Understanding and mitigating systemic risk is an important ongoing challenge in financial networks. We study an approach to rescue a bank in distress based on the idea of claims trading , a notion defined in Chapter 11 of the U.S. Bankruptcy Code. We formalize the idea in the context of the seminal model of financial networks by Eisenberg and Noe [ 9 ]. Given two banks v and w , we consider the operation that w takes over some claims of v and in return gives liquidity to v (or creditors of v ) to ultimately rescue v (or mitigate contagion effects). We study structural properties and computational complexity of decision and optimization problems arising from this trading operation. When trading claims for which v is the creditor, we show that there is no trade in which both banks v and w strictly improve their assets. While deciding the existence of a trade, in which v profits strictly and w remains indifferent, can be NP -hard, we provide FPTAS’es to approximate such a trade, even when banks settle their debt using general monotone payment functions. When trading claims with a common debtor, we show positive and negative results that crucially depend on the payment functions.
This is a corrigendum for the article "Lower Bounds for Planar Arithmetic Circuits" published in ACM Trans. Comput. Theory 18, 1, Article 9 (February 2026), 23 pages.
This article studies the hazard-free formula complexity of Boolean functions. Hazard-free formulas are De Morgan formulas that may take input variables with an unknown value, but produce a Boolean output whenever it can be determined by the known bits. Our first result concerns the hazard-derivative approach of Ikenmeyer et al. (J. ACM, 2019) for proving lower bounds in the hazard-free model. We show that this approach yields optimal bounds only for unate functions, i.e., the monotone formula complexity of a hazard-derivative equals the hazard-free formula complexity of the function itself. Our second result proves that the hazard-free formula complexity of a uniformly random Boolean function is at most \(2^{(1+o(1))n}\) . Previously, no better upper bound than \(O(3^n)\) was known. Notably, unlike in the general Boolean case, where the typical formula complexity is derived from that of the multiplexer function with n -bit selector, the hazard-free formula complexity of a random function is smaller than the optimal hazard-free formula for the multiplexer by an exponential factor in n . We provide two proofs of this fact. The first is direct, bounding the number of prime implicants of a random Boolean function and constructing a DNF of the claimed size. The second introduces a new and independently interesting result: a weak converse to the hazard-derivative lower bound method, which gives an upper bound on the hazard-free complexity of a function in terms of the monotone complexity of a subset of its hazard-derivatives. Additionally, we explore the hazard-free formula complexity of block composition and obtain a hazard-free analogue of the result of Karchmer, Raz, and Wigderson (Computational Complexity, 1995) in the monotone setting. Our result implies a stronger lower bound on the hazard-free formula depth of the block composition of the set covering function with the multiplexer function than the bound obtained via the hazard-derivative method.
Using spectral techniques, H. Huang proved that every subgraph of Hn, the hypercube of dimension n, induced on more than half the vertices has maximum degree at least root n. Combined with earlier work, this completed a proof of the sensitivity conjecture. In this work we show how to derive Huang's result using linear dependency and independence of vectors associated with the vertices of the hypercube. Our approach leads to several improvements of Huang's result. In particular, we prove that in any induced subgraph of Hn with more than half the number of vertices, there are two vertices, one of odd parity and the other of even parity, each with at least n vertices at distance at most 2. As an application, we show that for any Boolean function f, the polynomial degree off is bounded above by s0(f) s1(f), a statement which implies the sensitivity conjecture (but not immediately implied by the sensitivity conjecture). Using these linear dependencies, we show structural relations about the neighborhoods on the induced subgraphs at distance at most three. A key implement in Huang's proof is to assign signs (+, -) to the edges of Hn such that the product of the signs on each 4-cycle is-. With the set of negative edges being called a signature, one may observe that there are a total of 22n-1 such signatures on Hn satisfying this condition and that the symmetric difference of any two such signatures is an edge cut. A question of high interest then is to find the smallest size among all these signatures. This is known as the frustration index in the study of signed graphs. Here we provide lower and upper bounds for this parameter, observing that the two bounds match when n is a power of 4. We then establish a strong connection with other studies: On one hand with a question of Erd & odblac;s on the number of edges of a largest 4-cycle free subgraph of the hypercube. On the other hand with Ambainis functions which are used to show a separation between degree and adversary lower bounds on query complexity.
We design a deterministic subexponential time algorithm that takes as input a multivariate polynomial f computed by a constant-depth circuit over rational numbers, and outputs a list L of circuits (of unbounded depth and possibly with division gates) that contains all irreducible factors of f computable by constant-depth circuits. This list L might also include circuits that are spurious: they either do not correspond to factors of f or are not even well-defined, e.g. the input to a division gate is a sub-circuit that computes the identically zero polynomial. The key technical ingredient of our algorithm is a notion of the pseudo-resultant of f and a factor g , which serves as a proxy for the resultant of g and f / g , with the advantage that the circuit complexity of the pseudo-resultant is comparable to that of the circuit complexity of f and g . This notion, which might be of independent interest, together with the recent results of Limaye, Srinivasan and Tavenas [19] helps us derandomize one key step of multivariate polynomial factorization algorithms — that of deterministically finding a good starting point for Newton Iteration for the case when the input polynomial as well as the irreducible factor of interest have small constant-depth circuits.
