Majority illusion is a phenomenon in social networks wherein the decision by the majority of the network is not the same as one's personal social circle's majority, leading to an incorrect perception of the majority in a large network. We present polynomial-time algorithms which completely eliminate majority illusion by altering as few connections in the network as possible. Eliminating majority illusion ensures each neighbourhood in the network has at least a 1/2-fraction of the majority winner. This result is surprising as partially eliminating majority illusion is NP-hard. We generalize the majority illusion problem to an arbitrary fraction p and show that the problem of ensuring all neighbourhoods in the network contain at least a p-fraction of nodes consistent with a given preference is NP-hard, for nearly all values of p.

A graph $G=(V,E)$ is a $k$-leaf power if there is a tree $T$ whose leaves are the vertices of $G$ with the property that a pair of leaves $u$ and $v$ induce an edge in $G$ if and only if they are distance at most $k$ apart in $T$. For $k\le 4$, it is known that there exists a finite set $F_k$ of graphs such that the class $L(k)$ of $k$-leaf power graphs is characterized as the set of strongly chordal graphs that do not contain any graph in $F_k$ as an induced subgraph. We prove no such characterization holds for $k\ge 5$. That is, for any $k\ge 5$, there is no finite set $F_k$ of graphs such that $L(k)$ is equivalent to the set of strongly chordal graphs that do not contain as an induced subgraph any graph in $F_k$.

A cornerstone of social choice theory is Condorcet’s paradox which says that in an election where n voters rank m candidates it is possible that, no matter which candidate is declared the winner, a majority of voters would have preferred an alternative candidate. Instead, can we always choose a small committee of winning candidates that is preferred to any alternative candidate by a majority of voters? Elkind, Lang, and Saffidine raised this question and called such a committee a Condorcet winning set. They showed that winning sets of size 2 may not exist, but sets of size logarithmic in the number of candidates always do. In this work, we show that Condorcet winning sets of size 6 always exist, regardless of the number of candidates or the number of voters. More generally, we show that if α/1 − lnα ≥ 2/k + 1, then there always exists a committee of size k such that less than an α fraction of the voters prefer an alternate candidate. These are the first nontrivial positive results that apply for all k ≥ 2. Our proof uses the probabilistic method and the minimax theorem, inspired by recent work on approximately stable committee selection. We construct a distribution over committees that performs sufficiently well (when compared against any candidate on any small subset of the voters) so that this distribution must contain a committee with the desired property in its support.

We consider the design of a positioning system where a robot determines its position from local observations. This is a well-studied problem of considerable practical importance and mathematical interest. The dominant paradigm derives from the classical theory of de Bruijn sequences, where the robot has access to a window within a larger code and can determine its position if these windows are distinct. We propose an alternative model in which the robot has more limited observational powers, which we argue is more realistic in terms of engineering: the robot does not have access to the full pattern of colours (or letters) in the window, but only to the intensity of each colour (or the number of occurrences of each letter). This leads to a mathematically interesting problem with a different flavour to that arising in the classical paradigm, requiring new construction techniques. The parameters of our construction are optimal up to a constant factor, and computing the position requires only a constant number of arithmetic operations.

In the famous network creation game of Fabrikant et al. a set of agents play a game to build a connected graph. The $n$ agents form the vertex set $V$ of the graph and each vertex $v\in V$ buys a set $E_v$ of edges inducing a graph $G=(V,\bigcup\limits_{v\in V} E_v)$. The private objective of each vertex is to minimize the sum of its building cost (the cost of the edges it buys) plus its connection cost (the total distance from itself to every other vertex). Given a cost of $α$ for each individual edge, a long-standing conjecture, called the tree conjecture, states that if $α> n$ then every Nash equilibrium graph in the game is a spanning tree. After a plethora of work, it is known that the conjecture holds for any $α>3n-3$. In this paper we prove the tree conjecture holds for $α>2n$. This reduces by half the open range for $α$ with only $[n, 2n)$ remaining in order to settle the conjecture.

