Parallel Algorithms for the Edge-Coloring and Edge-Coloring Update Problems

LetG(V,E) be a simple undirected graph with a maximum vertex degree Δ(G) (or Δ for short), |V| =nand |E| =m. An edge-coloring ofGis an assignment to each edge inGa color such that all edges sharing a common vertex have different colors. The minimum number of colors needed is denoted by χ′(G) (called thechromatic index). For a simple graphG, it is known that Δ ≤ χ′(G) ≤ Δ + 1. This paper studies two edge-coloring problems. The first problem is to perform edge-coloring for an existing edge-colored graphGwith Δ + 1 colors stemming from the addition of a new vertex intoG. The proposed parallel algorithm for this problem runs inO(Δ^3/2log³Δ + Δ logn) time usingO(max{nΔ, Δ³}) processors. The second problem is to color the edges of a given uncolored graphGwith Δ + 1 colors. For this problem, our first parallel algorithm requiresO(Δ^5.5log³Δ logn+ Δ⁵log⁴n) time andO(max{n²Δ,nΔ³}) processors, which is a slight improvement on the algorithm by H. J. Karloff and D. B. Shmoys [J. Algorithms8 (1987), 39–52]. Their algorithm costsO(Δ⁶log⁴n) time andO(n²Δ) processors if we use the fastest known algorithm for finding maximal independent sets by M. Goldberg and T. Spencer [SIAM J. Discrete Math.2 (1989), 322–328]. Our second algorithm requiresO(Δ^4.5log³Δ logn+ Δ⁴log⁴n) time andO(max{n²,nΔ³}) processors. Finally, we present our third algorithm by incorporating the second algorithm as a subroutine. This algorithm requiresO(Δ^3.5log³Δ logn+ Δ³log⁴n) time andO(max{n²log Δ,nΔ³}) processors, which improves, by anO(Δ^2.5) factor in time, on Karloff and Shmoys' algorithm. All of these algorithms run in the COMMON CRCW PRAM model.     

Fully Dynamic Geometric Spanners

In this paper we present an algorithm for maintaining a spanner over a dynamic set of points in constant doubling dimension metric spaces. For a set S of points in ? ^d , a t-spanner is a sparse graph on the points of S such that there is a path in the spanner between any pair of points whose total length is at most t times the distance between the points. We present the first fully dynamic algorithm for maintaining a spanner whose update time depends solely on the number of points in S. In particular, we show how to maintain a (1+ε)-spanner with O(n/ε ^d ) edges, where points can be inserted to S in an amortized update time of O(log?n) and deleted from S in an amortized update time of $\tilde{O}(n^{1/3})$ . As a by-product of our techniques we obtain a simple incremental algorithm for constructing a (1+ε)-spanner with O(n/ε ^d ) edges in constant doubling dimension metric spaces whose running time is O(nlog?n).     

On the complexity of finding balanced oneway cuts

A bisection of an n-vertex graph is a partition of its vertices into two sets S and T, each of size n/2. The bisection cost is the number of edges connecting the two sets. In directed graphs, the cost is the number of arcs going from S to T. Finding a minimum cost bisection is NP-hard for both undirected and directed graphs. For the undirected case, an approximation of ratio O(log²n) is known. We show that directed minimum bisection is not approximable at all. More specifically, we show that it is NP-hard to tell whether there exists a directed bisection of cost 0, which we call oneway bisection. In addition, we study the complexity of the problem when some slackness in the size of S is allowed, namely, (1/2−ε)n?|S|?(1/2+ε)n. We show that the problem is solvable in polynomial time when , and provide evidence that the problem is not solvable in polynomial time when ε=o(1/(logn)⁴).     

