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International audience; We devise a constant-factor approximation algorithm for the maximization version of the edge-disjoint paths problem if the supply graph together with the demand edges form a planar graph. By planar duality this is equivalent to packing cuts in a planar graph such that each cut contains exactly one demand edge. We also show that the natural linear programming relaxations have constant integrality gap, yielding an approximate max-multiflow min-multicut theorem.

We study the detection and the reconstruction of a large very dense subgraph in a social graph with $n$ nodes and $m$ edges given as a stream of edges, when the graph follows a power law degree distribution, in the regime when $m=O(n. \log n)$. A subgraph $S$ is very dense if it has $\Omega(|S|^2)$ edges. We uniformly sample the edges with a Reservoir of size $k=O(\sqrt{n}.\log n)$. Our detection algorithm checks whether the Reservoir has a giant component. We show that if the graph contains a very dense subgraph of size $\Omega(\sqrt{n})$, then the detection algorithm is almost surely correct. On the other hand, a random graph that follows a power law degree distribution almost surely has no large very dense subgraph, and the detection algorithm is almost surely correct. We define a new model of random graphs which follow a power law degree distribution and have large very dense subgraphs. We then show that on this class of random graphs we can reconstruct a good approximation of the very dense subgraph with high probability. We generalize these results to dynamic graphs defined by sliding windows in a stream of edges., Comment: 32 pages

A top-list is a possibly incomplete ranking of elements: only a subset of the elements are ranked, with all unranked elements tied for last. Top-list aggregation, a generalization of the well-known rank aggregation problem, takes as input a collection of top-lists and aggregates them into a single complete ranking, aiming to minimize the number of upsets (pairs ranked in opposite order in the input and in the output). In this paper, we give simple approximation algorithms for top-list aggregation. * We generalize the footrule algorithm for rank aggregation. * Using inspiration from approval voting, we define the score of an element as the frequency with which it is ranked, i.e. appears in an input top-list. We reinterpret Ailon's RepeatChoice algorithm for top-list aggregation using the score of an element and its average rank given that it is ranked. * Using average ranks, we generalize and analyze Borda's algorithm for rank aggregation. * We design a simple 2-phase variant of the Generalized Borda's algorithm, roughly sorting by scores and breaking ties by average ranks. * We then design another 2-phase variant in which in order to break ties we use, as a black box, the Mathieu-Schudy PTAS for rank aggregation, yielding a PTAS for top-list aggregation. * Finally, we discuss the special case in which all input lists have constant length., Comment: To appear in SODA'20

Sherali and Adams (SIAM J Discrete Math 3:411–430, 1990) and Lovasz and Schrijver (SIAM J Optim 1:166–190, 1991) developed systematic procedures to construct the hierarchies of relaxations known as lift-and-project methods. They have been proven to be a strong tool for developing approximation algorithms, matching the best relaxations known for problems like Max-Cut and Sparsest-Cut. In this work we provide lower bounds for these hierarchies when applied over the configuration LP for the problem of scheduling identical machines to minimize the makespan. First we show that the configuration LP has an integrality gap of at least 1024/1023 by providing a family of instances with 15 different job sizes. Then we show that for any integer n there is an instance with n jobs in this family such that after $$\varOmega (n)$$ rounds of the Sherali–Adams ( $$\text {SA}$$ ) or the Lovasz–Schrijver ( $$\text {LS}_+$$ ) hierarchy the integrality gap remains at least 1024/1023.

We give a randomized O(n polylog n)-time approximation scheme for the Steiner forest problem in the Euclidean plane. For every fixed eps > 0 and given n terminals in the plane with connection requests between some pairs of terminals, our scheme finds a (1 + eps)-approximation to the minimum-length forest that connects every requested pair of terminals., This version is more recent than that appearing in the FOCS proceedings. The partition step has been corrected and the overall presentation has been clarified and formalized. This paper has been accepted to TALG

In the capacitated vehicle routing problem we are given the locations of customers and depots, along with a vehicle of capacity $$k$$k. The objective is to find a minimum length collection of tours covering all customers such that each tour starts and end at a depot and visits at most $$k$$k customers. The problem is a generalization of the traveling salesman problem. We present a quasipolynomial time approximation scheme for the Euclidean setting of the problem when all points lie in $$\mathbb {R}^d$$Rd for constant dimension $$d$$d.

We study online packing and covering problems, stated as zero-one linear programs. This class of problems includes maximum cut, bipartite matching, and set cover. In the packing problem, variables arrive in an online fashion. Upon arrival of v, the weight of v in the objective function and in any constraints involving v is revealed and the algorithm may update its solution. In the covering problem, constraints arrive in an online fashion. We allow the relaxation that the algorithm may change the value of each variable a constant number of times, k. We give an algorithm for OnlinePacking(k) and prove that for any @e>0, there exists some k equal to [email protected]?(1/@e) such that the algorithm gives a ([email protected]) approximation for OnlinePacking(k). We then show near-tightness for several cases. We also prove that no deterministic algorithm exists for OnlineCovering(k) that achieves a competitive ratio less than 3/2.

