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.
A catalytic Turing machine is a model of computation that is created by equipping a Turing machine with an additional auxiliary tape, which is initially filled with arbitrary content; the machine can read or write on the auxiliary tape during the computation, but it is constrained to halt with the same content in the auxiliary tape as it initially had. This article, studies the power of some natural variants of catalytic Turing machines with O(log n)-size work tape and a polynomial-size auxiliary tape. We first define the notion of unambiguous catalytic Turing machines and prove that under a standard derandomization assumption, the class of problems solved by unambiguous catalytic Turing machines is the same as the class of problems solved by nondeterministic catalytic Turing machines. We then introduce the notion of randomized catalytic Turing machines and show that the resulting complexity class CBPL is contained in the class ZPP. We also explore the notion of symmetricity in the context of catalytic computation and prove that, under the same assumption as before, randomized catalytic Turing machines, symmetric catalytic Turing machines, and deterministic catalytic Turing machines that run in polynomial time are equally powerful.
Merge Resolution ( MRes [6]) is a recently introduced proof system for false QBFs. Unlike other known QBF proof systems, it builds winning strategies for the universal player (countermodels) within the proofs as merge maps. Merge maps are deterministic branching programs in which isomorphism checking is efficient, as a result MRes is a polynomial time verifiable proof system. In this paper, we introduce a family of proof systems MRes- \(\mathcal {R} \) in which the information of countermodels are stored in any pre-fixed complete representation \(\mathcal {R} \) . Hence, corresponding to each possible complete representation \(\mathcal {R} \) , we have a sound and refutationally complete QBF-proof system in MRes- \(\mathcal {R} \) . To handle these arbitrary representations, we introduce consistency checking rules in MRes- \(\mathcal {R} \) instead of the isomorphism checking in MRes . As a result, these proof systems are not polynomial time verifiable (Non-P). Consequently, the paper shows that using merge maps is too restrictive and with a slight change in rules, it can be replaced with arbitrary representations leading to several interesting proof systems. We relate these new systems with the implicit proof system from the algorithm in [11], which was designed to solve DQBFs (Dependency QBFs) using clause-strategy pairs like MRes . We use the OBDD (Ordered Binary Decision Diagrams) representation suggested in [11] and deduce that ‘ordered’ version of the proof systems in MRes- \(\mathcal {R} \) along with some additional restrictions are polynomial time verifiable. Demonstrating the strength of these systems, we show that the regular versions of systems in MRes- \(\mathcal {R} \) are exponentially stronger than regular MRes . We also lift the lower bound result of regular MRes ([7]) by showing that the completion principle formulas ( CR n ) from [25] which are shown to be hard for regular MRes in [7], are also hard for any regular proof system in MRes- \(\mathcal {R} \) . Therefore, the paper lifts the lower bound of regular MRes to an entire class of proof systems, which use various complete representations, including those undiscovered, instead of only merge maps. Thereby proving that the hardness of CR n formulas is intact even after changing the weak isomorphism checking in MRes to the stronger consistency checking in MRes- \(\mathcal {R} \) .
