Computer Science

We consider the complexity properties of modern puzzle games, Hexiom, Cut the Rope and Back to Bed. The complexity of games plays an important role in the type of experience they provide to players. Back to Bed is shown to be PSPACE-Hard and the first two are shown to be NP-Hard. These results give further insight into the structure of these games and the resulting constructions may be useful in further complexity studies.

We develop a polynomial method on finite fields to amplify the hardness of spare sets in nondeterministic time complexity classes on a randomized streaming model. One of our results shows that if there exists a 2 n o(1) -sparse set in NTIME( 2 n o(1) ) that does not have any randomized streaming algorithm with n o(1) updating time, and n o(1) space, then NEXP≠BPP , where a f(n) -sparse set is a language that has at most f(n) strings of length n . We also show that if MCSP is ZPP -hard under polynomial time truth-table reductions, then EXP≠ZPP .

Goldmann and Russell (2002) initiated the study of the complexity of the equation satisfiability problem in finite groups by showing that it is in P for nilpotent groups while it is NP-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups of Fitting length two. In this work, we present the first lower bounds for the equation satisfiability problem in finite solvable groups: under the assumption of the exponential time hypothesis, we show that it cannot be in P for any group of Fitting length at least four and for certain groups of Fitting length three. Moreover, the same hardness result applies to the equation identity problem.

We study the complexity of finding an optimal hierarchical clustering of an unweighted similarity graph under the recently introduced Dasgupta objective function. We introduce a proof technique, called the normalization procedure, that takes any such clustering of a graph G and iteratively improves it until a desired target clustering of G is reached. We use this technique to show both a negative and a positive complexity result. Firstly, we show that in general the problem is NP-complete. Secondly, we consider min-well-behaved graphs, which are graphs H having the property that for any k the graph H(k) being the join of k copies of H has an optimal hierarchical clustering that splits each copy of H in the same optimal way. To optimally cluster such a graph H(k) we thus only need to optimally cluster the smaller graph H . Co-bipartite graphs are min-well-behaved, but otherwise they seem to be scarce. We use the normalization procedure to show that also the cycle on 6 vertices is min-well-behaved.

Higher order random walks (HD-walks) on high dimensional expanders have played a crucial role in a number of recent breakthroughs in theoretical computer science, perhaps most famously in the recent resolution of the Mihail-Vazirani conjecture (Anari et al. STOC 2019), which focuses on HD-walks on one-sided local-spectral expanders. In this work we study the spectral structure of walks on the stronger two-sided variant, which capture wide generalizations of important objects like the Johnson and Grassmann graphs. We prove that the spectra of these walks are tightly concentrated in a small number of strips, each of which corresponds combinatorially to a level in the underlying complex. Moreover, the eigenvalues corresponding to these strips decay exponentially with a measure we term the depth of the walk. Using this spectral machinery, we characterize the edge-expansion of small sets based upon the interplay of their local combinatorial structure and the global decay of the walk's eigenvalues across strips. Variants of this result for the special cases of the Johnson and Grassmann graphs were recently crucial both for the resolution of the 2-2 Games Conjecture (Khot et al. FOCS 2018), and for efficient algorithms for affine unique games over the Johnson graphs (Bafna et al. Arxiv 2020). For the complete complex, our characterization admits a low-degree Sum of Squares proof. Building on the work of Bafna et al., we provide the first polynomial time algorithm for affine unique games over the Johnson scheme. The soundness and runtime of our algorithm depend upon the number of strips with large eigenvalues, a measure we call High-Dimensional Threshold Rank that calls back to the seminal work of Barak, Raghavendra, and Steurer (FOCS 2011) on unique games and threshold rank.

In this paper we study polynomials in VP e (polynomial-sized formulas) and in Σ?Σ (polynomial-size depth- 3 circuits) whose orbits, under the action of the affine group GL aff n (F) , are dense in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms. As VP= VNC 2 , our results for VP e translate immediately to VP with a quasipolynomial blow up in parameters. If any of our hitting or interpolating sets could be made robust then this would immediately yield a hitting set for the superclass in which the relevant class is dense, and as a consequence also a lower bound for the superclass. Unfortunately, we also prove that the kind of constructions that we have found (which are defined in terms of k -independent polynomial maps) do not necessarily yield robust hitting sets.

We show that, while Minesweeper is NP-complete, its hyperbolic variant is in P. Our proof does not rely on the rules of Minesweeper, but is valid for any puzzle based on satisfying local constraints on a graph embedded in the hyperbolic plane.

The Ideal Membership Problem (IMP) tests if an input polynomial f∈F[ x 1 ,…, x n ] with coefficients from a field F belongs to a given ideal I⊆F[ x 1 ,…, x n ] . It is a well-known fundamental problem with many important applications, though notoriously intractable in the general case. In this paper we consider the IMP for polynomial ideals encoding combinatorial problems and where the input polynomial f has degree at most d=O(1) (we call this problem IMP d ). A dichotomy result between ``hard'' (NP-hard) and ``easy'' (polynomial time) IMPs was recently achieved for Constraint Satisfaction Problems over finite domains [Bulatov FOCS'17, Zhuk FOCS'17] (this is equivalent to IMP 0 ) and IMP d for the Boolean domain [Mastrolilli SODA'19], both based on the classification of the IMP through functions called polymorphisms. The complexity of the IMP d for five polymorphisms has been solved in [Mastrolilli SODA'19] whereas for the ternary minority polymorphism it was incorrectly declared to have been resolved by a previous result. As a matter of fact the complexity of the IMP d for the ternary minority polymorphism is open. In this paper we provide the missing link by proving that the IMP d for Boolean combinatorial ideals whose constraints are closed under the minority polymorphism can be solved in polynomial time. This result, along with the results in [Mastrolilli SODA'19], completes the identification of the precise borderline of tractability for the IMP d for constrained problems over the Boolean domain. This paper is motivated by the pursuit of understanding the issue of bit complexity of Sum-of-Squares proofs raised by O'Donnell [ITCS'17]. Raghavendra and Weitz [ICALP'17] show how the IMP d tractability for combinatorial ideals implies bounded coefficients in Sum-of-Squares proofs.

As it is well known, the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical 2-dimensional Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent Cai-Fürer-Immerman construction shows that even the multidimensional version of the algorithm does not suffice for graphs with color multiplicity 4. We give an efficient decision procedure that, given a graph G of color multiplicity 4, recognizes whether or not G is identifiable by 2-WL, that is, whether or not 2-WL distinguishes G from any non-isomorphic graph. In fact, we solve the much more general problem of recognizing whether or not a given coherent configuration of maximum fiber size 4 is separable. This extends our recognition algorithm to graphs of color multiplicity 4 with directed and colored edges. Our decision procedure is based on an explicit description of the class of graphs with color multiplicity 4 that are not identifiable by 2-WL. The Cai-Fürer-Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai-Fürer-Immerman construction is not ad hoc. Our classification reveals also other types of graphs that are hard for 2-WL. One of them arises from patterns known as ( n 3 ) -configurations in incidence geometry.

Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size. In a recent work of Chatterjee and the authors (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.