International audience; We give the first polynomial-time approximation schemes (PTASs) for the following problems: (1) uniform facility location in edge-weighted planar graphs; (2) $k$-median and $k$-means in edge-weighted planar graphs; and (3) $k$-means in Euclidean space of bounded dimension. Our first and second results extend to minor-closed families of graphs. All our results extend to cost functions that are the $p$th power of the shortest-path distance. The algorithm is local search, where the local neighborhood of a solution $S$ consists of all solutions obtained from $S$ by removing and adding $1/\varepsilon^{\Theta(1)}$ centers.

We present a framework for addressing several problems on weighted planar graphs and graphs of bounded genus. With that framework, we derive polynomial-time approximation schemes for the following problems in planar graphs or graphs of bounded genus: edge-weighted tree cover and tour cover; vertex-weighted connected dominating set, max-weight-leaf spanning tree, and connected vertex cover. In addition, we obtain a polynomial-time approximation scheme for feedback vertex set in planar graphs. These are the first polynomial-time approximation schemes for all those problems in weighted embedded graphs. (For unweighted versions of some of these problems, polynomial-time approximation schemes were previously given using bidimensionality.)

We prove that the randomized competitive ratio of online chain partitioning of posets equals the deterministic competitive ratio.

We give a Polynomial-Time Approximation Scheme (PTAS) for the Steiner tree problem in planar graphs. The running time is O ( n log n ).

In this paper, we show how we can derive lower bounds and also compute the exact distortion for the line embeddings of some special metrics, especially trees and graphs with certain structure. Using linear programming to formulate a simpler version of the problem gives an interesting intuition and direction concerning the computation of general lower bounds for distortion into the line. We also show that our lower bounds on special cases of metrics are a lot better than previous lower bounds.

We examine the classic on-line bipartite matching problem studied by Karp, Vazirani, and Vazirani [8] and provide a simple proof of their result that the Ranking algorithm for this problem achieves a competitive ratio of 1 -- 1/ e .

We study the problem of reconstructing a hidden graph given access to a distance oracle. We design randomized algorithms for the following problems: reconstruction of a degree bounded graph with query complexity $\tilde{O}(n^{3/2})$; reconstruction of a degree bounded outerplanar graph with query complexity $\tilde{O}(n)$; and near-optimal approximate reconstruction of a general graph.

A permutation $a_1a_2\ldots a_n$ is indecomposable if there does not exist $p

The mean weight of a cycle in an edge-weighted graph is the sum of the cycle's edge weights divided by the cycle's length. We study the minimum mean-weight cycle on the complete graph on n vertices, with random i.i.d. edge weights drawn from an exponential distribution with mean 1. We show that the probability of the min mean weight being at most c/n tends to a limiting function of c which is analytic for c1/e. We further show that if the min mean weight is 1/(en), then the relevant cycle almost always has mean weight (1+o(1))/(en) and length at least (2/pi^2-o(1)) log^2 n log log n., 21 pages, 1 figure

We present three semi-streaming algorithms for Maximum Bipartite Matching with one and two passes. Our one-pass semi-streaming algorithm is deterministic and returns a matching of size at least 1/2 + 0.005 times the optimal matching size in expectation, assuming that edges arrive one by one in (uniform) random order. Our first two-pass algorithm is randomized and returns a matching of size at least 1/2 + 0.019 times the optimal matching size in expectation (over its internal random coin flips) for any arrival order. These two algorithms apply the simple Greedy matching algorithm several times on carefully chosen subgraphs as a subroutine. Furthermore, we present a two-pass deterministic algorithm for any arrival order returning a matching of size at least 1/2 + 0.019 times the optimal matching size. This algorithm is built on ideas from the computation of semi-matchings.

We study the problem of producing a global ranking of items given pairwise ranking information, when the items to be ranked arrive in an online fashion. We study both the maximization and the minimization versions of the problem on tournaments (max acyclic subgraph, feedback arc set). We also study the case when the items arrive in random order.

In this paper, we study the integrality gap of the Knapsack linear program in the Sherali-Adams and Lasserre hierarchies. First, we show that an integrality gap of 2 - e persists up to a linear number of rounds of Sherali-Adams, despite the fact that Knapsack admits a fully polynomial time approximation scheme [24, 30]. Second, we show that the Lasserre hierarchy closes the gap quickly. Specifically, after t rounds of Lasserre, the integrality gap decreases to t/(t - 1). This answers the open question in [9]. Also, to the best of our knowledge, this is the first positive result that uses more than a small number of rounds in the Lasserre hierarchy. Our proof uses a decomposition theorem for the Lasserre hierarchy, which may be of independent interest.

