Search

The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The connectivity augmentation problem (CAP) is arguably one of the most basic problems in this area: given a k(-edge)-connected graph G and a set of extra edges (links), select a minimum cardinality subset A of links such that adding A to G increases its edge connectivity to k+1. Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation factor for this NP-hard problem is 2, and this can be achieved with multiple approaches (the first such result is in [G. N. Frederickson and J\'aj\'a, SIAM J. Comput., 10 (1981), pp. 270--283]. It is known [E. A. Dinitz, A. V. Karzanov, and M. V. Lomonosov, Studies in Discrete Optimization, Nauka, Moscow, 1976, pp. 290--306] that CAP can be reduced to the case k = 1, also known as the tree augmentation problem (TAP) for odd k, and to the case k = 2, also known as the cactus augmentation problem (CacAP) for even k. Prior to the conference version of this paper [J. Byrka, F. Grandoni, and A. Jabal Ameli, STOC'20, ACM, New York, 2020, pp. 815--825], several better than 2 approximation algorithms were known for TAP, culminating with a recent 1.458 approximation [F. Grandoni, C. Kalaitzis, and R. Zenklusen, STOC'18, ACM, New York, 1918, pp. 632--645]. However, for CacAP the best known approximation was 2. In this paper we breach the 2 approximation barrier for CacAP, hence, for CAP, by presenting a polynomial-time 2 ln(4) 967 1120 + \varepsilon < 1.91 approximation. From a technical point of view, our approach deviates quite substantially from previous work. In particular, the better-than-2 approximation algorithms for TAP either exploit greedy-style algorithms or are based on rounding carefully designed LPs. We instead use a reduction to the Steiner tree problem which was previously used in parameterized algorithms [Basavaraju et al., ICALP'14, Springer, Berlin, 2014, pp. 800--811]. This reduction is not approximation preserving, and using the current best approximation factor for a Steiner tree [Byrka et al., J. ACM, 60 (2013), 6] as a black box would not be good enough to improve on 2. To achieve the latter goal, we "open the box" and exploit the specific properties of the instances of a Steiner tree arising from CacAP. In our opinion this connection between approximation algorithms for survivable network design and Steiner-type problems is interesting, and might lead to other results in the area. [ABSTRACT FROM AUTHOR]

Abstract. In the distributed all-pairs shortest paths problem, every node in the weighted undirected distributed network (the CONGEST model) needs to know the distance from every other node using least number of communication rounds (typically called time complexity). The problem admits a (1 + (1))-approximation 9(n)-time algorithm and a nearly tight 12(n) lower bound [D. Nanongkai, STOC'14, ACM, New York, 2014, pp. 565-573; C. Lenzen and B. Patt-Shamir, PODC'15, ACM, New York, 2015, pp. 153-162]. (9, 0 and n hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios (C. Lenzen and D. Peleg, PODC, ACM, New York, 2013, pp. 375-382; S. Holzer and R. Wattenofer, PODC, ACM, New York, 2012, pp. 355-364; D. Peleg, L. Roditty, and E. Tal, ICALP, Springer, Berlin, 2012, pp. 660-672; D. Nanongkai, STOC, ACM, New York, 2014, pp. 565-573-672). For the exact case, Elkin [STOC'17, ACM, New York, 2017, pp. 757-790] presented an 0(n5/3 log2/3 n) time bound, which was later improved to O(n5/4) [C.-C. Huang, D. Nanongkai, T. Saranurak, FOCS'17, IEEE Computer Society, Los Alamitos, CA, 2017, pp. 168-179]. It was shown that any superlinear lower bound (in n) requires a new technique [K. Censor-Hillel, S. Khoury, A. Paz, DISC'17, LIPIcs Leibniz Int. Proc. Inform., Vol. 91, Schloss-Dagstuhl, Wadern, Germany, 2017, 10], but otherwise it remained widely open whether there exists a O(n)-time algorithm for the exact case, which would match the best possible approximation algorithm. This paper resolves this question positively: we present a randomized (Las Vegas) 0(n)-time algorithm, matching the lower bound up to polylogarithmic factors. Like the previous 0(n5/4) bound, our result works for directed graphs with zero (and even negative) edge weights. In addition to the improved running time, our algorithm works in a more general setting than that required by the previous 0(n5/4) bound; in our setting (i) the communication is only along edge directions (as opposed to bidirectional), and (ii) edge weights are arbitrary (as opposed to integers in {1, 2, ..., poly(n)}). As far as we know, ours is the first o(n²) algorithm that only requires unidirectional communication. For arbitrary weights, the previous state-of-the-art required 0(n4/3) time [U. Agarwal and V. Ramachandran, IPDPS 2019, IEEE Computer Society, Los Alamitos, CA, 2019, and SPAA 2020, ACM, New York, 2020, pp. 11-21]. Our algorithm is extremely simple and relies on a new technique called random filtered broadcast. Given any sets of nodes A, B C V and assuming that every b E B knows all distances from nodes in A, and every node v E V knows all distances from nodes in B, we want every v E V to know DistThroughB (a, v) = minbEB dist(a, b) + dist(b, v) for every a E A. Previous works typically solve this problem by broadcasting all knowledge of every b E B, causing superlinear edge congestion and time. We show a randomized algorithm that can reduce edge congestions and thus solve this problem in 0(n) expected time. [ABSTRACT FROM AUTHOR]