We present a simple algorithm based on the inclusion-exclusion principle for computing the sum of the coefficients of all multilinear monomials in a polynomial computed by an arithmetic circuit over any field F. This problem is formally denoted as (k, n)-MLC and is defined as follows: Given an arithmeticicircuit computing a homogeneous polynomial f is an element of F(x1,...,x] of degree k, where F is any field, compite the sum of the coefficients of all multilinear monomials in f. We provide a deterministic algorithm for the (k, n)-MLC problem with a running time of O(n <^> (k / 2) * 2 <^> (O(k)) * s <^> (O(1))) where s is the size of the circuit. Our algorithm is also polynomial space-bounded. The (k, n)-MLC problem was first studied by Koutis and Williams [15-17, 22] and later studied by Arvind et al., [3], where they obtain a deterministic algorithm with a running time of O*((2) 50logk))1.2 using Hadamard product of polynomials. Pratt [18] proposed a deterministic algorithm that runs in O(n) time, but this approach relies on a non-trivial Waring decomposition of symmetric polynomials over the rationals, which does not extend to small finite fields. Our method is notable for its simplicity and general applicability to all fields. In contrast to existing works, we introduce the complement of the (k,n)-MLC problem, which we call the (k, n)-NMLC problem. This approach looks at finding the sum of the coefficients of all non-multilinear monomials, which gives us a new way to think about the (k, n)-MLC problem. This new method results in a simple algorithm and helps explain where the leading exponent k/2 comes from. We develop a deterministic algorithm for the (k, n)-NMLC problem that runs in O(n <^> (k / 2) * 2 <^> (O(k)) * s) time and uses polynomial space. The algorithm for the (k, n)-MLC problem is straightforward: it subtracts the sum of the coefficients of all non-multilinear monomials (the result from the (k, n)-NMLC problem) from the total sum of all monomial coefficients in the polynomial, which includes both multilinear and non-multilinear monomials. We apply these results to obtain a deterministic algorithm for the exact counting version of the r-simple k-path problem, as introduced in References [1, 13], and also to a generalization of the (k, n)-MLC problem, which we call the (r, k, n)-MLC problem. This involves computing the sum of the coefficients of all monomials in the polynomial produced by the given arithmetic circuit, where the degree of each variable is at most r. The O notation suppresses polynomial factors. k/2 (2) () Sigma
Using the algebraic approach to promise constraint satisfaction problems, we establish complexity classifications of three natural variants of hypergraph colourings: standard nonmonochromatic colourings, conflict-free colourings, and linearly-ordered colourings. Firstly, we show that finding an l-colouring of a k-colourable r-uniform hypergraph is NP-hard for all constant 2 <= k <= l and r >= 3. This provides a shorter proof of a celebrated result by Dinur et al. [FOCS'02/Combinatorica'05]. Secondly, we show that finding an l-conflict-free colouring of an r-uniform hypergraph that admits a k-conflict-free colouring is NP-hard for all constant 2 <= k <= l and r >= 4, except for r = 4 and k = 2 (and any l); this case is solvable in polynomial time. The case of r = 3 is the standard nonmonochromatic colouring, and the case of r = 2 is the notoriously difficult open problem of approximate graph colouring. Thirdly, we show that finding an l-linearly-ordered colouring of an r-uniform hypergraph that admits a k-linearly-ordered colouring is NP-hard for all constant 3 <= k <= l and r >= 4, thus improving on the results of Nakajima and Zivny [ICALP'22/ACM TocT'23].
Let G be a weighted graph embedded in a metric space (M, d(M)). The vertices of G correspond to the points in M, with the weight of each edge uv being the distance d(M) (u, v) between their respective points in M. The dilation (or stretch) of G is defined as the minimum factor t such that, for any pair of vertices u, v, the distance between u and v - represented by the weight of a shortest u, v-path - is at most t . d(M) (u, v). We study DILATION t-AUGMENTATION, where the objective is, given a metric M, a graph G, and numerical values k and t, to determine whether G can be transformed into a graph with dilation t by adding at most k edges. Our primary focus is on the scenario where the metric M is the shortest path metric of an unweighted graph Gamma. Even in this specific case, DILATION t-AUGMENTATION remains computationally challenging. In particular, the problem is W[2]-hard parameterized by k when Gamma is a complete graph, already for t = 2. Our main contribution lies in providing new insights into the impact of combinations of various parameters on the computational complexity of the problem. We establish the following. - The parameterized dichotomy of the problem with respect to dilation t, when the graph G is sparse: Parameterized by k, the problem is FPT for graphs excluding a biclique K-d,K-d as a subgraph for t <= 2 and the problem is W[1]-hard for t >= 3 even if G is a forest consisting of disjoint stars. - The problem is FPT parameterized by the combined parameter k + t + Delta, where Delta is the maximum degree of the graph G or Gamma.