We study the fair division of a collection of mindivisible goods amongst a set of nagents. Whilst envy-free allocations typically do not exist in the indivisible-goods setting, envy-freeness can be achieved if some amount of a divisible good (money) is introduced. Specifically, Halpern and Shah (SAGT 2019, pp.374-389) showed that, given additive valuation functions where the marginal value of each good is at most one dollar for each agent, there always exists an envy-free allocation requiring a subsidy of at most (n-1)·m dollars. The authors also conjectured that a subsidy of $n-1$ dollars is sufficient for additive valuations. We prove this conjecture. In fact, a subsidy of at most one dollar per agent is sufficient to guarantee the existence of an envy-free allocation. Further, we prove that for general monotonic valuation functions an envy-free allocation always exists with a subsidy of at most 2(n-1) dollars per agent. In particular, the total subsidy required for monotonic valuations is independent of the number of goods.

We present a factor 4/3 approximation algorithm for the problem of finding a minimum 2-edge connected spanning subgraph of a given undirected multigraph. The algorithm is based upon a reduction to a restricted class of graphs. In these graphs, the approximation algorithm constructs a 2-edge connected spanning subgraph by modifying the smallest 2-edge cover.

We present tight bounds on the social welfare ratio of the combinatorial clock auction under the basic price increment rule used in practice. Specifically, under the assumption of truthful bidding, we prove that the welfare ratio is Θ(m^\frac23)$ in the case of unit-demand bidders, and Θ(m)$ for general bidders. We explain how changes to the price increment rule effect the welfare guarantee, and extend our results to settings with strategic bidding.

Since the 1990s spectrum auctions have been implemented world-wide. This has provided for a practical examination of an assortment of auction mechanisms and, amongst these, two simultaneous ascending price auctions have proved to be extremely successful. These are the simultaneous multiround ascending auction (SMRA) and the combinatorial clock auction (CCA). It has long been known that, for certain classes of valuation functions, the SMRA provides good theoretical guarantees on social welfare. However, no such guarantees were known for the CCA. In this paper, we show that CCA does provide strong guarantees on social welfare provided the price increment and stopping rule are well-chosen. This is very surprising in that the choice of price increment has been used primarily to adjust auction duration and the stopping rule has attracted little attention. The main result is a polylogarithmic approximation guarantee for social welfare when the maximum number of items demanded by a bidder is fixed. Specifically, we show that either the revenue of the CCA is at least an -fraction of the optimal welfare or the welfare of the CCA is at least an -fraction of the optimal welfare, where n is the number of bidders and m is the number of items. As a corollary, the welfare ratio – the worst case ratio between the social welfare of the optimum allocation and the social welfare of the CCA allocation – is at most O( 2 · log n·· log2 m). We emphasize that this latter result requires no assumption on bidders valuation functions. Finally, we prove that such a dependence on is necessary. In particular, we show that the welfare ratio of the CCA is at least .

We prove that for any constant k and any epsilon < 1, there exist bimatrix win-lose games for which every epsilon-WSNE requires supports of cardinality greater than k. To do this, we provide a graph-theoretic characterization of win-lose games that possess epsilon-WSNE with constant cardinality supports. We then apply a result in additive number theory of Haight to construct win-lose games that do not satisfy the requirements of the characterization. These constructions disprove graph theoretic conjectures of Daskalakis, Mehta and Papadimitriou and Myers.

A single-sink confluent flow is a routing of multiple demands to a sink r such that any flow exiting a node v must use a single arc. Hence, a confluent flow routes on a tree within the network. In uncapacitated (or uniform-capacity) networks, there is an O(1)-approximation algorithm for demand maximization and a logarithmic approximation algorithm for congestion minimization [6]. We study the case of capacitated networks, where each node v has its own capacity μ(v). Indeed, it was recently shown that demand maximization is in approximable to within polynomial factors in capacitated networks [20]. We circumvent this lower bound in two ways. First, we prove that there is a polylogarithmic approximation algorithm for demand maximization in networks that satisfy the ubiquitous no-bottleneck assumption (NBA). Second, we show a bicriteria result for capacitated networks without the NBA: there is a polylog factor approximation guarantee for demand maximization provided we allow congestion 2. We model the capacitated confluent flows problem using a multilayer linear programming formulation. At the heart of our approach for demand maximization is a rounding procedure for flows on multilayer networks which can be viewed as a proposal algorithm for an extension of stable matchings. In addition, the demand maximization algorithms require, as a subroutine, an algorithm for approximate congestion minimization in a special class of capacitated networks that may be of independent interest. Specifically, we present a polylogarithmic approximation algorithm for congestion minimization in monotonic networks - those networks with the property that μ(u) ≤ μ(v) for each arc (u, v).

