Alexis Kaporis

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Papers by Alexis Kaporis

Algorithmic game theory : 5th international symposium, SAGT 2011, Barcelona, Spain, October 22 - 23, 2012 ; proceedings

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Dynamic Interpolation Search revisited

Information & Computation, Feb 1, 2020

A new dynamic Interpolation Search (IS) data structure is presented that achieves O (log log n) s... more A new dynamic Interpolation Search (IS) data structure is presented that achieves O (log log n) search time with high probability on unknown continuous or even discrete input distributions with measurable probability of element collisions, including power law and Binomial distributions. No such previous result holds for IS when the probability of element collisions is measurable. Moreover, our data structure exhibits O (1) search time with high probability (w.h.p.) for a wide class of input distributions that contains all those for which o(log log n) expected search time was previously known.

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The Probabilistic Analysis of a Greedy Satisfiability Algorithm

Lecture Notes in Computer Science, 2002

On input a random 3-CNF formula of clauses-to-variables ratio r3 apply repeatedly the following s... more On input a random 3-CNF formula of clauses-to-variables ratio r3 apply repeatedly the following simple heuristic: Set to True a literal that appears in the maximum number of clauses, irrespectively of their size and the number of occurrences of the negation of the literal (ties are broken randomly; 1-clauses when they appear get priority). We prove that for r3 < 3.42 this heuristic succeeds with probability asymptotically bounded away from zero. Previously, heuristics of increasing sophistication were shown to succeed for r3 < 3.26. We improve up to r3 < 3.52 by further exploiting the degree of the negation of the evaluated to True literal.

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Efficient Methods for Selfish Network Design

Lecture Notes in Computer Science, 2009

Intuitively, Braess's paradox states that destroying a part of a network may improve the common l... more Intuitively, Braess's paradox states that destroying a part of a network may improve the common latency of selfish flows at Nash equilibrium. Such a paradox is a pervasive phenomenon in real-world networks. Any administrator that wants to improve equilibrium delays in selfish networks, is facing some basic questions:-Is the network paradox-ridden?-How can we delete some edges to optimize equilibrium flow delays?-How can we modify edge latencies to optimize equilibrium flow delays? Unfortunately, such questions lead to NP-hard problems in general. In this work, we impose some natural restrictions on our networks, e.g. we assume strictly increasing linear latencies. Our target is to formulate efficient algorithms for the three questions above. We manage to provide:-A polynomial-time algorithm that decides if a network is paradox-ridden, when latencies are linear and strictly increasing.-A reduction of the problem of deciding if a network with (arbitrary) linear latencies is paradox-ridden to the problem of generating all optimal basic feasible solutions of a Linear Program that describes the optimal traffic allocations to the edges with constant latency.-An algorithm for finding a subnetwork that is almost optimal wrt equilibrium latency. Our algorithm is subexponential when the number of paths is polynomial and each path is of polylogarithmic length.-A polynomial-time algorithm for the problem of finding the best subnetwork which outperforms any known approximation for the case of strictly increasing linear latencies.-A polynomial-time method that turns the optimal flow into a Nash flow by deleting the edges not used by the optimal flow, and performing minimal modifications on the latencies of the remaining ones. Our results provide a deeper understanding of the computational complexity of recognizing the Braess's paradox most severe manifestations, and our techniques show novel ways of using the probabilistic method and of exploiting convex separable quadratic programs.

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Stackelberg Games: The Price of Optimum