A number of complexity measures for Boolean functions have previously been introduced. These include (1) sensitivity, (2) block sensitivity, (3) witness complexity, (4) subcube partition complexity and (5) algorithmic complexity. Each of these is concerned with "worst-case" inputs. It has been shown that there is "asymptotic separation" between these complexity measures and very recently, due to the work of Huang, it has been established that they are all "polynomially related". In this paper, we study the notion of distributional complexity where the input bits are independent and one considers all of the above notions in expectation. We obtain a number of results concerning distributional complexity measures, among others addressing the above concepts of "asymptotic separation" and being "polynomially related" in this context. We introduce a new distributional complexity measure, local witness complexity, which only makes sense in the distributional context and we also study a new version of algorithmic complexity which involves partial information. Many interesting examples are presented including some related to percolation. The latter connects a number of the recent developments in percolation theory over the last two decades with the study of complexity measures in theoretical computer science.
Given an (infinite) relational structure S, we say that a finite structure C is a smallest finite factor of S if for every finite structure A there is a homomorphism S-A if and only if there is a homomorphism C-A. In this brief note we prove that if CSP(S) is in GMSNP, then S has a smallest finite factor C, and moreover, CSP(C) reduces in polynomial time to CSP(S). We discuss two nice applications of this result. First, we see that if a finite promise constraint satisfaction problem PCSP(A, B) has a tractable GMSNP sandwich, then it has a tractable finite sandwich. We also show that if G is a non-bipartite (possibly infinite) graph with finite chromatic number, and CSP(G) is in GMSNP, then CSP(G) in NP-complete, partially answering a question recently asked by Bodirsky and Guzm & aacute;n-Pro.
Representing graphs by their homomorphism counts has led to the beautiful theory of homomorphism indistinguishability in recent years. Moreover, homomorphism counts have promising applications in database theory and machine learning, where one would like to answer queries or classify graphs solely based on the representation of a graph Gas a finite vector of homomorphism counts from some fixed finite set of graphs to G. We study the computational complexity of the arguably most fundamental computational problem associated to these representations, the homomorphism reconstructibility problem: given a finite sequence of graphs and a corresponding vector of natural numbers, decide whether there exists a graph G that realises the given vector as the homomorphism counts from the given graphs. We show that this problem yields a natural example of an NP#P-hard problem, which still can be NP-hard when restricted to a fixed number of input graphs of bounded treewidth and a fixed input vector of natural numbers, or alternatively, when restricted to a finite input set of graphs. We further show that, when restricted to a finite input set of graphs and given an upper bound on the order of the graph G as additional input, the problem cannot be NP-hard unless P = NP. For this regime, we obtain partial positive results. We also investigate the problem's parameterised complexity and provide fpt-algorithms for the case that a single graph is given and that multiple graphs of the same order with subgraph instead of homomorphism counts are given.
In recent years the framework of learning from label proportions (LLP) has been gaining importance in machine learning. In this setting, the training examples are aggregated into subsets or bags and only the average label per bag is available for learning an example-level predictor. This generalizes traditional PAC learning which is the special case of unit-sized bags. The computational learning aspects of LLP were studied in recent works (Saket, NeurIPS'21; Saket, NeurIPS'22) which showed algorithms and hardness for learning halfspaces in the LLP setting. In this work we focus on the intractability of LLP learning Boolean functions. Our first result shows that given a collection of bags of size at most $2$ which are consistent with an OR function, it is NP-hard to find a CNF of constantly many clauses which satisfies any constant-fraction of the bags. This is in contrast with the work of (Saket, NeurIPS'21) which gave a $(2/5)$-approximation for learning ORs using a halfspace. Thus, our result provides a separation between constant clause CNFs and halfspaces as hypotheses for LLP learning ORs. Next, we prove the hardness of satisfying more than $1/2 + o(1)$ fraction of such bags using a $t$-DNF (i.e. DNF where each term has $\leq t$ literals) for any constant $t$. In usual PAC learning such a hardness was known (Khot-Saket, FOCS'08) only for learning noisy ORs. We also study the learnability of parities and show that it is NP-hard to satisfy more than $(q/2^{q-1} + o(1))$-fraction of $q$-sized bags which are consistent with a parity using a parity, while a random parity based algorithm achieves a $(1/2^{q-2})$-approximation.