A classical branch of graph algorithms is graph transversals, where one seeks a minimum-weight subset of nodes in a node-weighted graph G which intersects all copies of subgraphs F from a fixed family\scrF. Many such graph transversal problems have been shown to admit polynomial-time approximation schemes (PTASs) for planar input graphs G, using a variety of techniques like the shifting technique [B. S. Baker, J. ACM, 41 (1994), pp. 153--180], bidimensionality [F. V. Fomin et al., Bidimensionality and EPTAS, in Proceedings of SODA 2011, ACM, New York, SIAM, Philadelphia, 2011, pp. 748--759], or connectivity domination [V. Cohen-Addad et al., Approximating connectivity domination in weighted bounded-genus graphs, in Proceedings of STOC 2016, ACM, New York, 2016, pp. 584--597]. These techniques do not seem to apply to graph transversals with parity constraints, which have recently received significant attention, but for which no PTASs are known. In the Even Cycle Transversal (ECT) problem, the goal is to find a minimum-weight hitting set for the set of even cycles in an undirected graph. For ECT, Fiorini, Joret, and Pietropaoli [Hitting diamonds and growing cacti, in Proceedings of IPCO 2010, Lecture Notes in Comput. Sci. 6080, Springer, Berlin, 2010, pp. 191--204] showed that the integrality gap of the standard covering LP relaxation is\Theta (log n), and that adding sparsity inequalities reduces the integrality gap to 10. Our main result is a primal-dual algorithm that yields a 47/7\approx 6.71-approximation for ECT on node-weighted planar graphs, and an integrality gap upper bound of the same value for the standard LP relaxation on node-weighted planar graphs. [ABSTRACT FROM AUTHOR]

Persuasion, defined as the act of exploiting an informational advantage in order to inuence the decisions of others, is ubiquitous. Indeed, persuasive communication has been estimated to account for almost a third of all economic activity in the U.S. This paper examines persuasion through a computational lens, focusing on what is perhaps the most basic and fundamental model in this space: the celebrated Bayesian persuasion model of Kamenica and Gentzkow [Am. Econ. Rev., 101 (2011), pp. 2590-2615]. Here there are two players, a sender and a receiver. The receiver must take one of a number of actions with an a priori unknown payoff, and the sender has access to additional information regarding the payoffs of the various actions for both players. The sender can commit to revealing a noisy signal regarding the realization of the payoffs of various actions, and would like to do so to maximize her own payoff in expectation assuming that the receiver rationally acts to maximize his own payoff. When the payoffs of various actions follow a joint distribution (the common prior), the sender's problem is nontrivial, and its computational complexity depends on the representation of this prior. We examine the sender's optimization task in three of the most natural input models for this problem, and essentially pin down its computational complexity in each. When the payoff distributions of the different actions are independently and identically distributed (i.i.d.) and given explicitly, we exhibit a polynomial-time (exact) algorithmic solution, and a "simple" (1-1=e)-approximation algorithm. Our optimal scheme for the i.i.d. setting involves an analogy to auction theory, and makes use of Border's characterization of the space of reduced-forms for single-item auctions. When action payoffs are independent but nonidentical with marginal distributions given explicitly, we show that it is #P-hard to compute the optimal expected sender utility. In doing so, we rule out a generalized Border's theorem, in the sense of Gopalan, Nisan, and Roughgarden [Public projects, boolean functions, and the borders of Border's theorem, in Proceedings of the Sixteenth ACM Conference on Economics and Computation, EC '15, ACM, New York, 2015, p. 395], for this setting. Finally, we consider a general (possibly correlated) joint distribution of action payoffs presented by a black box sampling oracle, and exhibit a fully polynomial-time approximation scheme (FPTAS) with a bicriteria guarantee. Our FPTAS is based on Monte Carlo sampling, and its analysis relies on the principle of deferred decisions. Moreover, we show that this result is the best possible in the black-box model for information-theoretic reasons. [ABSTRACT FROM AUTHOR]

We study a central problem in algorithmic mechanism design: constructing truthful mechanisms for welfare maximization in combinatorial auctions with submodular bidders. Dobzinski, Nisan, and Schapira provided the first mechanism that guarantees a nontrivial approximation ratio of O(log²m) [STOC'06, ACM, New York, 2006, pp. 644-652], where m is the number of items. This approximation ratio was subsequently improved to O(logmlog logm) [S. Dobzinski, APPROX'07, Springer, Berlin, 2007, pp. 89-103] and then to O(logm) [P. Krysta and B. Vöcking, ICALP'12, Springer, Heidelberg, 2012, pp. 636-647]. In this paper we develop the first mechanism that breaks the logarithmic barrier. Specifically, the mechanism provides an approximation ratio of O(√logm). Similarly to previous constructions, our mechanism uses polynomially many value and demand queries and, in fact, provides the same approximation ratio for the larger class of XOS (also known as fractionally subadditive) valuations. We also develop a computationally efficient implementation of the mechanism for combinatorial auctions with budget additive bidders. Although, in general, computing a demand query is NP-hard for budget additive valuations, we observe that the specific form of demand queries that our mechanism uses can be efficiently computed when bidders are budget additive. [ABSTRACT FROM AUTHOR]