Journal of the ACM (JACM)

Komargodski, Ilan; Naor, Moni; Yogev, Eylon

doi: 10.1145/3341106pmid: N/A

Ramsey theory assures us that in any graph there is a clique or independent set of a certain size, roughly logarithmic in the graph size. But how difficult is it to find the clique or independent set? If the graph is given explicitly, then it is possible to do so while examining a linear number of edges. If the graph is given by a black-box, where to figure out whether a certain edge exists the box should be queried, then a large number of queries must be issued. But what if one is given a program or circuit for computing the existence of an edge? This problem was raised by Buss and Goldberg and Papadimitriou in the context of TFNP, search problems with a guaranteed solution. We examine the relationship between black-box complexity and white-box complexity for search problems with guaranteed solution such as the above Ramsey problem. We show that under the assumption that collision-resistant hash function exists (which follows from the hardness of problems such as factoring, discrete-log, and learning with errors) the white-box Ramsey problem is hard and this is true even if one is looking for a much smaller clique or independent set than the theorem guarantees. This is also true for the colorful Ramsey problem where one is looking, say, for a monochromatic triangle. In general, one cannot hope to translate all black-box hardness for TFNP into white-box hardness: we show this by adapting results concerning the random oracle methodology and the impossibility of instantiating it. Another model we consider is that of succinct black-box, where the complexity of an algorithm is measured as a function of the description size of the object in the box (and no limitation on the computation time). In this case, we show that for all TFNP problems there is an efficient algorithm with complexity proportional to the description size of the object in the box times the solution size. However, for promise problems this is not the case. Finally, we consider the complexity of graph property testing in the white-box model. We show a property that is hard to test even when one is given the program for computing the graph (under the appropriate assumptions such as hardness of Decisional Diffie-Hellman). The hard property is whether the graph is a two-source extractor.

Bernstein, Aaron; Holm, Jacob; Rotenberg, Eva

doi: 10.1145/3344999pmid: N/A

In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one-by-one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O(log 2 n) replacements per insertion, where n is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for any replacement strategy, almost matching the (log n) lower bound. The previous best strategy known achieved amortized O(√ n) replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than the trivial O(n) bound was known except in special cases. Our analysis immediately implies the same upper bound of O(log 2 n) reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices. We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load L, then the SAP protocol makes amortized O(min L log2 n , √ nlog n) reassignments. We also show that this is close to tight, because (min L, √ n) reassignments can be necessary.

Lenzen, Christoph; Rybicki, Joel

doi: 10.1145/3339471pmid: N/A

We give fault-tolerant algorithms for establishing synchrony in distributed systems in which each of the n nodes has its own clock. Our algorithms operate in a very strong fault model: we require self-stabilisation, i.e., the initial state of the system may be arbitrary, and there can be up to f< n/3 ongoing Byzantine faults, i.e., nodes that deviate from the protocol in an arbitrary manner. Furthermore, we assume that the local clocks of the nodes may progress at different speeds (clock drift) and communication has bounded delay. In this model, we study the pulse synchronisation problem, where the task is to guarantee that eventually all correct nodes generate well-separated local pulse events (i.e., unlabelled logical clock ticks) in a synchronised manner. Compared to prior work, we achieve exponential improvements in stabilisation time and the number of communicated bits, and give the first sublinear-time algorithm for the problem: In the deterministic setting, the state-of-the-art solutions stabilise in time (f) and have each node broadcast (f log f) bits per time unit. We exponentially reduce the number of bits broadcasted per time unit to (log f) while retaining the same stabilisation time. In the randomised setting, the state-of-the-art solutions stabilise in time (f) and have each node broadcast O(1) bits per time unit. We exponentially reduce the stabilisation time to polylog f while each node broadcasts polylog f bits per time unit. These results are obtained by means of a recursive approach reducing the above task of self-stabilising pulse synchronisation in the bounded-delay model to non-self-stabilising binary consensus in the synchronous model. In general, our approach introduces at most logarithmic overheads in terms of stabilisation time and broadcasted bits over the underlying consensus routine.

Romashchenko, Andrei; Zimand, Marius

doi: 10.1145/3356867pmid: N/A

We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal, up to logarithmic precision, to the length of the longest shared secret key that two parties—one having x and the complexity profile of the pair and the other one having y and the complexity profile of the pair—can establish via a probabilistic protocol with interaction on a public channel. For > 2, the longest shared secret that can be established from a tuple of strings (x1, , x) by parties—each one having one component of the tuple and the complexity profile of the tuple—is equal, up to logarithmic precision, to the complexity of the tuple minus the minimum communication necessary for distributing the tuple to all parties. We establish the communication complexity of secret key agreement protocols that produce a secret key of maximal length for protocols with public randomness. We also show that if the communication complexity drops below the established threshold, then only very short secret keys can be obtained.

