Abstract: We perform an extensive experimental study of several dynamic algorithms for transitive closure. In particular, we implemented algorithms given by Italiano, Yellin, Cicerone et al., and two recent randomizedalgorithms by Henzinger and King. We propose a fine-tuned version of Italiano's algorithms as well as a new variant of them, both of which were always faster than any of the other implementations of the dynamic algorithms. We also considered simple-minded algorithms that were easy to implement and likely to be fast in practice. Wetested and compared the above implementations on random inputs, on non-random inputs that are worst-case inputs for the dynamic algorithms, and on an input motivated by a real-world graph.

Abstract: We study the following Constrained Bipartite Edge Coloring problem: We are given a bipartite graph G=(U,V,E) of maximum degree I with n vertices, in which some of the edges have been legally colored with c colors. We wish to complete the coloring of the edges of G minimizing the total number of colors used. The problem has been proved to be NP-hard even for bipartite graphs of maximum degree three. Two special cases of the problem have been previously considered and tight upper and ower bounds on the optimal number of colors were proved. The upper bounds led to 3/2-approximation algorithms for both problems. In this paper we present a randomized (1.37+o(1))-approximation algorithm for the general problem in the case where max{l,c} = {\`u}(ln n). Our techniques are motivated by recent works on the Circular Arc Coloring problem and are essentially different and simpler than the existing ones.

Abstract: Motivated by the wavelength assignment problem in WDM optical networks, we study path coloring problems in graphs. Given a set of paths P on a graph G, the path coloring problem is to color the paths of P so that no two paths traversing the same edge of G are assigned the same color and the total number of colors used is minimized. The problem has been proved to be NP-hard even for trees and rings.
Using optimal solutions to fractional path coloring, a natural relaxation of path coloring, on which we apply a randomized rounding technique combined with existing coloring algorithms, we obtain new upper bounds on the minimum number of colors sufficient to color any set of paths on any graph. The upper bounds are either existential or constructive.
The existential upper bounds significantly improve existing ones provided that the cost of the optimal fractional path coloring is sufficiently large and the dilation of the set of paths is small. Our algorithmic results include improved approximation algorithms for path coloring in rings and in bidirected trees. Our results extend to variations of the original path coloring problem arizing in multifiber WDM optical networks.

Abstract: We study the problem of scheduling permanent jobs on un-
related machines when the objective is to minimize the Lp
norm of the machine loads. The problem is known as load
balancing under the Lp norm. We present an improved up-
per bound for the greedy algorithm through simple analy-
sis; this bound is also shown to be best possible within the
class of deterministic online algorithms for the problem. We
also address the question whether randomization helps on-
line load balancing under Lp norms on unrelated machines;
this is a challenging question which is open for more than a
decade even for the L2 norm. We provide a positive answer
to this question by presenting the ¯rst randomized online
algorithms which outperform deterministic ones under any
(integral) Lp norm for p = 2; :::; 137. Our algorithms es-
sentially compute in an online manner a fractional solution
to the problem and use the fractional values to make ran-
dom choices. The local optimization criterion used at each
step is novel and rather counterintuitive: the values of the
fractional variables for each job correspond to °ows at an ap-
proximate Wardrop equilibrium for an appropriately de¯ned
non-atomic congestion game. As corollaries of our analysis
and by exploiting the relation between the Lp norm and the
makespan of machine loads, we obtain new competitive algo-
rithms for online makespan minimization, making progress
in another longstanding open problem.

Abstract: We study the problem of scheduling permanent jobs on un-
related machines when the objective is to minimize the Lp
norm of the machine loads. The problem is known as load
balancing under the Lp norm. We present an improved up-
per bound for the greedy algorithm through simple analy-
sis; this bound is also shown to be best possible within the
class of deterministic online algorithms for the problem. We
also address the question whether randomization helps on-
line load balancing under Lp norms on unrelated machines;
this is a challenging question which is open for more than a
decade even for the L2 norm. We provide a positive answer
to this question by presenting the ¯rst randomized online
algorithms which outperform deterministic ones under any
(integral) Lp norm for p = 2; :::; 137. Our algorithms es-
sentially compute in an online manner a fractional solution
to the problem and use the fractional values to make ran-
dom choices. The local optimization criterion used at each
step is novel and rather counterintuitive: the values of the
fractional variables for each job correspond to °ows at an ap-
proximate Wardrop equilibrium for an appropriately de¯ned
non-atomic congestion game. As corollaries of our analysis
and by exploiting the relation between the Lp norm and the
makespan of machine loads, we obtain new competitive algo-
rithms for online makespan minimization, making progress
in another longstanding open problem.