We examine cooperative games where the viability of a coalition is determined by whether or not its members have the ability to communicate amongst themselves independently of non-members. This necessary condition for viability was proposed by Myerson [1977] and is modeled via an interaction graph G=(V,E); a coalition S ⊆ V is then viable if and only if the induced graph G[S] is connected. The non-emptiness of the core of a coalition game can be tested by a well-known covering LP. Moreover, the integrality gap of its dual packing LP defines exactly the multiplicative least-core and the relative cost of stability of the coalition game. This gap is upper bounded by the packing-covering ratio which, for graphical coalition games, is known to be at most the treewidth of the interaction graph plus one [Meir et al. 2013].

Given a consumer data-set, the axioms of revealed preference proffer a binary test for rational behaviour. A natural (non-binary) measure of the degree of rationality exhibited by the consumer is the minimum number of data points whose removal induces a rationalisable data-set. We study the computational complexity of the resultant consumer rationality problem in this paper. This problem is, in the worst case, equivalent (in terms of approximation) to the directed feedback vertex set problem. Our main result is to obtain an exact threshold on the number of commodities that separates easy cases and hard cases. Specifically, for two-commodity markets the consumer rationality problem is polynomial time solvable; we prove this via a reduction to the vertex cover problem on perfect graphs. For three-commodity markets, however, the problem is NP-complete; we prove this using a reduction from planar 3-sat that is based upon oriented-disc drawings.

The simultaneous multiple-round auction (SMRA) and the combinatorial clock auction (CCA) are the two primary mechanisms used to sell bandwidth. Recently, it was shown that the CCA provides good welfare guarantees for general classes of valuation functions [7]. This motivates the question of whether similar welfare guarantees hold for the SMRA in the case of general valuation functions. We show the answer is no. But we prove that good welfare guarantees still arise if the degree of complementarities in the bidder valuations are bounded. In particular, if bidder valuations functions are $$\alpha $$ -near-submodular then, under truthful bidding, the SMRA has a welfare ratio (the worst case ratio between the social welfare of the optimal allocation and the auction allocation) of at most $$(1+\alpha )$$ . However, for $$\alpha >1$$ , this is a bicriteria guarantee, to obtain good welfare under truthful bidding requires relaxing individual rationality. We prove this bicriteria guarantee is asymptotically (almost) tight. Finally, we examine what strategies are required to ensure individual rationality in the SMRA with general valuation functions. First, we provide a weak characterization, namely secure bidding, for individual rationality. We then show that if the bidders use a profit-maximizing secure bidding strategy the welfare ratio is at most $$1+\alpha $$ . Consequently, by bidding securely, it is possible to obtain the same welfare guarantees as truthful bidding without the loss of individual rationality.

Combinatorial auctions are multiple-item auctions in which bidders may place bids on any package (subset) of goods. This additional expressibility produces benefits that have led to combinatorial auctions becoming extremely important both in practice and in theory. In the computer science community, auction design has focused primarily on computational practicality and incentive compatibility. The latter concerns mechanisms that are resistant to bidders misrepresenting themselves via a single false identity; however, with modern forms of bid submission, such as electronic bidding, other types of cheating have become feasible. Prominent amongst them is false-name bidding; that is, bidding under pseudonyms. For example, the ubiquitous Vickrey-Clarke-Groves (VCG) mechanism is incentive compatible and produces optimal allocations, but it is not false-name-proof–bidders can increase their utility by submitting bids under multiple identifiers. Thus, there has recently been much interest in the design and analysis of false-name-proof auction mechanisms. These false-name-proof mechanisms, however, have polynomially small efficiency guarantees: they can produce allocations with very low economic efficiency/social welfare. In contrast, we show that, provided the degree to which different goods are complementary is bounded (as is the case in many important, practical auctions), the VCG mechanism gives a constant efficiency guarantee. Constant efficiency guarantees hold even at equilibria where the agents bid in a manner that is not individually rational. Thus, while an individual bidder may personally benefit greatly from making false-name bids, this will have only a small detrimental effect on the objective of the auctioneer: maximizing economic efficiency. So, from the auctioneer's viewpoint the VCG mechanism remains preferable to false-name-proof mechanisms.

