331 research outputs found

    The Complexity of Rationalizing Network Formation

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    We study the complexity of rationalizing network formation. In this problem we fix an underlying model describing how selfish parties (the vertices) produce a graph by making individual decisions to form or not form incident edges. The model is equipped with a notion of stability (or equilibrium), and we observe a set of "snapshots" of graphs that are assumed to be stable. From this we would like to infer some unobserved data about the system: edge prices, or how much each vertex values short paths to each other vertex. We study two rationalization problems arising from the network formation model of Jackson and Wolinsky [14]. When the goal is to infer edge prices, we observe that the rationalization problem is easy. The problem remains easy even when rationalizing prices do not exist and we instead wish to find prices that maximize the stability of the system. In contrast, when the edge prices are given and the goal is instead to infer valuations of each vertex by each other vertex, we prove that the rationalization problem becomes NP-hard. Our proof exposes a close connection between rationalization problems and the Inequality-SAT (I-SAT) problem. Finally and most significantly, we prove that an approximation version of this NP-complete rationalization problem is NP-hard to approximate to within better than a 1/2 ratio. This shows that the trivial algorithm of setting everyone's valuations to infinity (which rationalizes all the edges present in the input graphs) or to zero (which rationalizes all the non-edges present in the input graphs) is the best possible assuming P ≠ NP To do this we prove a tight (1/2 + δ) -approximation hardness for a variant of I-SAT in which all coefficients are non-negative. This in turn follows from a tight hardness result for MAX-LlN_(R_+) (linear equations over the reals, with non-negative coefficients), which we prove by a (non-trivial) modification of the recent result of Guruswami and Raghavendra [10] which achieved tight hardness for this problem without the non-negativity constraint. Our technical contributions regarding the hardness of I-SAT and MAX-LIN_(R_+) may be of independent interest, given the generality of these problem

    Non-Abelian Analogs of Lattice Rounding

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    Lattice rounding in Euclidean space can be viewed as finding the nearest point in the orbit of an action by a discrete group, relative to the norm inherited from the ambient space. Using this point of view, we initiate the study of non-abelian analogs of lattice rounding involving matrix groups. In one direction, we give an algorithm for solving a normed word problem when the inputs are random products over a basis set, and give theoretical justification for its success. In another direction, we prove a general inapproximability result which essentially rules out strong approximation algorithms (i.e., whose approximation factors depend only on dimension) analogous to LLL in the general case.Comment: 30 page

    Cryptography from tensor problems

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    We describe a new proposal for a trap-door one-way function. The new proposal belongs to the "multivariate quadratic" family but the trap-door is different from existing methods, and is simpler

    Max-3-Lin over Non-Abelian Groups with Universal Factor Graphs

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    Factor graph of an instance of a constraint satisfaction problem with n variables and m constraints is the bipartite graph between [m] and [n] describing which variable appears in which constraints. Thus, an instance of a CSP is completely defined by its factor graph and the list of predicates. We show inapproximability of Max-3-LIN over non-abelian groups (both in the perfect completeness case and in the imperfect completeness case), with the same inapproximability factor as in the general case, even when the factor graph is fixed. Along the way, we also show that these optimal hardness results hold even when we restrict the linear equations in the Max-3-LIN instances to the form x? y? z = g, where x,y,z are the variables and g is a group element. We use representation theory and Fourier analysis over non-abelian groups to analyze the reductions

    Computational Topology and the Unique Games Conjecture

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    Covering spaces of graphs have long been useful for studying expanders (as "graph lifts") and unique games (as the "label-extended graph"). In this paper we advocate for the thesis that there is a much deeper relationship between computational topology and the Unique Games Conjecture. Our starting point is Linial\u27s 2005 observation that the only known problems whose inapproximability is equivalent to the Unique Games Conjecture - Unique Games and Max-2Lin - are instances of Maximum Section of a Covering Space on graphs. We then observe that the reduction between these two problems (Khot-Kindler-Mossel-O\u27Donnell, FOCS \u2704; SICOMP \u2707) gives a well-defined map of covering spaces. We further prove that inapproximability for Maximum Section of a Covering Space on (cell decompositions of) closed 2-manifolds is also equivalent to the Unique Games Conjecture. This gives the first new "Unique Games-complete" problem in over a decade. Our results partially settle an open question of Chen and Freedman (SODA, 2010; Disc. Comput. Geom., 2011) from computational topology, by showing that their question is almost equivalent to the Unique Games Conjecture. (The main difference is that they ask for inapproximability over Z_2, and we show Unique Games-completeness over Z_k for large k.) This equivalence comes from the fact that when the structure group G of the covering space is Abelian - or more generally for principal G-bundles - Maximum Section of a G-Covering Space is the same as the well-studied problem of 1-Homology Localization. Although our most technically demanding result is an application of Unique Games to computational topology, we hope that our observations on the topological nature of the Unique Games Conjecture will lead to applications of algebraic topology to the Unique Games Conjecture in the future

    Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

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    Recent inapproximability results of Sly (2010), together with an approximation algorithm presented by Weitz (2006) establish a beautiful picture for the computational complexity of approximating the partition function of the hard-core model. Let λc(TΔ)\lambda_c(T_\Delta) denote the critical activity for the hard-model on the infinite Δ\Delta-regular tree. Weitz presented an FPTAS for the partition function when λ<λc(TΔ)\lambda<\lambda_c(T_\Delta) for graphs with constant maximum degree Δ\Delta. In contrast, Sly showed that for all Δ3\Delta\geq 3, there exists ϵΔ>0\epsilon_\Delta>0 such that (unless RP=NP) there is no FPRAS for approximating the partition function on graphs of maximum degree Δ\Delta for activities λ\lambda satisfying λc(TΔ)<λ<λc(TΔ)+ϵΔ\lambda_c(T_\Delta)<\lambda<\lambda_c(T_\Delta)+\epsilon_\Delta. We prove that a similar phenomenon holds for the antiferromagnetic Ising model. Recent results of Li et al. and Sinclair et al. extend Weitz's approach to any 2-spin model, which includes the antiferromagnetic Ising model, to yield an FPTAS for the partition function for all graphs of constant maximum degree Δ\Delta when the parameters of the model lie in the uniqueness regime of the infinite tree TΔT_\Delta. We prove the complementary result that for the antiferrogmanetic Ising model without external field that, unless RP=NP, for all Δ3\Delta\geq 3, there is no FPRAS for approximating the partition function on graphs of maximum degree Δ\Delta when the inverse temperature lies in the non-uniqueness regime of the infinite tree TΔT_\Delta. Our results extend to a region of the parameter space for general 2-spin models. Our proof works by relating certain second moment calculations for random Δ\Delta-regular bipartite graphs to the tree recursions used to establish the critical points on the infinite tree.Comment: Journal version (no changes

    On the complexity of finding and counting solution-free sets of integers

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    Given a linear equation L\mathcal{L}, a set AA of integers is L\mathcal{L}-free if AA does not contain any `non-trivial' solutions to L\mathcal{L}. This notion incorporates many central topics in combinatorial number theory such as sum-free and progression-free sets. In this paper we initiate the study of (parameterised) complexity questions involving L\mathcal{L}-free sets of integers. The main questions we consider involve deciding whether a finite set of integers AA has an L\mathcal{L}-free subset of a given size, and counting all such L\mathcal{L}-free subsets. We also raise a number of open problems.Comment: 27 page
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