Proceedings Of STOC Research Articles

Approximately Counting and Sampling Small Witnesses Using a Colorful Decision Oracle

In this paper, we design efficient algorithms to approximately count the number of edges of a given $k$-hypergraph, and to sample an approximately uniform random edge. The hypergraph is not given explicitly and can be accessed only through its colorful independence oracle: The colorful independence oracle returns yes or no depending on whether a given subset of the vertices contains an edge that is colorful with respect to a given vertex-coloring. Our results extend and/or strengthen recent results in the graph oracle literature due to Beame et al. [ACM Trans. Algorithms, 16 (2020), 52], Dell and Lapinskas [Proceedings of STOC, ACM, 2018, pp. 281--288], and Bhattacharya et al. [Proceedings of ISAAC, 2019]. Our results have consequences for approximate counting/sampling: We can turn certain kinds of decision algorithms into approximate counting/sampling algorithms without causing much overhead in the running time. We apply this approximate counting/sampling-to-decision reduction to key problems in fine-grained complexity (such as $k$-SUM, $k$-OV, and weighted $k$-Clique) and parameterized complexity (such as induced subgraphs of size $k$ or weight-$k$ solutions to constraint satisfaction problems).

A Faster Isomorphism Test for Graphs of Small Degree

In a recent breakthrough, Babai [Proceedings of STOC, ACM, New York, 2016, pp. 684--697] gave a quasipolynomial-time graph isomorphism test. In this work, we give an improved isomorphism test for g...

An Almost Optimal Algorithm for Computing Nonnegative Rank

Here, we give an algorithm for deciding if the nonnegative rank of a matrix $M$ of dimension $m \times n$ is at most $r$ which runs in time $(nm)^{O(r^2)}$. This is the first exact algorithm that runs in time singly exponential in $r$. This algorithm (and earlier algorithms) are built on methods for finding a solution to a system of polynomial inequalities (if one exists). Notably, the best algorithms for this task run in time exponential in the number of variables but polynomial in all of the other parameters (the number of inequalities and the maximum degree). Hence, these algorithms motivate natural algebraic questions whose solution have immediate algorithmic implications: How many variables do we need to represent the decision problem, and does $M$ have nonnegative rank at most $r$? A naive formulation uses $nr + mr$ variables and yields an algorithm that is exponential in $n$ and $m$ even for constant $r$. Arora et al. [Proceedings of STOC, 2012, pp. 145--162] recently reduced the number of variables to $2r^2 2^r$, and here we exponentially reduce the number of variables to $2r^2$ and this yields our main algorithm. In fact, the algorithm that we obtain is nearly optimal (under the exponential time hypothesis) since an algorithm that runs in time $(nm)^{o(r)}$ would yield a subexponential algorithm for $3$-SAT [Proceedings of STOC, 2012, pp. 145--162]. Our main result is based on establishing a normal form for nonnegative matrix factorization---which in turn allows us to exploit algebraic dependence among a large collection of linear transformations with variable entries. Additionally, we also demonstrate that nonnegative rank cannot be certified by even a very large submatrix of $M$, and this property also follows from the intuition gained from viewing nonnegative rank through the lens of systems of polynomial inequalities.

Fully Dynamic Maximal Matching in $O(\log n)$ Update Time

We present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our algorithm is randomized and it takes expected amortized $O(\log n)$ time for each edge update, where $n$ is the number of vertices in the graph. While there exists a trivial $O(n)$ time algorithm for each edge update, the previous best known result for this problem is due to Ivkovicź and Lloyd [Lecture Notes in Comput. Sci. 790, Springer-Verlag, London, 1994, pp. 99--111]. For a graph with $n$ vertices and $m$ edges, they gave an $O( {(n+ m)}^{0.7072})$ update time algorithm which is sublinear only for a sparse graph. For the related problem of maximum matching, Onak and Rubinfeld [Proceedings of STOC'10, Cambridge, MA, 2010, pp. 457--464] designed a randomized algorithm that achieves expected amortized $O(\log^2 n)$ time for each update for maintaining a $c$-approximate maximum matching for some unspecified large constant $c$. In contrast, we can maintain a factor 2 approximate maximum matching in expected amortized $O(\log n )$ time per update as a direct corollary of the maximal matching scheme. This in turn also implies a 2-approximate vertex cover maintenance scheme that takes expected amortized $O(\log n )$ time per update.