The Directed Steiner Tree (DST) problem is a cornerstone problem in network design. We focus on the generalization of the problem with higher connectivity requirements. The problem with one root and two sinks is APX-hard. The problem with one root and many sinks is as hard to approximate as the directed Steiner forest problem, and the latter is well known to be as hard to approximate as the label cover problem. Utilizing previous techniques, we strengthen these results and extend them to undirected graphs. Specifically, we give an Ω( k ϵ ) hardness bound for the rooted k -connectivity problem in undirected graphs. As a consequence, we obtain an Ω( k ϵ ) hardness bound for the undirected subset k -connectivity problem. Additionally, we give a result on the integrality ratio of the natural linear programming relaxation of the directed rooted k -connectivity problem.

We consider the problem of edge clique cover on sparse networks and study an application to the identification of overlapping protein complexes for a network of binary protein-protein interactions. We first give an algorithm whose running time is linear in the size of the graph, provided the treewidth is bounded. We then provide an algorithm for planar graphs with bounded branchwidth upon which we build a PTAS for planar graphs. Empirical studies show that our algorithms are both efficient and practical on actual simulated and biological networks, and that the clique covers obtained on real networks yield biological insights.

The Directed Steiner Tree (DST) problem is a cornerstone problem in network design. We focus on the generalization of the problem with higher connectivity requirements. The problem with one root and two sinks is APX-hard. The problem with one root and many sinks is as hard to approximate as the directed Steiner forest problem, and the latter is well known to be as hard to approximate as the label cover problem. Utilizing previous techniques (due to others), we strengthen these results and extend them to undirected graphs. Specifically, we give an Ω(k∊) hardness bound for the rooted k-connectivity problem in undirected graphs; this addresses a recent open question of Khanna. As a consequence, we also obtain the Ω(k∊) hardness of the undirected subset k-connectivity problem. Additionally, we give a result on the integrality ratio of the natural linear programming relaxation of the directed rooted k-connectivity problem.

Many genome-wide datasets are routinely generated to study different aspects of biological systems, but integrating them to obtain a coherent view of the underlying biology remains a challenge. We propose simultaneous clustering of multiple networks as a framework to integrate large-scale datasets on the interactions among and activities of cellular components. Specifically, we develop an algorithm JointCluster that finds sets of genes that cluster well in multiple networks of interest, such as coexpression networks summarizing correlations among the expression profiles of genes and physical networks describing protein-protein and protein-DNA interactions among genes or gene-products. Our algorithm provides an efficient solution to a well-defined problem of jointly clustering networks, using techniques that permit certain theoretical guarantees on the quality of the detected clustering relative to the optimal clustering. These guarantees coupled with an effective scaling heuristic and the flexibility to handle multiple heterogeneous networks make our method JointCluster an advance over earlier approaches. Simulation results showed JointCluster to be more robust than alternate methods in recovering clusters implanted in networks with high false positive rates. In systematic evaluation of JointCluster and some earlier approaches for combined analysis of the yeast physical network and two gene expression datasets under glucose and ethanol growth conditions, JointCluster discovers clusters that are more consistently enriched for various reference classes capturing different aspects of yeast biology or yield better coverage of the analysed genes. These robust clusters, which are supported across multiple genomic datasets and diverse reference classes, agree with known biology of yeast under these growth conditions, elucidate the genetic control of coordinated transcription, and enable functional predictions for a number of uncharacterized genes.

We present an O (√opt)-approximation algorithm for the maximum leaf spanning arborescence problem, where opt is the number of leaves in an optimal spanning arborescence. The result is based upon an O (1)-approximation algorithm for a special class of directed graphs called willows. Incorporating the method for willow graphs as a subroutine in a local improvement algorithm gives the bound for general directed graphs.