Springer eBooks, 2014

Thresholds of Random k-Sat

Springer eBooks, 2016

Consider n Boolean variables V D fx1; : : : ; xng and the corresponding set of 2n literals L D fx... more Consider n Boolean variables V D fx1; : : : ; xng and the corresponding set of 2n literals L D fx1x1 : : : ; xn; xng. A k-clause is a disjunction of k literals of distinct underlying variables. A random formula n;m in k conjunctive normal form (k-CNF) is the conjunction of m clauses, each selected in a uniformly random and independent way among the 2 n k possible k-clauses on n variables in V . The density rk of a k-CNF formula n;m is the clauses-to-variables ratio m/n. It was conjectured that for each k 2 there exists a critical density rk such that asymptotically almost all (a.a.a.) k-CNF formulas with density r rk / are satisfiable (unsatisfiable, respectively). So far, the conjecture has been proved only for k D 2 [3, 11]. For k 3, the conjecture still remains open but is supported by experimental evidence [14] as well as by theoretical, but non-rigorous, work based on statistical physics [15]. The value of the putative threshold r3 is estimated to be around 4.27. Approximate values of the putative threshold for larger values of k have also been computed. As far as rigorous results are concerned, Friedgut [10] proved that for each k 3, there exists a sequence rk .n/ such that for any > 0, a.a.a. k-CNF formulas n; b.r k .n/ /nc ( n; b.r k .n/C /nc) are satisfiable (unsatisfiable, respectively). The convergence of the sequence rk .n/; n D 0; 1; : : : for k 3 remains open. Let now

The Impact of Social Ignorance on Weighted Congestion Games

Lecture Notes in Computer Science, 2009

We consider weighted linear congestion games, and investigate how social ignorance, namely lack o... more We consider weighted linear congestion games, and investigate how social ignorance, namely lack of information about the presence of some players, affects the inefficiency of pure Nash equilibria (PNE) and the convergence rate of the ε-Nash dynamics. To this end, we adopt the model of graphical linear congestion games with weighted players, where the individual cost and the strategy selection of each player only depends on his neighboring players in the social graph. We show that such games admit a potential function, and thus a PNE. Our main result is that the impact of social ignorance on the Price of Anarchy (PoA) and the Price of Stability (PoS) is naturally quantified by the independence number α(G) of the social graph G. In particular, we show that the PoA grows roughly as α(G)(α(G) + 2), which is essentially tight as long as α(G) does not exceed half the number of players, and that the PoS lies between α(G) and 2α(G). Moreover, we show that the ε-Nash dynamics reaches an α(G)(α(G) + 2)-approximate configuration in time that is polynomial and does not directly depend on the social graph. For unweighted graphical linear games with symmetric strategies, we show that the ε-Nash dynamics converges to an ε-approximate PNE in time that is polynomial and exceeds the corresponding time for symmetric linear games by a factor at most as large as the number of players.

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On the Hardness of Network Design for Bottleneck Routing Games

arXiv (Cornell University), Jul 22, 2012

In routing games, the selfish behavior of the players may lead to a degradation of the network pe... more In routing games, the selfish behavior of the players may lead to a degradation of the network performance at equilibrium. In more than a few cases however, the equilibrium performance can be significantly improved if we remove some edges from the network. This counterintuitive fact, widely known as Braess's paradox, gives rise to the (selfish) network design problem, where we seek to recognize routing games suffering from the paradox, and to improve their equilibrium performance by edge removal. In this work, we investigate the computational complexity and the approximability of the network design problem for non-atomic bottleneck routing games, where the individual cost of each player is the bottleneck cost of her path, and the social cost is the bottleneck cost of the network, i.e. the maximum latency of a used edge. We first show that bottleneck routing games do not suffer from Braess's paradox either if the network is series-parallel, or if we consider only subpath-optimal Nash flows. On the negative side, we prove that even for games with strictly increasing linear latencies, it is NP-hard not only to recognize instances suffering from the paradox, but also to distinguish between instances for which the Price of Anarchy (PoA) can decrease to 1 and instances for which the PoA cannot be improved by edge removal, even if their PoA is as large as Ω(n 0.121). This implies that the network design problem for linear bottleneck routing games is NP-hard to approximate within a factor of O(n 0.121−ε), for any constant ε > 0. The proof is based on a recursive construction of hard instances that carefully exploits the properties of bottleneck routing games, and may be of independent interest. On the positive side, we present an algorithm for finding a subnetwork that is almost optimal w.r.t. the bottleneck cost of its worst Nash flow, when the worst Nash flow in the best subnetwork routes a non-negligible amount of flow on all used edges. We show that the running time is essentially determined by the total number of paths in the network, and is quasipolynomial when the number of paths is quasipolynomial.

