Computer Science

There has been significant recent progress on algorithms for approximating graph spanners, i.e., algorithms which approximate the best spanner for a given input graph. Essentially all of these algorithms use the same basic LP relaxation, so a variety of papers have studied the limitations of this approach and proved integrality gaps for this LP in a variety of settings. We extend these results by showing that even the strongest lift-and-project methods cannot help significantly, by proving polynomial integrality gaps even for n Ω(ϵ) levels of the Lasserre hierarchy, for both the directed and undirected spanner problems. We also extend these integrality gaps to related problems, notably Directed Steiner Network and Shallow-Light Steiner Network.

We revisit known constructions of efficient learning algorithms from various notions of constructive circuit lower bounds such as distinguishers breaking pseudorandom generators or efficient witnessing algorithms which find errors of small circuits attempting to compute hard functions. As our main result we prove that if it is possible to find efficiently, in a particular interactive way, errors of many p-size circuits attempting to solve hard problems, then p-size circuits can be PAC learned over the uniform distribution with membership queries by circuits of subexponential size. The opposite implication holds as well. This provides a new characterisation of learning algorithms and extends the natural proofs barrier of Razborov and Rudich. The proof is based on a method of exploiting Nisan-Wigderson generators introduced by Krají?ek (2010) and used to analyze complexity of circuit lower bounds in bounded arithmetic. An interesting consequence of known constructions of learning algorithms from circuit lower bounds is a learning speedup of Oliveira and Santhanam (2016). We present an alternative proof of this phenomenon and discuss its potential to advance the program of hardness magnification.

We develop algorithms for writing a polynomial as sums of powers of low degree polynomials. Consider an n -variate degree- d polynomial f which can be written as f= c 1 Q m 1 +…+ c s Q m s , where each c i ∈ F × , Q i is a homogeneous polynomial of degree t , and tm=d . In this paper, we give a poly((ns ) t ) -time learning algorithm for finding the Q i 's given (black-box access to) f , if the Q ′ i s satisfy certain non-degeneracy conditions and n is larger than d 2 . The set of degenerate Q i 's (i.e., inputs for which the algorithm does not work) form a non-trivial variety and hence if the Q i 's are chosen according to any reasonable (full-dimensional) distribution, then they are non-degenerate with high probability (if s is not too large). Our algorithm is based on a scheme for obtaining a learning algorithm for an arithmetic circuit model from a lower bound for the same model, provided certain non-degeneracy conditions hold. The scheme reduces the learning problem to the problem of decomposing two vector spaces under the action of a set of linear operators, where the spaces and the operators are derived from the input circuit and the complexity measure used in a typical lower bound proof. The non-degeneracy conditions are certain restrictions on how the spaces decompose.

In this lecture note we give Liu-Chen-Servedio-Sheng-Xie's (LCSSX) lower bound for property testing in the non-adaptive distribution-free.

These lecture notes are intended as a supplement to Moore and Mertens' The Nature of Computation or as a standalone resource, and are available to anyone who wants to use them. Comments are welcome, and please let me know if you use these notes in a course. There are 61 exercises. I emphasize that automata are elementary playgrounds where we can explore the issues of deterministic and nondeterministic computation. Unlike P vs. NP, we can prove that nondeterminism is equivalent to determinism, or strictly more powerful than determinism, in finite-state and push-down automata respectively. I also correct several historical and aesthetic injustices: in particular, the Myhill-Nerode theorem and the idea of building minimal DFAs from equivalence classes of prefixes is restored to its rightful place above the Pumping Lemma for regular languages. I also discuss the Pumping Lemma for context-free languages, and briefly discuss counter automata, queue automata, and the connection between unambiguous context-free languages and algebraic generating functions.

We study the computational complexity of approximating the partition function of the ferromagnetic Ising model with the external field parameter λ on the unit circle in the complex plane. Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics, but have also been key in the recent deterministic approximation scheme for all |λ|≠1 by Liu, Sinclair, and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle, and on the tantalising question of what happens around λ=1 , where on one hand the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability, and on the other hand the presence of Lee-Yang zeros alludes to computational hardness. Our main result establishes a sharp computational transition at the point λ=1 , and more generally on the entire unit circle. For an integer Δ≥3 and edge interaction parameter b∈(0,1) we show #P-hardness for approximating the partition function on graphs of maximum degree Δ on the arc of the unit circle where the Lee-Yang zeros are dense. This result contrasts with known approximation algorithms when |λ|≠1 or when λ is in the complementary arc around 1 of the unit circle. Our work thus gives a direct connection between the presence/absence of Lee-Yang zeros and the tractability of efficiently approximating the partition function on bounded-degree graphs.

In this article, we study a variant of the minimum dominating set problem known as the minimum liar's dominating set (MLDS) problem. We prove that the MLDS problem is NP-hard in unit disk graphs. Next, we show that the recent sub-quadratic time 11 2 -factor approximation algorithm \cite{bhore} for the MLDS problem is erroneous and propose a simple O(n+m) time 7.31-factor approximation algorithm, where n and m are the number of vertices and edges in the input unit disk graph, respectively. Finally, we prove that the MLDS problem admits a polynomial-time approximation scheme.

We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve two open problems: * We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there are no refutations in subexponential length and subpolynomial line space if coefficients are restricted to be of polynomial magnitude. * We give the first explicit separation between monotone Boolean formulas and monotone real formulas. Specifically, we give an explicit family of functions that can be computed with monotone real formulas of nearly linear size but require monotone Boolean formulas of exponential size. Previously only a non-explicit separation was known. An important technical ingredient, which may be of independent interest, is that we show that the Nullstellensatz degree of refuting the pebbling formula over a DAG G over any field coincides exactly with the reversible pebbling price of G. In particular, this implies that the standard decision tree complexity and the parity decision tree complexity of the corresponding falsified clause search problem are equal.

Proving super polynomial size lower bounds for various classes of arithmetic circuits computing explicit polynomials is a very important and challenging task in algebraic complexity theory. We study representation of polynomials as sums of weaker models such as read once formulas (ROFs) and read once oblivious algebraic branching programs (ROABPs). We prove: (1) An exponential separation between sum of ROFs and read- k formulas for some constant k . (2) A sub-exponential separation between sum of ROABPs and syntactic multilinear ABPs. Our results are based on analysis of the partial derivative matrix under different distributions. These results highlight richness of bounded read restrictions in arithmetic formulas and ABPs. Finally, we consider a generalization of multilinear ROABPs known as strict-interval ABPs defined in [Ramya-Rao, MFCS2019]. We show that strict-interval ABPs are equivalent to ROABPs upto a polynomial size blow up. In contrast, we show that interval formulas are different from ROFs and also admit depth reduction which is not known in the case of strict-interval ABPs.

The k -dimensional Weisfeiler-Leman algorithm is a powerful tool in graph isomorphism testing. For an input graph G , the algorithm determines a canonical coloring of s -tuples of vertices of G for each s between 1 and k . We say that a numerical parameter of s -tuples is k -WL-invariant if it is determined by the tuple color. As an application of Dvořák's result on k -WL-invariance of homomorphism counts, we spot some non-obvious regularity properties of strongly regular graphs and related graph families. For example, if G is a strongly regular graph, then the number of paths of length 6 between vertices x and y in G depends only on whether or not x and y are adjacent (and the length 6 is here optimal). Or, the number of cycles of length 7 passing through a vertex x in G is the same for every x (where the length 7 is also optimal).