A d-furcated flow is a network flow whose support graph has maximum out degree d. Take a single-sink multi-commodity flow problem on any network and with any set of routing demands. Then we show that the existence of feasible fractional flow with node congestion one implies the existence of a d-furcated flow with congestion at most 1+1/(d-1), for d ≥ 2. This result is tight, and sothe congestion gap for d-furcated flows is bounded andexactly equal to 1+ 1/(d-1). For the case d=1 (confluent flows), it is known that the congestion gap is unbounded, namely Θ(log n). Thus, allowing single-sink multicommodity network flows to increase their maximum out degree from one to two virtually eliminates this previously observed congestion gap.

A flow of a commodity is said to be confluent if at any node all the flow of the commodity leaves along a single edge. In this article, we study single-commodity confluent flow problems, where we need to route given node demands to a single destination using a confluent flow. Single- and multi-commodity confluent flows arise in a variety of application areas, most notably in networking; in fact, most flows in the Internet are (multi-commodity) confluent flows since Internet routing is destination based. We present near-tight approximation algorithms, hardness results, and existence theorems for minimizing congestion in single-commodity confluent flows. The maximum edge congestion of a single-commodity confluent flow occurs at one of the incoming edges of the destination. Therefore, finding a minimum-congestion confluent flow is equivalent to the following problem: given a directed graph G with k sinks and non-negative demands on all the nodes of G , determine a confluent flow that routes every node demand to some sink such that the maximum congestion at a sink is minimized. The main result of this article is a polynomial-time algorithm for determining a confluent flow with congestion at most 1 + ln( k ) in G , if G admits a splittable flow with congestion at most 1. We complement this result in two directions. First, we present a graph G that admits a splittable flow with congestion at most 1, yet no confluent flow with congestion smaller than H k , the k th harmonic number, thus establishing tight upper and lower bounds to within an additive constant less than 1. Second, we show that it is NP-hard to approximate the congestion of an optimal confluent flow to within a factor of (log 2 k )/2, thus resolving the polynomial-time approximability to within a multiplicative constant. We also consider a demand maximization version of the problem. We show that if G admits a splittable flow of congestion at most 1, then a variant of the congestion minimization algorithm yields a confluent flow in G with congestion at most 1 that satisfies 1/3 fraction of total demand. We show that the gap between confluent flows and splittable flows is much smaller, if the underlying graph is k -connected. In particular, we prove that k -connected graphs with k sinks admit confluent flows of congestion less than C + d max , where C is the congestion of the best splittable flow, and d max is the maximum demand of any node in G . The proof of this existence theorem is non-constructive and relies on topological techniques introduced by Lovász.

We study undirected networks with edge costs that satisfy the triangle inequality. Let n denote the number of nodes. We present an $O(1)$-approximation algorithm for a generalization of the metric-cost subset k-node-connectivity problem. Our approximation guarantee is proved via lower bounds that apply to the simple edge-connectivity version of the problem, where the requirements are for edge-disjoint paths rather than for openly node-disjoint paths. A corollary is that, for metric costs and for each $k=1,2,\dots,n-1$, there exists a k-node connected graph whose cost is within a factor of ${ 22\/}$ of the cost of any simple k-edge connected graph. Based on our $O(1)$-approximation algorithm, we present an $O(\log r_{\max})$-approximation algorithm for the metric-cost node-connectivity survivable network design problem, where $r_{\max}$ denotes the maximum requirement over all pairs of nodes. Our results contrast with the case of edge costs of 0 or 1, where Kortsarz, Krauthgamer, and Lee. [SIAM J. Comput., 33 (2004), pp. 704–720] recently proved, assuming NP$\nsubseteq\;$DTIME($n^{polylog(n)}$), a hardness-of-approximation lower bound of $2^{\log^{1-\epsilon}n}$ for the subset k-node-connectivity problem, where $\epsilon$ denotes a small positive number.

We consider Nash equilibria in 2-player random games and analyze a simple Las Vegas algorithm for finding an equilibrium. The algorithm is combinatorial and always finds a Nash equilibrium; on m /spl times/ n payoff matrices, it runs in time O(m/sup 2/n log log n + n/sup 2/m log log m) with high probability. Our main tool is a polytope formulation of equilibria.