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Dynamic 3-Sided Planar Range Queries with Expected Doubly Logarithmic Time

Springer eBooks, 2009

This work studies the problem of 2-dimensional searching for the 3-sided range query of the form ... more This work studies the problem of 2-dimensional searching for the 3-sided range query of the form [a, b] × (−∞, c] in both main and external memory, by considering a variety of input distributions. We present three sets of solutions each of which examines the 3-sided problem in both RAM and I/O model respectively. The presented data structures are deterministic and the expectation is with respect to the input distribution: (1) Under continuous µ-random distributions of the x and y coordinates, we present a dynamic linear main memory solution, which answers 3sided queries in O(log n + t) worst case time and scales with O(log log n) expected with high probability update time, where n is the current number of stored points and t is the size of the query output. We externalize this solution, gaining O(log B n + t/B) worst case and O(logBlogn) amortized expected with high probability I/Os for query and update operations respectively, where B is the disk block size. (2)Then, we assume that the inserted points have their x-coordinates drawn from a class of smooth distributions, whereas the y-coordinates are arbitrarily distributed. The points to be deleted are selected uniformly at random among the inserted points. In this case we present a dynamic linear main memory solution that supports queries in O(log log n + t) expected time with high probability and updates in O(log log n) expected amortized time, where n is the number of points stored and t is the size of the output of the query. We externalize this solution, gaining O(log log B n + t/B) expected I/Os with high probability for query operations and O(log B log n) expected amortized I/Os for update operations, where B is the disk block size. The space remains linear O(n/B).

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Resolving Braess’s Paradox in Random Networks

Algorithmica, Jun 28, 2016

Braess's paradox states that removing a part of a network may improve the players' latency at equ... more Braess's paradox states that removing a part of a network may improve the players' latency at equilibrium. In this work, we study the approximability of the best subnetwork problem for the class of random Gn,p instances proven prone to Braess's paradox by (Roughgarden & Valiant, RSA '10) and (Chung & Young WINE '10-Chung, Young & Zhao, RSA '12). Our main contribution is a polynomialtime approximation-preserving reduction of the best subnetwork problem for such instances to the corresponding problem in a simplified network where all neighbors of source s and destination t are directly connected by 0 latency edges. Building on this, we consider two cases, either when the total rate r is sufficiently low, or, when r is sufficiently high. In the first case of low r = O(n+), here n+ is the maximum degree of {s, t}, we obtain an approximation scheme that for any constant ε > 0 and with high probability, computes a subnetwork and an ε-Nash flow with maximum latency at most (1 + ε)L * + ε, where L * is the equilibrium latency of the best subnetwork. Our approximation scheme runs in polynomial time if the random network has average degree O(poly(ln n)) and the traffic rate is O(poly(ln ln n)), and in quasipolynomial time for average degrees up to o(n) and traffic rates of O(poly(ln n)). Finally, in the second case of high r = Ω(n+), we compute in strongly polynomial time a subnetwork and an ε-Nash flow with maximum latency at most (1 + 2ε + o(1))L * .

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Atomic Congestion Games: Fast, Myopic and Concurrent

Theory of computing systems, Mar 5, 2009

We study here the effect of concurrent greedy moves of players in atomic congestion games where n... more We study here the effect of concurrent greedy moves of players in atomic congestion games where n selfish agents (players) wish to select a resource each (out of m resources) so that her selfish delay there is not much. Such games usually admit a global potential that decreases by sequential and selfishly improving moves. However, concurrent moves may not always lead to global convergence. On the other hand, concurrent play is desirable because it might essentially improve the system convergence time to some balanced state. The problem of "maintaining" global progress while allowing concurrent play is exactly what is examined and answered here. We examine two orthogonal settings : (i) A game where the players decide their moves without global information, each acting "freely" by sampling resources randomly and locally deciding to migrate (if the new resource is better) via a random experiment. Here, the resources can have quite arbitrary latency that is load dependent. (ii) An "organised" setting where the players are prepartitioned into selfish groups (coalitions) and where each coalition does an improving coalitional move. Here the concurrency is among the members of the coalition. In this second setting, the resources have latency functions that are only linearly dependent on the load, since this is the only case so far where a global potential exists. In both cases (i), (ii) we show that the system converges to an "approximate" equilibrium very fast (in logarithmic rounds where the logarithm is taken on the maximum value of the global potential). This is interesting, since two quite orthogonal settings lead to the same result. Our work considers concurrent selfish play for arbitrary latencies for the first time. Also, this is the first time where fast coalitional convergence to an approximate equilibrium is shown. All our results refer to atomic games (ie players are finite and distinct).