Iwata, Satoru; Oki, Taihei; Takamatsu, Mizuyo

doi: 10.1145/3341499pmid: N/A

Differential-algebraic equations (DAEs) are widely used for the modeling of dynamical systems. The difficulty in numerically solving a DAE is measured by its differentiation index. For highly accurate simulation of dynamical systems, it is important to convert high-index DAEs into low-index DAEs. Most of the existing simulation software packages for dynamical systems are equipped with an index-reduction algorithm given by Mattsson and Söderlind. Unfortunately, this algorithm fails if there are numerical cancellations. These numerical cancellations are often caused by accurate constants in structural equations. Distinguishing those accurate constants from generic parameters that represent physical quantities, Murota and Iri introduced the notion of a mixed matrix as a mathematical tool for faithful model description in a structural approach to systems analysis. For DAEs described with the use of mixed matrices, efficient algorithms to compute the index have been developed by exploiting matroid theory. This article presents an index-reduction algorithm for linear DAEs whose coefficient matrices are mixed matrices, i.e., linear DAEs containing physical quantities as parameters. Our algorithm detects numerical cancellations between accurate constants and transforms a DAE into an equivalent DAE to which Mattsson–Söderlind’s index-reduction algorithm is applicable. Our algorithm is based on the combinatorial relaxation approach, which is a framework to solve a linear algebraic problem by iteratively relaxing it into an efficiently solvable combinatorial optimization problem. The algorithm does not rely on symbolic manipulations but on fast combinatorial algorithms on graphs and matroids. Our algorithm is proved to work for any linear DAEs whose coefficient matrices are mixed matrices. Furthermore, we provide an improved algorithm under an assumption based on dimensional analysis of dynamical systems. Through numerical experiments, it is confirmed that our algorithms run sufficiently fast for large-scale DAEs and output DAEs such that physical meanings of coefficients are easy to interpret. Our algorithms can also be applied to nonlinear DAEs by regarding nonlinear terms as parameters.

Harris, David G.; Srinivasan, Aravind

doi: 10.1145/3342222pmid: N/A

The resampling algorithm of Moser and Tardos is a powerful approach to develop constructive versions of the Lovász Local Lemma. We generalize this to partial resampling: When a bad event holds, we resample an appropriately random subset of the variables that define this event rather than the entire set, as in Moser and Tardos. This is particularly useful when the bad events are determined by sums of random variables. This leads to several improved algorithmic applications in scheduling, graph transversals, packet routing, and so on. For instance, we settle a conjecture of Szabó and Tardos (2006) on graph transversals asymptotically and obtain improved approximation ratios for a packet routing problem of Leighton, Maggs, and Rao (1994).

S., Karthik C.; Laekhanukit, Bundit; Manurangsi, Pasin

doi: 10.1145/3325116pmid: N/A

We study the parameterized complexity of approximating the k-Dominating Set (DomSet) problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a dominating set of size at most F(k) k whenever the graph G has a dominating set of size k. When such an algorithm runs in time T(k) poly (n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for k-DomSet. Whether such an algorithm exists is listed in the seminal book of Downey and Fellows (2013) as one of the “most infamous” open problems in parameterized complexity. This work gives an almost complete answer to this question by showing the non-existence of such an algorithm under W[1] FPT and further providing tighter running time lower bounds under stronger hypotheses. Specifically, we prove the following for every computable functions T, F and every constant ε > 0: Assuming W[1] FPT, there is no F(k)-FPT-approximation algorithm for k-DomSet. Assuming the Exponential Time Hypothesis (ETH), there is no F(k)-approximation algorithm for k-DomSet that runs in T(k) no(k) time. Assuming the Strong Exponential Time Hypothesis (SETH), for every integer k 2, there is no F(k)-approximation algorithm for k-DomSet that runs in T(k) nk ε time. Assuming the k-SUM Hypothesis, for every integer k 3, there is no F(k)-approximation algorithm for k-DomSet that runs in T(k) n k/2 ε time. Previously, only constant ratio FPT-approximation algorithms were ruled out under sf W[1] FPT and (log1/4 &minus ε k)-FPT-approximation algorithms were ruled out under ETH [Chen and Lin, FOCS 2016]. Recently, the non-existence of an F(k)-FPT-approximation algorithm for any function F was shown under Gap-ETH [Chalermsook et al., FOCS 2017]. Note that, to the best of our knowledge, no running time lower bound of the form n&delta k for any absolute constant > 0 was known before even for any constant factor inapproximation ratio. Our results are obtained by establishing a connection between communication complexity and hardness of approximation, generalizing the ideas from a recent breakthrough work of Abboud et al. [FOCS 2017]. Specifically, we show that to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis we rely on. Each of these communication problems turns out to be either a well-studied problem or a variant of one; this allows us to easily apply known techniques to solve them.