Abstract: We address an important communication issue arising in
wireless cellular networks that utilize frequency division
multiplexing (FDM) technology. In such networks, many
users within the same geographical region (cell) can communicate
simultaneously with other users of the network
using distinct frequencies. The spectrum of the available
frequencies is limited; thus, efficient solutions to the call
controlproblemareessential.Theobjectiveofthecallcontrol
problem is, given a spectrum of available frequencies
and users that wish tocommunicate, to maximize the benefit,
i.e., the number of users that communicate without
signalinterference.Weconsidercellularnetworksofreuse
distance k ≥ 2 and we study the online version of the
problem using competitive analysis. In cellular networks
of reuse distance 2, the previously best known algorithm
that beats the lower bound of 3 on the competitiveness
of deterministic algorithms, works on networks with one
frequency, achieves a competitive ratio against oblivious
adversaries, which is between 2.469 and 2.651, and uses
a number of random bits at least proportional to the size
of the network.We significantly improve this result by presentingaseriesofsimplerandomizedalgorithmsthathave
competitiveratiossignificantlysmallerthan3,workonnetworks
with arbitrarily many frequencies, and use only a
constant number of random bits or a comparable weak
random source. The best competitiveness upper bound
we obtain is 16/7 using only four random bits. In cellular
networks of reuse distance k > 2, we present simple
randomized online call control algorithms with competitive
ratios, which significantly beat the lower bounds on
the competitiveness of deterministic ones and use only
O(log k )randombits. Also,weshownewlowerboundson
thecompetitivenessofonlinecallcontrolalgorithmsincellularnetworksofanyreusedistance.
Inparticular,weshow
thatnoonline algorithm can achieve competitive ratio better
than 2, 25/12, and 2.5, in cellular networks with reuse
distancek ∈ {2, 3, 4},k = 5,andk ≥ 6, respectively.

Abstract: In this paper we consider communication issues arising in mobile networks that utilize Frequency Division Multiplexing (FDM) technology. In such networks, many users within the same geographical region can communicate simultaneously with other users of the network using distinct frequencies. The spectrum of available frequencies is limited; thus, efficient solutions to the frequency allocation and the call control problem are essential. In the frequency allocation problem, given users that wish to communicate, the objective is to minimize the required spectrum of frequencies so that communication can be established without signal interference. The objective of the call control problem is, given a spectrum of available frequencies and users that wish to communicate, to maximize the number of users served. We consider cellular, planar, and arbitrary network topologies. In particular, we study the on-line version of both problems using competitive analysis. For frequency allocation in cellular networks, we improve the best known competitive ratio upper bound of 3 achieved by the folklore Fixed Allocation algorithm, by presenting an almost tight competitive analysis for the greedy algorithm; we prove that its competitive ratio is between 2.429 and 2.5. For the call control problem, we present the first randomized algorithm that beats the deterministic lower bound of 3 achieving a competitive ratio of 2.934 in cellular networks. Our analysis has interesting extensions to arbitrary networks. Also, using Yao's Minimax Principle, we prove two lower bounds of 1.857 and 2.086 on the competitive ratio of randomized call control algorithms for cellular and arbitrary planar networks, respectively.

Abstract: In this paper we consider communication issues arising in cellular (mobile) networks that utilize frequency division multiplexing (FDM) technology. In such networks, many users within the same geographical region can communicate simultaneously with other users of the network using distinct frequencies. The spectrum of available frequencies is limited; thus, efficient solutions to the frequency-allocation and the call-control problems are essential. In the frequency-allocation problem, given users that wish to communicate, the objective is to minimize the required spectrum of frequencies so that communication can be established without signal interference. The objective of the call-control problem is, given a spectrum of available frequencies and users that wish to communicate, to maximize the number of users served. We consider cellular, planar, and arbitrary network topologies.
In particular, we study the on-line version of both problems using competitive analysis. For frequency allocation in cellular networks, we improve the best known competitive ratio upper bound of 3 achieved by the folklore Fixed Allocation algorithm, by presenting an almost tight competitive analysis for the greedy algorithm; we prove that its competitive ratio is between 2.429 and 2.5 . For the call-control problem, we present the first randomized algorithm that beats the deterministic lower bound of 3 achieving a competitive ratio between 2.469 and 2.651 for cellular networks. Our analysis has interesting extensions to arbitrary networks. Also, using Yao's Minimax Principle, we prove two lower bounds of 1.857 and 2.086 on the competitive ratio of randomized call-control algorithms for cellular and arbitrary planar networks, respectively.

