algorithm for the three-dimensional Tarski problem. We also give a new decomposition theorem for k -dimensional Tarski problems which, in combination with our new algorithm for three dimensions, gives an O (log 2 ⌈k/3⌉ n ) query algorithm for the k -dimensional problem.

Dang et al. have given an algorithm that can find a Tarski fixed point in a k-dimensional lattice of width n using O(log^k n) queries [Chuangyin Dang et al., 2020]. Multiple authors have conjectured that this algorithm is optimal [Chuangyin Dang et al., 2020; Kousha Etessami et al., 2020], and indeed this has been proven for two-dimensional instances [Kousha Etessami et al., 2020]. We show that these conjectures are false in dimension three or higher by giving an O(log² n) query algorithm for the three-dimensional Tarski problem, which generalises to give an O(log^{k-1} n) query algorithm for the k-dimensional problem when k ≥ 3.

We study search problems that can be solved by performing Gradient Descent on a bounded convex polytopal domain and show that this class is equal to the intersection of two well-known classes: PPAD and PLS. As our main underlying technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a continuously differentiable function over the domain [0,1]2 is PPAD ∩ PLS-complete. This is the first natural problem to be shown complete for this class. Our results also imply that the class CLS (Continuous Local Search) - which was defined by Daskalakis and Papadimitriou as a more “natural” counterpart to PPAD ∩ PLS and contains many interesting problems - is itself equal to PPAD ∩ PLS.

We study the problem of finding an exact solution to the Consensus Halving problem. While recent work has shown that the approximate version of this problem is PPA -complete [29] , [30] , we show that the exact version is much harder. Specifically, finding a solution with n agents and n cuts is FIXP -hard, and deciding whether there exists a solution with fewer than n cuts is ETR -complete. Along the way, we define a new complexity class, called BU , which captures all problems that can be reduced to solving an instance of the Borsuk-Ulam problem exactly. We show that FIXP ⊆ BU ⊆ TFETR and that LinearBU = PPA , where LinearBU is the subclass of BU in which the Borsuk-Ulam instance is specified by a linear arithmetic circuit .

We prove that it is PPAD-hard to compute a Nash equilibrium in a tree polymatrix game with twenty actions per player. This is the first PPAD hardness result for a game with a constant number of actions per player where the interaction graph is acyclic. Along the way we show PPAD-hardness for finding an ε-fixed point of a 2D-LinearFIXP instance, when ε is any constant less than (√2 - 1)/2 ≈ 0.2071. This lifts the hardness regime from polynomially small approximations in k-dimensions to constant approximations in two-dimensions, and our constant is substantial when compared to the trivial upper bound of 0.5.

The main result of this paper is that computing the value of a one-clock priced timed game (OCPTG) is PSPACE-hard. Along the way, we provide a family of OCPTGs that have an exponential number of event points. Both results hold even in very restricted classes of games such as DAGs with treewidth three. Finally, we provide a number of positive results, including polynomial-time algorithms for even more restricted classes of OCPTGs such as trees.

Abstract In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly polynomial time algorithms for finding best responses in $$L_1$$ L1 and $$L_2^2$$ L22 biased games, and then use these algorithms to provide strongly polynomial algorithms that find 2/3 and 5/7 approximate equilibria for these norms, respectively.

The complexity class CLS was proposed by Daskalakis and Papadimitriou in 2011 to understand the complexity of important NP search problems that admit both path following and potential optimizing algorithms. Here we identify a subclass of CLS – called UniqueEOPL – that applies a more specific combinatorial principle that guarantees unique solutions. We show that UniqueEOPL contains several important problems such as the P-matrix Linear Complementarity Problem, finding fixed points of Contraction Maps, and solving Unique Sink Orientations (USOs). We identify a problem – closely related to solving contraction maps and USOs – that is complete for UniqueEOPL.

We present a new, distributed method to compute approximate Nash equilibria in bimatrix games. In contrast to previous approaches that analyze the two payoff matrices at the same time (for example, by solving a single LP that combines the two players’ payoffs), our algorithm first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then computes an approximate Nash equilibrium using only limited communication between the players. Our method gives improved bounds on the complexity of computing approximate Nash equilibria in a number of different settings. Firstly, it gives a polynomial-time algorithm for computing approximate well supported Nash equilibria (WSNE) that always finds a 0.6528-WSNE, beating the previous best guarantee of 0.6608. Secondly, since our algorithm solves the two LPs separately, it can be applied to give an improved bound in the limited communication setting, giving a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and finds a 0.6528-WSNE, which beats the previous best known guarantee of 0.732. It can also be applied to the case of approximate Nash equilibria, where we obtain a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and always finds a 0.382-approximate Nash equilibrium, which improves the previous best guarantee of 0.438. Finally, the method can also be applied in the query complexity setting to give an algorithm that makes $$O(n \log n)$$ payoff queries and always finds a 0.6528-WSNE, which improves the previous best known guarantee of 2/3.

In this paper, we study the problem of deciding the winner of reachability switching games. We study zero-, one-, and two-player variants of these games. We show that the zero-player case is NL-hard, the one-player case is NP-complete, and that the two-player case is PSPACE-hard and in EXPTIME. For the zero-player case, we also show P-hardness for a succinctly-represented model that maintains the upper bound of NP n coNP. For the one- and two-player cases, our results hold in both the natural, explicit model and succinctly-represented model. We also study the structure of winning strategies in these games, and in particular we show that exponential memory is required in both the one- and two-player settings.