The 2-Orthogonal Vectors (2-0V) problem is the following: given two tuples A and B of n Boolean vectors, each of dimension d, decide if there exist vectors u E A, and v is an element of B, such that u and v are orthogonal. This problem, and its generalization k-OV defined analogously for k tuples, are central problems in the area of fine-grained complexity. One of the major conjectures in fine-grained complexity is that k-OV cannot be solved by a randomised algorithm in 11k-'poly(d) time for any constant epsilon > 0. In this paper, we are interested in unconditional lower bounds against k-epsilon, but for weaker models of computation than the general Turing Machine. In particular, we are interested in circuit lower bounds to computing k-OV by Boolean circuit families of depth 3 of the form OR-AND -OR, or equivalently, a disjunction of CNFs. We show that for all k <= d, any disjunction of t-CNFs computing k-OV requires size Omega((n/t)(k)). In particular, when k is a constant, any disjunction of k-CNFs computing k-OV needs to use Omega(n(k)) CNFs. This matches the brute-force construction, and for each fixed k > 2, this is the first unconditional Omega(n(k)) lower bound against k-OV for a computation model that can compute it in size O(n(k)). Our results partially resolve a conjecture by Kane and Williams [17] (page 12, conjecture 10) about depth-3 AC(0) circuits computing 2-OV. As a secondary result, we show an exponential lower bound on the size of AND o OR o AND circuits computing 2-OV when d is very large. Since 2-OV reduces to k-OV by projections trivially, this lower bound works against k-OV as well.
Generalised hypertree width ($ghw$) is a hypergraph parameter that is central to the tractability of many prominent problems with natural hypergraph structure. Computing $ghw$ of a hypergraph is notoriously hard. The decision version of the problem, checking whether $ghw(H) \leq k$, is paraNP-hard when parameterised by $k$. Furthermore, approximation of $ghw$ is at least as hard as approximation of Set-Cover, which is known to not admit any fpt approximation algorithms. Research in the computation of ghw so far has focused on identifying structural restrictions to hypergraphs -- such as bounds on the size of edge intersections -- that permit XP algorithms for $ghw$. Yet, even under these restrictions that problem has so far evaded any kind of fpt algorithm. In this paper we make the first step towards fpt algorithms for $ghw$ by showing that the parameter can be approximated in fpt time for graphs of bounded edge intersection size. In concrete terms we show that there exists an fpt algorithm, parameterised by $k$ and $d$, that for input hypergraph $H$ with maximal cardinality of edge intersections $d$ and integer $k$ either outputs a tree decomposition with $ghw(H) \leq 4k(k+d+1+)(2k-1)$, or rejects, in which case it is guaranteed that $ghw(H) > k$. Thus, in the special case, of hypergraphs of bounded edge intersection, we obtain an fpt $O(k^3)$-approximation algorithm for $ghw$.
Vertex splitting is a graph operation that replaces a vertex v with two nonadjacent new vertices u, w and makes each neighbor of v adjacent with one or both of u or w. Vertex splitting has been used in contexts from circuit design to statistical analysis. In this work, we generalize from specific vertex-splitting problems and systematically explore the computational complexity of achieving a given graph property Pi by a limited number of vertex splits, formalized as the problem Pi VERTEX SPLITTING (Pi-VS). We focus on hereditary graph properties and contribute four groups of results: First, we classify the classical complexity of Pi-VS for graph properties characterized by forbidden subgraphs of order at most 3. Second, we provide a framework that allows one to show NP-completeness whenever one can construct a combination of a forbidden subgraph and prescribed vertex splits that satisfy certain conditions. Using this framework we show NP-completeness when Pi is characterized by sufficiently well-connected forbidden subgraphs. In particular, we show that F-FREE-VS is NP-complete for each biconnected graph F. Third, we study infinite families of forbidden subgraphs, obtaining NP-completeness for BIPARTITE-VS and PERFECT-VS, contrasting the known result that Pi-VS is in P if Pi is the set of all cycles. Finally, we contribute to the study of the parameterized complexity of Pi-VS with respect to the number of allowed splits. We show para-NP-hardness for K-3-FREE-VS and derive an XP-algorithm when each vertex is only allowed to be split at most once, showing that the ability to split a vertex more than once is a key driver of the problems' complexity.