Quasi-Polynomial Time Approximation Scheme for Weighted Geometric Set Cover on Pseudodisks and Halfspaces

Weighted geometric set-cover problems arise naturally in several geometric and nongeometric settings (e.g., the breakthrough of Bansal and Pruhs [Proceedings of FOCS, 2010, pp. 407--414] reduces a wide class of machine scheduling problems to weighted geometric set cover). More than two decades of research has succeeded in settling the $(1+\epsilon)$-approximability status for most geometric set-cover problems, except for some basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan [Proceedings of STOC'10, 2010, pp. 641--648] presented a clever quasi-sampling technique, which together with improvements by Chan et al. [Proceedings of SODA, 2012, pp. 1576--1585], yielded an $O(1)$-approximation algorithm. Even for the unweighted case, a polynomial time approximation scheme (PTAS) for a fundamental class of objects called pseudodisks (which includes halfspaces, disks, unit-height rectangles, translates of convex sets, etc.) is curren...

Fixed-Parameter Tractability of Multicut in Directed Acyclic Graphs

The Multicut problem, given a graph G, a set of terminal pairs $\mathcal{T}=\{(s_i,t_i)\ |\ 1\leq i\leq r\}$, and an integer $p$, asks whether one can find a cutset consisting of at most $p$ nonterminal vertices that separates all the terminal pairs, i.e., after removing the cutset, $t_i$ is not reachable from $s_i$ for each $1\leq i\leq r$. The fixed-parameter tractability of Multicut in undirected graphs, parameterized by the size of the cutset only, has been recently proved by Marx and Razgon [SIAM J. Comput., 43 (2014), pp. 355--388] and, independently, by Bousquet, Daligault, and Thomassé [Proceedings of STOC, ACM, 2011, pp. 459--468], after resisting attacks as a long-standing open problem. In this paper we prove that Multicut is fixed-parameter tractable on directed acyclic graphs when parameterized both by the size of the cutset and the number of terminal pairs. We complement this result by showing that this is implausible for parameterization by the size of the cutset only, as this version of the problem remains $W[1]$-hard.

Lower Bounds for Depth-4 Formulas Computing Iterated Matrix Multiplication

We study the arithmetic complexity of iterated matrix multiplication. We show that any multilinear homogeneous depth-4 arithmetic formula computing the product of $d$ generic matrices of size $n \times n$, $\mathrm{IMM}_{n,d}$, has size $n^{\Omega(\sqrt{d})}$ as long as $d = n^{O(1)}$. This improves the result of Nisan and Wigderson [Comput. Complexity, 6 (1997), pp. 217--234] for depth-4 set-multilinear formulas. We also study $\Sigma\Pi^{[O(d/t)]}\Sigma\Pi^{[t]}$ formulas, which are depth-4 formulas with the stated bounds on the fan-ins of the $\Pi$ gates. A recent depth reduction result of Tavenas [Lecture Notes in Comput. Sci. 8087, 2013, pp. 813--824] shows that any $n$-variate degree $d = n^{O(1)}$ polynomial computable by a circuit of size $\mathop{\mathrm{poly}}(n)$ can also be computed by a depth-4 $\Sigma\Pi^{[O(d/t)]}\Sigma\Pi^{[t]}$ formula of top fan-in $n^{O(d/t)}$. We show that any such formula computing $\mathrm{IMM}_{n,d}$ has top fan-in $n^{\Omega({d/t})}$, proving the optimality of Tavenas' result. This also strengthens a result of Kayal, Saha, and Saptharishi [Proceedings of STOC, 2014, pp. 146--153], which gives a similar lower bound for an explicit polynomial in VNP.