Parity games play an important role in model checking and synthesis. In their paper, Calude et al. have recently shown that these games can be solved in quasi-polynomial time. We show that their algorithm can be implemented efficiently: we use their data structure as a progress measure, allowing for a backward implementation instead of a complete unravelling of the game. To achieve this, a number of changes have to be made to their techniques, where the main one is to add power to the antagonistic player that allows for determining her rational move without changing the outcome of the game. We provide a first implementation for a quasi-polynomial algorithm, test it on small examples, and provide a number of side results, including minor algorithmic improvements, a quasi bi-linear complexity in the number of states and edges for a fixed number of colours, and matching lower bounds for the algorithm of Calude et al.

Strategy improvement is a widely-used and well-studied class of algorithms for solving graph-based infinite games. These algorithms are parametrized by a switching rule, and one of the most natural rules is “all switches” which switches as many edges as possible in each iteration. Continuing a recent line of work, we study all-switches strategy improvement from the perspective of computational complexity. We consider two natural decision problems, both of which have as input a game G, a starting strategy s, and an edge e. The problems are: 1. The edge switch problem, namely, is the edge e ever switched by all-switches strategy improvement when it is started from s on game G? 2. The optimal strategy problem, namely, is the edge e used in the final strategy that is found by strategy improvement when it is started from s on game G? We show PSPACE-completeness of the edge switch problem and optimal strategy problem for the following settings: Parity games with the discrete strategy improvement algorithm of Vöge and Jurdziński; mean-payoff games with the gain-bias algorithm [11, 33]; and discounted-payoff games and simple stochastic games with their standard strategy improvement algorithms. We also show PSPACE-completeness of an analogous problem to edge switch for the bottom-antipodal algorithm for Acyclic Unique Sink Orientations on Cubes.

The simplex method is a well-studied and widely-used pivoting method for solving linear programs. When Dantzig originally formulated the simplex method, he gave a natural pivot rule that pivots into the basis a variable with the most violated reduced cost. In their seminal work, Klee and Minty showed that this pivot rule takes exponential time in the worst case. We prove two main results on the simplex method. Firstly, we show that it is PSPACE-complete to find the solution that is computed by the simplex method using Dantzig's pivot rule. Secondly, we prove that deciding whether Dantzig's rule ever chooses a specific variable to enter the basis is PSPACE-complete. We use the known connection between Markov decision processes (MDPs) and linear programming, and an equivalence between Dantzig's pivot rule and a natural variant of policy iteration for average-reward MDPs. We construct MDPs and then show PSPACE-completeness results for single-switch policy iteration, which in turn imply our main results for the simplex method.

We study the deterministic and randomized query complexity of finding approximate equilibria in a k x k bimatrix game. We show that the deterministic query complexity of finding an ε-Nash equilibrium when ε < 1/2 is Ω(k2), even in zero-one constant-sum games. In combination with previous results, this provides a complete characterization of the deterministic query complexity of approximate Nash equilibria. We also study randomized querying algorithms. We give a randomized algorithm for finding a (3--√5/2 + ε)-Nash equilibrium using O(k . log k/ε2) payoff queries, which shows that the 1/2 barrier for deterministic algorithms can be broken by randomization. For well-supported Nash equilibria (WSNE), we first give a randomized algorithm for finding an ε-WSNE of a zero-sum bimatrix game O(k . log k/ε4) payoff queries, and we then use this to obtain a randomized algorithm for finding a (2/3 + ε)-WSNE in a general bimatrix game using O(k . log k /ε2) payoff queries. Finally, we initiate the study of lower bounds against randomized algorithms in the context of bimatrix games, by showing that randomized algorithms require Omega(k2) payoff queries in order to find a 1/6k-Nash equilibrium, even in zero-one constant-sum games. In particular, this rules out query-efficient randomized algorithms for finding exact Nash equilibria.

A recent body of experimental literature has studied empirical game-theoretical analysis, in which we have partial knowledge of a game, consisting of observations of a subset of the pure-strategy profiles and their associated payoffs to players. The aim is to find an exact or approximate Nash equilibrium of the game, based on these observations. It is usually assumed that the strategy profiles may be chosen in an on-line manner by the algorithm. We study a corresponding computational learning model, and the query complexity of learning equilibria for various classes of games. We give basic results for bimatrix and graphical games. Our focus is on symmetric network congestion games. For directed acyclic networks, we can learn the cost functions (and hence compute an equilibrium) while querying just a small fraction of pure-strategy profiles. For the special case of parallel links, we have the stronger result that an equilibrium can be identified while only learning a small fraction of the cost values.

We study the time-bounded reachability problem for continuous time Markov decision processes (CTMDPs) and games (CTMGs). Existing techniques for this problem use discretization techniques to break time into discrete intervals, and optimal control is approximated for each interval separately. Current techniques provide an accuracy of O(\epsilon^2) on each interval, which leads to an infeasibly large number of intervals. We propose a sequence of approximations that achieve accuracies of O(\epsilon^3), O(\epsilon^4), and O(\epsilon^5), that allow us to drastically reduce the number of intervals that are considered. For CTMDPs, the resulting algorithms are comparable to the heuristic approach given by Buckholz and Schulz, while also being theoretically justified. All of our results generalise to CTMGs, where our results yield the first practically implementable algorithms for this problem. We also provide positional strategies for both players that achieve similar error bounds.