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.

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 propose a unifying framework for smoothed analysis of combinatorial local optimization problems, and show how a diverse selection of problems within the complexity class PLS can be cast within this model. This abstraction allows us to identify key structural properties, and corresponding parameters, that determine the smoothed running time of local search dynamics. We formalize this via a black-box tool that provides concrete bounds on the expected maximum number of steps needed until local search reaches an exact local optimum. This bound is particularly strong, in the sense that it holds for any starting feasible solution, any choice of pivoting rule, and does not rely on the choice of specific noise distributions that are applied on the input, but it is parameterized by just a global upper bound ϕ on the probability density. The power of this tool can be demonstrated by instantiating it for various PLS-hard problems of interest to derive efficient smoothed running times (as a function of ϕ and the input size). Most notably, we focus on the important local optimization problem of finding pure Nash equilibria in Congestion Games, that has not been studied before from a smoothed analysis perspective. Specifically, we propose novel smoothed analysis models for general and Network Congestion Games, under various representations, including explicit, step-function, and polynomial resource latencies. We study PLS-hard instances of these problems and show that their standard local search algorithms run in polynomial smoothed time. Further applications of our framework to a wide range of additional combinatorial problems can be found in the full version of our paper.

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.

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.

This paper explores the use of the Directional Changes (DC) paradigm for financial forecasting. DC is an event-based alternative to the traditional approach of time-series with fixed intervals. In the DC approach, price movements are recorded when specific events occur, rather than in fixed time intervals, while significant price changes are identified using a threshold. Here, we consider a more general model that allows multiple weighted thresholds, and propose three novel trading strategies built within the DC paradigm. To optimize the weights of the thresholds, we use a genetic algorithm and manage to find strategies that outperform previously known single-threshold strategies under the common efficiency metrics. Furthermore, our method manages to create profitable trading strategies that outperform some traditional ones, such as buy-and-hold, MACD, and RSI.

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.

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 .

Abstract We study a security game over a network played between adefenderandkattackers. Every attacker chooses, probabilistically, a node of the network to damage. The defender chooses, probabilistically as well, a connected induced subgraph of the network of $$\lambda $$ λ nodes to scan and clean. Each attacker wishes to maximize the probability of escaping her cleaning by the defender. On the other hand, the goal of the defender is to maximize the expected number of attackers that she catches. This game is a generalization of the model from the seminal paper of Mavronicolas et al. Mavronicolas et al. (in: International symposium on mathematical foundations of computer science, MFCS, pp 717–728, 2006). We are interested in Nash equilibria of this game, as well as in characterizingdefense-optimalnetworks which allow for the bestequilibrium defense ratio; this is the ratio ofkover the expected number of attackers that the defender catches in equilibrium. We provide a characterization of the Nash equilibria of this game and defense-optimal networks. The equilibrium characterizations allow us to show that even if the attackers are centrally controlled the equilibria of the game remain the same. In addition, we give an algorithm for computing Nash equilibria. Our algorithm requires exponential time in the worst case, but it is polynomial-time for $$\lambda $$ λ constantly close to 1 orn. For the special case of tree-networks, we further refine our characterization which allows us to derive a polynomial-time algorithm for deciding whether a tree is defense-optimal and if this is the case it computes a defense-optimal Nash equilibrium. On the other hand, we prove that it is $${\mathtt {NP}}$$ NP -hard to find a best-defense strategy if the tree is not defense-optimal. We complement this negative result with a polynomial-time constant-approximation algorithm that computes solutions that are close to optimal ones for general graphs. Finally, we provide asymptotically (almost) tight bounds for thePrice of Defensefor any $$\lambda $$ λ ; this is the worst equilibrium defense ratio over all graphs.