Impossibility of Differentially Private Universally Optimal Mechanisms

The notion of a universally utility-maximizing privacy mechanism was introduced by Ghosh, Roughgarden, and Sundararajan [Proceedings of STOC, 2009]. These are mechanisms that guarantee optimal utility to a large class of information consumers, simultaneously, while preserving privacy. They demonstrated, quite surprisingly, a case where such a universally utility-maximizing privacy mechanism exists, when the information consumers are Bayesian. This result was later extended by Gupte and Sundararajan [Proceedings of PODS, 2010] to risk-averse consumers. Both positive results deal with mechanisms (approximately) computing a single count query (i.e., the number of individuals satisfying a specific property in a given population). We show that such universally optimal mechanisms do not exist for some natural extensions of count queries, both for Bayesian and risk-averse consumers. For the Bayesian case, we go further and give a characterization of those functions that admit universally optimal mechanisms, showing that a universally optimal mechanism exists, essentially, only for a (single) count query. At the heart of our proof is a representation of a query function by its privacy constraint graph whose edges correspond to values resulting by applying the query function to neighboring databases.

The complexity of temporal constraint satisfaction problems

A temporal constraint language is a set of relations that has a first-order definition in(Q;<), the dense linear order of the rational numbers. We present a complete complexity classification of the constraint satisfaction problem (CSP) for temporal constraint languages: if the constraint language is contained in one out of nine temporal constraint languages, then the CSP can be solved in polynomial time; otherwise, the CSP is NP-complete. Our proof combines model-theoretic concepts with techniques from universal algebra, and also applies the so-called product Ramsey theorem, which we believe will useful in similar contexts of constraint satisfaction complexity classification. An extended abstract of this article appeared in the proceedings of STOC'08.

Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms

There exist fast variants of the gcd algorithm which are all based on principles due to Knuth and Schönhage. On inputs of size n , these algorithms use a Divide and Conquer approach, perform FFT multiplications with complexity μ ( n ) and stop the recursion at a depth slightly smaller than lg n . A rough estimate of the worst-case complexity of these fast versions provides the bound O ( μ ( n ) log n ) . Even the worst-case estimate is partly based on heuristics and is not actually proven. Here, we provide a precise probabilistic analysis of some of these fast variants, and we prove that their average bit-complexity on random inputs of size n is Θ ( μ ( n ) log n ) , with a precise remainder term, and estimates of the constant in the Θ -term. Our analysis applies to any cases when the cost μ ( n ) is of order Ω ( n log n ) , and is valid both for the FFT multiplication algorithm of Schönhage–Strassen, but also for the new algorithm introduced quite recently by Fürer [Fürer, M., 2007. Faster integer Multiplication. In: Proceedings of STOC’07. pp. 57–66]. We view such a fast algorithm as a sequence of what we call interrupted algorithms, and we obtain two main results about the (plain) Euclid Algorithm, which are of independent interest. We precisely describe the evolution of the distribution of numbers during the execution of the (plain) Euclid Algorithm, and we exhibit an (unexpected) density ψ which plays a central rôle since it always appears at the beginning of each recursive call. This strong regularity phenomenon proves that the interrupted algorithms are locally “similar” to the total algorithm. This ultimately leads to the precise evaluation of the average bit-complexity of these fast algorithms. This work uses various tools, and is based on a precise study of generalised transfer operators related to the dynamical system underlying the Euclid Algorithm.

A faster off-line algorithm for the TCP acknowledgement problem

In a recent paper [Proceedings of STOC'98, 1998, pp. 389–398], Dooly, Goldman and Scott study a problem that is motivated by the networking problem of dynamically adjusting delays of acknowledgements in the Transmission Control Protocol (TCP). Among other results, they give an O( n 2) off-line algorithm for computing the optimal way of acknowledging n packet arrivals and departures. In this brief note, we observe that there is a faster off-line algorithm for this problem with time complexity O( n).