We introduce the concept of a sink equilibrium. A sink equilibrium is a strongly connected component with no outgoing arcs in the strategy profile graph associated with a game. The strategy profile graph has a vertex set induced by the set of pure strategy profiles; its arc set corresponds to transitions between strategy profiles that occur with nonzero probability. (Here our focus will just be on the special case in which the strategy profile graph is actually a best response graph; that is, its arc set corresponds exactly to best response moves that result from myopic or greedy behaviour). We argue that there is a natural convergence process to sink equilibria in games where agents use pure strategies. This leads to an alternative measure of the social cost of a lack of coordination, the price of sinking, which measures the worst case ratio between the value of a sink equilibrium and the value of the socially optimal solution. We define the value of a sink equilibrium to be the expected social value of the steady state distribution induced by a random walk on that sink. We illustrate the value of this measure in three ways. Firstly, we show that it may more accurately reflects the inefficiency of uncoordinated solutions in competitive games when the use of pure strategies is the norm. In particular, we give an example (a valid-utility game) in which the game converges to solutions which are a factor n worse than socially optimal. The price of sinking is indeed n, but the price of anarchy is close to 1. Secondly, sink equilibria always exist. Thus, even in games in which pure strategy Nash equilibria (PSNE) do not exist, we can still calculate the price of sinking. Thirdly, we show that bounding the price of sinking can have important implications for the speed of convergence to socially good solutions in games where the agents make best response moves in a random order. We present two examples to illustrate our ideas. (i) Unsplittable selfish routing (and weighted congestion games):we prove that the price of sinking for the weighted unsplittable flow version of the selfish routing problem (for bounded-degree polynomial latency functions) is at most O(2/sup 2d/ d/sup 2d + 3/). In comparison, we give instances of these games without any PSNE. Moreover, our proof technique implies fast convergence to socially good (approximate) solutions. This is in contrast to the negative result of Fabrikant, Papadimitriou, and Talwar (2004) showing the existence of exponentially long best-response paths. (ii) Valid-utility games: we show that for valid-utility games the price of sinking is at most n+1; thus the worst case price of sinking in a valid-utility game is between it and n+1. We use our proof to show fast convergence to constant factor approximate solutions in basic-utility games. In addition, we present a hardness result which shows that, in general, there might be states that are exponentially far from any sink equilibrium in valid-utility games. We prove this by showing that the problem of finding a sink equilibrium (or a PSNE) in valid-utility games is PLS-complete.

We study undirected networks with edge costs that satisfy the triangle inequality. Let n denote the number of nodes. We present an O(1)-approximation algorithm for a generalization of the metric-cost subset k-node-connectivity problem. Our approximation guarantee is proved via lower bounds that apply to the simple edge-connectivity version of the problem, where the requirements are for edge-disjoint paths rather than for openly node-disjoint paths. A corollary is that, for metric costs and for each k=1,2,…,n-1, there exists a k-node connected graph whose cost is within a factor of 24 of the cost of any simple k-edge connected graph. This resolves an open question in the area. Based on our O(1)-approximation algorithm, we present an O(log rmax)-approximation algorithm for the node-connectivity survivable network design problem where rmax denotes the maximum requirement over all pairs of nodes. Our results contrast with the case of edge costs of zero or one, where Kortsarz et al. [20]recently proved, assuming NP⊈, quasi-P, a hardness-of-approximation lower bound of 2log 1-εn for the subset k-node-connectivity problem, where ε denotes a small positive number.

A flow is said to be confluent if at any node all the flow leaves along a single edge. Given a directed graph G with k sinks and non-negative demands on all the nodes of G, we consider the problem of determining a confluent flow that routes every node demand to some sink such that the maximum congestion at a sink is minimized. Confluent flows arise in a variety of application areas, most notably in networking; in fact, most flows in the Internet are confluent since Internet routing is destination based.We present near-tight approximation algorithms, hardness results, and existence theorems for confluent flows. The main result of this paper is a polynomial-time algorithm for determining a confluent flow with congestion at most 1 + ln(k) in G, if G admits a splittable flow with congestion at most 1. We complement this result in two directions. First, we present a graph G that admits a splittable flow with congestion at most 1, yet no confluent flow with congestion smaller than Hk, thus establishing tight upper and lower bounds to within an additive constant less than 1. Second, we show that it is NP-hard to approximate the congestion of an optimal confluent flow to within a factor of (lg k)/2, thus resolving the polynomial-time approximability to within a multiplicative constant. We also consider a demand maximization version of the problem. We show that if G admits a splittable flow of congestion at most 1, then a variant of the congestion minimization algorithm yields a confluent flow in G with congestion at most 1 that satisfies 1/3 fraction of total demand.We show that the gap between confluent flows and splittable flows is much smaller, if the underlying graph were k connected. In particular, we prove that k-connected graphs with k sinks admit confluent flows of congestion less than C + dmax, where C is the congestion of the best splittable flow, and dmax is the maximum demand of any node in G. The proof of this existence theorem is non-constructive and relies on topological techniques introduced in [16].