In this paper we study the interactions between so-called fractional relaxations of the integer programs (IPs) which encode homomorphism and isomorphism of relational structures. We give a combinatorial characterization of a certain natural linear programming (LP) relaxation of homomorphism in terms of fractional isomorphism. As a result, we show that the families of constraint satisfaction problems (CSPs) that are solvable by such linear program are precisely those that are closed under an equivalence relation which we call Weisfeiler-Leman invariance. We also generalize this result to the much broader framework of Promise Valued Constraint Satisfaction Problems, which brings together two well-studied extensions of the CSP framework. Finally, we consider the hierarchies of increasingly tighter relaxations of the homomorphism and isomorphism IPs obtained by applying the Sherali-Adams and Weisfeiler-Leman methods respectively. We extend our combinatorial characterization of the basic LP to higher levels of the Sherali-Adams hierarchy, and we generalize a well-known logical characterization of the Weisfeiler-Leman test from graphs to relational structures.
Arithmetic circuits are a natural well-studied model for computing multivariate polynomials over a field. In this article, we study planar arithmetic circuits. These are circuits whose underlying graph is planar. In particular, we prove an Q(n log n) lower bound on the size of planar arithmetic circuits computing explicit bilinear forms on 2n variables. As a consequence, we get an Q(n log n) lower bound on the size of arithmetic formulas and planar algebraic branching programs computing explicit bilinear forms on 2n variables. This is the first such lower bound on the formula complexity of an explicit bilinear form. In the case of read-once planar circuits, we show Q(n2) size lower bounds for computing explicit bilinear forms on 2n variables. Furthermore, we prove fine separations between the various planar models of computations mentioned above. In addition to this, we look at multi-output planar circuits and show Q(n4/3) size lower bound for computing an explicit linear transformation on n-variables. For a suitable definition of multi-output formulas, we extend the above result to get an Q(n2/ log n) size lower bound. As a consequence, we demonstrate that there exists an n-variate polynomial computable by an n1+o(1)-sized formula such that any multi-output planar circuit (resp., multi-output formula) simultaneously computing all its first-order partial derivatives requires size Q(n4/3) (resp., Q(n2/ log n)). This shows that a statement analogous to that of Baur, Strassen [3] does not hold in the case of planar circuits and formulas.
The online manipulation-resilient testing model, proposed by Kalemaj, Raskhodnikova and Varma (ITCS 2022 and Theory of Computing 2023), studies property testing in situations where access to the input degrades continuously and adversarially. Specifically, after each query made by the tester is answered, the adversary can intervene and either erase or corrupt t data points. In this work, we investigate a more nuanced version of the online model in order to overcome old and new impossibility results for the original model. We start by presenting an optimal tester for linearity and a lower bound for low-degree testing of Boolean functions in the original model. We overcome the lower bound by allowing batch queries, where the tester gets a group of queries answered between manipulations of the data. Our batch size is small enough so that function values for a single batch on their own give no information about whether the function is of low degree. Finally, to overcome the impossibility results of Kalemaj et al. for sortedness and the Lipschitz property of sequences, we extend the model to include t<1, i.e., adversaries that make less than one erasure per query. For sortedness, we characterize the rate of erasures for which online testing can be performed, exhibiting a sharp transition from optimal query complexity to impossibility of testability (with any number of queries). Our online tester works for a general class of local properties of sequences. One feature of our results is that we get new (and in some cases, simpler) optimal algorithms for several properties in the standard property testing model.