Abstract: We consider the problem of searching for a piece of
information in a fully interconnected computer network
(also called a complete network or clique) by exploiting
advice about its location from the network nodes. Each
node contains a database that ?knows? what kind of
documents or information are stored in other nodes
(e.g., a node could be a Web server that answers queries
about documents stored on the Web). The databases in
each node, when queried, provide a pointer that leads to
the node that contains the information. However, this
information is up-to-date (or correct) with some
bounded probability. While, in principle, one may always
locate the information by simply visiting the network
nodes in some prescribed ordering, this requires a time
complexity in the order of the number of nodes of the
network. In this paper, we provide algorithms for locating
an information node in the complete communication
network, which take advantage of advice given from
network nodes. The nodes may either give correct advice,
by pointing directly to the information node, or give
wrong advice, by pointing elsewhere. On the lowerbounds?
side, we show that no fixed-memory (i.e., with
memory independent of the network size) deterministic
algorithm may locate the information node in a constant
(independent of the network size) expected number of
steps. Moreover, if p (1/n) is the probability that a
node of an n-node clique gives correct advice, we show
that no algorithm may locate the information node in an
expected number of steps less than 1/p o(1). To study
how the expected number of steps is affected by the
amount of memory allowed to the algorithms, we give a
memoryless randomized algorithm with expected number
of steps 4/p o(1/p) o(1) and a 1-bit randomized
algorithm requiring on the average at most 2/p o(1)
steps. In addition, in the memoryless case, we also
prove a 4/p lower bound for the expected number of
steps in the case where the nodes giving faulty advice
may decide on the content of this advice in any possible
way and not merely at random (adversarial fault model).
Finally, for the case where faulty nodes behave randomly,
we give an optimal, unlimited memory deterministic
algorithm with expected number of steps bounded
from above by 1/p o(1/p) 1.

Abstract: The voting rules proposed by Dodgson and Young are both
designed to nd the alternative closest to being a Condorcet
winner, according to two dierent notions of proximity; the
score of a given alternative is known to be hard to compute
under either rule.
In this paper, we put forward two algorithms for ap-
proximating the Dodgson score: an LP-based randomized
rounding algorithm and a deterministic greedy algorithm,
both of which yield an O(logm) approximation ratio, where
m is the number of alternatives; we observe that this result
is asymptotically optimal, and further prove that our greedy
algorithm is optimal up to a factor of 2, unless problems in
NP have quasi-polynomial time algorithms. Although the
greedy algorithm is computationally superior, we argue that
the randomized rounding algorithm has an advantage from
a social choice point of view.
Further, we demonstrate that computing any reasonable
approximation of the ranking produced by Dodgson's rule
is NP-hard. This result provides a complexity-theoretic
explanation of sharp discrepancies that have been observed
in the Social Choice Theory literature when comparing
Dodgson elections with simpler voting rules.
Finally, we show that the problem of calculating the
Young score is NP-hard to approximate by any factor. This
leads to an inapproximability result for the Young ranking.

Abstract: We consider in this paper the problem of scheduling a set of independent
parallel tasks (jobs) with respect to two criteria, namely,
the makespan (time of the last finishing job) and the minsum (average
completion time). There exist several algorithms with a good
performance guaranty for one of these criteria. We are interested
here in studying the optimization of both criteria simultaneously.
The numerical values are given for the moldable task model, where
the execution time of a task depends on the number of processors
alloted to it. The main result of this paper is to derive explicitly
a family of algorithms guaranteed for both the minsum and the
makespan. The performance guaranty of these algorithms is better
than the best algorithms known so far. The Guaranty curve
of the family is the set of all points (x; y) such that there is an
algorithm with guarantees x on makespan and y on the minsum.
When the ratio on the minsum increases, the curve tends to the
best ratio known for the makespan for moldable tasks (3=2). One
extremal point of the curves is a (3;6)-approximation algorithm.
Finally a randomized version is given, which improves this results
to (3;4.08).

Abstract: We address an important communication issue in wireless cellular networks that utilize Frequency Division Multiplexing (FDM) technology. In such networks, many users within the same geographical region (cell) can communicate simultaneously with other users of the network using distinct frequencies. The spectrum of the available frequencies is limited; thus, efficient solutions to the call control problem are essential. The objective of the call control problem is, given a spectrum of available frequencies and users that wish to communicate, to maximize the number of users that communicate without signal interference. We consider cellular networks of reuse distance kge 2 and we study the on-line version of the problem using competitive analysis.
In cellular networks of reuse distance 2, the previously best known algorithm that beats the lower bound of 3 on the competitiveness of deterministic algorithms works on networks with one frequency, achieves a competitive ratio against oblivious adversaries which is between 2.469 and 2.651, and uses a number of random bits at least proportional to the size of the network. We significantly improve this result by presenting a series of simple randomizedalgorithms that have competitive ratios smaller than 3, work on networks with arbitrarily many frequencies, and use only a constant number of random bits or a comparable weak random source. The best competitiveness upper bound we obtain is 7/3.
In cellular networks of reuse distance k>2, we present simple randomized on-line call control algorithms with competitive ratios which significantly beat the lower bounds on the competitiveness of deterministic ones and use only random bits. Furthermore, we show a new lower bound on the competitiveness of on-line call control algorithms in cellular networks of reuse distance kge 5.