We motivate and develop a natural bicriteria measure for assessing the quality of a clustering that avoids the drawbacks of existing measures. A simple recursive heuristic is shown to have poly-logarithmic worst-case guarantees under the new measure. The main result of the article is the analysis of a popular spectral algorithm. One variant of spectral clustering turns out to have effective worst-case guarantees; another finds a "good" clustering, if one exists.

We consider the problem of network design in transparent, or clear channel, optical networks associated with wavelength-division multiplexing (WDM). We focus on the class of traffic engineering models known as routing, wavelength, and capacity assignment problems. Here, in contrast to traditional networks, traffic flow paths must also be assigned an end-to-end wavelength. This additional requirement means that there can be an increased cost associated with optimal capacity allocations for such WDM-flows. In general, this can be arbitrarily worse than traditional network designs. We argue that in order to evaluate the benefit of different switch technologies, a good benchmark is to measure the increase in costs purely in terms of link capacity, we call this the cost of transparency. Experimental research shows that this cost is small in multifiber networks with modest switching functionality at the nodes. We present theoretical justification for why this occurs, and prove that in multiwavelength multifiber transparent networks the cost of transparency all but disappears if there is moderate traffic load. Our arguments are based on efficient heuristics that may also be useful for more complex network optimizations. This suggests that the cost savings from using wavelength converters is significant only in young networks with relatively few fibers lit. Such savings may, thus, be small relative to the initial capital expense involved in installing wavelength conversion.

We present an approximation algorithm for the problem of finding a minimum-cost k-vertex connected spanning subgraph, assuming that the number of vertices is at least 6k 2 . The approximation guarantee is six times the kth harmonic number (which is O(log k)), and this is also an upper bound on the integrality ratio for a standard linear programming relaxation.

We consider the following class of problems. The value of an outcome to a society is measured via a submodular utility function (submodularity has a natural economic interpretation: decreasing marginal utility). Decisions, however, are controlled by non-cooperative agents who seek to maximise their own private utility. We present, under basic assumptions, guarantees on the social performance of Nash equilibria. For submodular utility functions, any Nash equilibrium gives an expected social utility within a factor 2 of optimal, subject to a function-dependent additive term. For non-decreasing, submodular utility functions, any Nash equilibrium gives an expected social utility within a factor 1+/spl delta/ of optimal, where 0/spl les//spl delta//spl les/1 is a number based upon discrete curvature of the function. A condition under which all sets of social and private utility functions induce pure strategy Nash equilibria is presented. The case in which agents themselves make use of approximation algorithms in decision making is discussed and performance guarantees given. Finally we present specific problems that fall into our framework. These include competitive versions of the facility location problem and k-median problem, a maximisation version of the traffic routing problem studied by Roughgarden and Tardos (2000), and multiple-item auctions.

We present two new algorithms for the problem of nding a minimum-cost k-vertex connected spanning subgraph. The rst algorithm works on undirected graphs with at least 6k vertices and achieves an approximation of 6 times the kth harmonic number (which is O(log k)), The second algorithm works on any graph (directed or undirected) and gives an O( n=)-approximation algorithm for any > 0 and k (1 )n. These algorithms improve on the previous best approximation factor (more than k=2). The latter algorithm also extends to other problems in network design with vertex connectivity requirements. Our main tools are setpair relaxations, a theorem of Mader's (in the undirected case) and iterative rounding (general case).

We propose a new measure for assessing the quality of a clustering. A simple heuristic is shown to give worst-case guarantees under the new measure. Then we present two results regarding the quality of the clustering found by a popular spectral algorithm. One proffers worst case guarantees whilst the other shows that if there exists a "good" clustering then the spectral algorithm will find one close to it.