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Random input helps searching predecessors

We solve the dynamic Predecessor Problem with high probability (whp) in constant time, using only... more We solve the dynamic Predecessor Problem with high probability (whp) in constant time, using only $n^{1+\delta}$ bits of memory, for any constant $\delta > 0$. The input keys are random wrt a wider class of the well studied and practically important class of $(f_1, f_2)$-smooth distributions introduced in \cite{and:mat}. It achieves O(1) whp amortized time. Its worst-case time is $O(\sqrt{\frac{\log n}{\log \log n}})$. Also, we prove whp $O(\log \log \log n)$ time using only $n^{1+ \frac{1}{\log \log n}}= n^{1+o(1)}$ bits. Finally, we show whp $O(\log \log n)$ time using O(n) space.

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Emerge-Sort: Converging to Ordered Sequences by Simple Local Operators

In this paper we examine sorting on the assumption that we do not know in advance which way to so... more In this paper we examine sorting on the assumption that we do not know in advance which way to sort a sequence of numbers and we set at work simple local comparison and swap operators whose repeating application ends up in sorted sequences. These are the basic elements of Emerge-Sort, our approach to self-organizing sorting, which we then validate experimentally across a range of samples. Observing an O(n 2 ) run-time behaviour, we note that the n/logn delay coefficient that differentiates Emerge-Sort from the classical comparison based algorithms is an instantiation of the price of anarchy we pay for not imposing a sorting order and for letting that order emerge through the local interactions.

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Corrigendum to" A Note on the Non-Colorability Threshold of a Random Graph

Journal Publications

Emerge-Sort: Converging to Ordered Sequences by Simple Local Operators

Arxiv preprint arXiv:0812.1126, Dec 5, 2008

Abstract: In this paper we examine sorting on the assumption that we do not know in advance which... more Abstract: In this paper we examine sorting on the assumption that we do not know in advance which way to sort a sequence of numbers and we set at work simple local comparison and swap operators whose repeating application ends up in sorted sequences. These are the basic elements of Emerge-Sort, our approach to self-organizing sorting, which we then validate experimentally across a range of samples. Observing an O (n2) run-time behaviour, we note that the n/logn delay coefficient that differentiates Emerge-Sort from ...

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Selfish splittable flows and -completeness

Computer Science Review, 2011

A picturesque way to see a large network of links shared by many infinitesimally small selfish us... more A picturesque way to see a large network of links shared by many infinitesimally small selfish users is as a large pipeline infrastructure with users as liquid molecules flowing into it. When the owner of such a selfishly congested network tries to improve its flow speed, the common sense suggests to focus and fix links that seem older and slower.

Improved Bounds for Finger Search on a RAM

Algorithmica, 2013

We present a new finger search tree with O(1) worst-case update time and O(log log d) expected se... more We present a new finger search tree with O(1) worst-case update time and O(log log d) expected search time with high probability in the Random Access Machine (RAM) model of computation for a large class of input distributions. The parameter d represents the number of elements (distance) between the search element and an element pointed to by a finger, in a finger search tree that stores n elements. For the need of the analysis we model the updates by a "balls and bins" combinatorial game that is interesting in its own right as it involves insertions and deletions of balls according to an unknown distribution.

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A Note on the Non-Colorability Threshold of a Random Graph

The Electronic Journal of Combinatorics

In this paper we consider the problem of establishing a value $r_0$ such that almost all random g... more In this paper we consider the problem of establishing a value $r_0$ such that almost all random graphs with $n$ vertices and $rn$ edges, $r > r_0$, are asymptotically not 3-colorable. In our approach we combine the concept of rigid legal colorings introduced by Achlioptas and Molloy with the occupancy problem for random allocations of balls into bins. Using the sharp estimates obtained by Kamath et al. of the probability that no bin is empty after the random placement of the balls and exploiting the relationship between the placement of balls and the rigid legal colorings, we improve the value $r_0 = 2.522$ previously obtained by Achlioptas and Molloy to $r_0 = 2.495$.

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The Impact of Social Ignorance on Weighted Congestion Games

Proceedings of the 5th International Workshop on Internet and Network Economics, 2009

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Alexis Kaporis

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