We study the Sherali-Adams lift-and-project hierarchy of linear programming relaxations of the matching polytope. Our main result is an asymptotically tight expression 1+1/k for the integrality gap after k rounds of this hierarchy. The result is derived by a detailed analysis of the LP after k rounds applied to the complete graph K_{2d+1}. We give an explicit recurrence for the value of this LP, and hence show that its gap exhibits a "phase transition," dropping from close to its maximum value 1+1/2d to close to 1 around the threshold k=2d-Θ(√d). We also show that the rank of the matching polytope (i.e., the number of Sherali-Adams rounds until the integer polytope is reached) is exactly 2d-1.

A permutation $a_1a_2 \ldots a_n$ is $\textit{indecomposable}$ if there does not exist $p \lt n$ such that $a_1a_2 \ldots a_p$ is a permutation of $\{ 1,2, \ldots ,p\}$. We compute the asymptotic probability that a permutation of $\mathbb{S}_n$ with $m$ cycles is indecomposable as $n$ goes to infinity with $m/n$ fixed. The error term is $O(\frac{\log(n-m)}{ n-m})$. The asymptotic probability is monotone in $m/n$, and there is no threshold phenomenon: it degrades gracefully from $1$ to $0$. When $n=2m$, a slight majority ($51.1 \ldots$ percent) of the permutations are indecomposable. We also consider indecomposable fixed point free involutions which are in bijection with maps of arbitrary genus on orientable surfaces, for these involutions with $m$ left-to-right maxima we obtain a lower bound for the probability of being indecomposable., Une permutation $a_1a_2 \ldots a_n$ est $\textit{indécomposable}$, s’il n’existe pas de $p \lt n$ tel que $a_1a_2 \ldots a_p$ est une permutation de $\{ 1,2, \ldots ,p\}$. Nous calculons la probabilité pour qu’une permutation de $\mathbb{S}_n$ ayant $m$ cycles soit indécomposable et plus particulièrement son comportement asymptotique lorsque $n$ tend vers l’infini et que $m=n$ est fixé. Cette valeur décroît régulièrement de $1$ à $0$ lorsque $m=n$ croît, et il n’y a pas de phénomène de seuil. Lorsque $n=2m$, une faible majorité ($51.1 \ldots$ pour cent) des permutations sont indécomposables. Nous considérons aussi les involutions sans point fixe indécomposables qui sont en bijection avec les cartes de genre quelconque plongées dans une surface orientable, pour ces involutions ayant $m$ maxima partiels (ou records) nous obtenons une borne inférieure pour leur probabilité d’êtres indécomposables.

Motivated by a concrete problem and with the goal of understanding the sense in which the complexity of streaming algorithms is related to the complexity of formal languages, we investigate the problem Dyck(s) of checking matching parentheses, with $s$ different types of parenthesis. We present a one-pass randomized streaming algorithm for Dyck(2) with space $\Order(\sqrt{n}\log n)$, time per letter $\polylog (n)$, and one-sided error. We prove that this one-pass algorithm is optimal, up to a $\polylog n$ factor, even when two-sided error is allowed. For the lower bound, we prove a direct sum result on hard instances by following the "information cost" approach, but with a few twists. Indeed, we play a subtle game between public and private coins. This mixture between public and private coins results from a balancing act between the direct sum result and a combinatorial lower bound for the base case. Surprisingly, the space requirement shrinks drastically if we have access to the input stream in reverse. We present a two-pass randomized streaming algorithm for Dyck(2) with space $\Order((\log n)^2)$, time $\polylog (n)$ and one-sided error, where the second pass is in the reverse direction. Both algorithms can be extended to Dyck(s) since this problem is reducible to Dyck(2) for a suitable notion of reduction in the streaming model., Comment: 20 pages, 5 figures

We give an algorithm that, for any e > 0, any undirected planar graph G, and any set S of nodes of G, computes a (1+e)-optimal Steiner tree in G that spans the nodes of S. The algorithm takes time O(2poly(1/e)n log n).

International audience; Integer programs defined by two equations with two free integer variables and nonnegative continuous variables have three types of nontrivial facets: split, triangle or quadrilateral inequalities. In this paper, we compare the strength of these three families of inequalities. In particular we study how well each family approximates the integer hull. We show that, in a well defined sense, triangle inequalities provide a good approximation of the integer hull. The same statement holds for quadrilateral inequalities. On the other hand, the approximation produced by split inequalities may be arbitrarily bad.