Hardness results for approximate pure Horn CNF formulae minimization

We study the hardness of approximation of clause minimum and literal minimum representations of pure Horn functions in n Boolean variables. We show that unless P=NP, it is not possible to approximate in polynomial time the minimum number of clauses and the minimum number of literals of pure Horn CNF representations to within a factor of \(2^{\log^{1-o(1)} n}\) . This is the case even when the inputs are restricted to pure Horn 3-CNFs with O(n ^1+ε ) clauses, for some small positive constant ε. Furthermore, we show that even allowing sub-exponential time computation, it is still not possible to obtain constant factor approximations for such problems unless the Exponential Time Hypothesis turns out to be false.     

Approximation Schemes for Packing Splittable Items with Cardinality Constraints

We continue the study of bin packing with splittable items and cardinality constraints. In this problem, a set of n items must be packed into as few bins as possible. Items may be split, but each bin may contain at most?k (parts of) items, where k is some given parameter. Complicating the problem further is the fact that items may be larger than?1, which is the size of a bin. The problem is known to be strongly NP-hard for any fixed value of?k. We essentially close this problem by providing an efficient polynomial-time approximation scheme (EPTAS) for most of its versions. Namely, we present an efficient polynomial time approximation scheme for k=o(n). A?PTAS for k=Θ(n) does not exist unless P = NP. Additionally, we present dual approximation schemes for k=2 and for constant values of?k. Thus we show that for any ε>0, it is possible to pack the items into the optimal number of bins in polynomial time, if the algorithm may use bins of size 1+ε.     

Distributed deterministic edge coloring using bounded neighborhood independence

We study the edge-coloring problem in the message-passing model of distributed computing. This is one of the most fundamental problems in this area. Currently, the best-known deterministic algorithms for (2Δ ?1)-edge-coloring requires O(Δ) +  log* n time (Panconesi and Rizzi in Distrib Comput 14(2):97–100, 2001), where Δ is the maximum degree of the input graph. Also, recent results of Barenboim and Elkin (2010) for vertex-coloring imply that one can get an O(Δ)-edge-coloring in ${O(\Delta^{\epsilon}\cdot \log n)}$ time, and an ${O(\Delta^{1 + \epsilon})}$ -edge-coloring in O(log Δ log n) time, for an arbitrarily small constant ${\epsilon > 0}$ . In this paper we devise a significantly faster deterministic edge-coloring algorithm. Specifically, our algorithm computes an O(Δ)-edge-coloring in ${O(\Delta^{\epsilon}) + \log* n}$ time, and an ${O(\Delta^{1 + \epsilon})}$ -edge-coloring in O(log Δ) +  log* n time. This result improves the state-of-the-art running time for deterministic edge-coloring with this number of colors in almost the entire range of maximum degree Δ. Moreover, it improves it exponentially in a wide range of Δ, specifically, for 2 ^Ω(log*n) ≤ Δ ≤ polylog(n). In addition, for small values of Δ (up to log^{1 - δ} n, for some fixed δ > 0) our deterministic algorithm outperforms all the existing randomized algorithms for this problem. Also, our algorithm is the first O(Δ)-edge-coloring algorithm that has running time o(Δ) + log* n, for the entire range of Δ. All previous (deterministic and randomized) O(Δ)-edge-coloring algorithms require ${\Omega(\min \{\Delta, \sqrt{\log n}\ \})}$ time. On our way to these results we study the vertex-coloring problem on graphs with bounded neighborhood independence. This is a large family of graphs, which strictly includes line graphs of r-hypergraphs (i.e., hypergraphs in which each hyperedge contains r or less vertices) for r =  O(1), and graphs of bounded growth. We devise a very fast deterministic algorithm for vertex-coloring graphs with bounded neighborhood independence. This algorithm directly gives rise to our edge-coloring algorithms, which apply to general graphs. Our main technical contribution is a subroutine that computes an O(Δ/p)-defective p-vertex coloring of graphs with bounded neighborhood independence in O(p ²) + log* n time, for a parameter p, 1 ≤ p ≤ Δ. In all previous efficient distributed routines for m-defective p-coloring the product m· p is super-linear in Δ. In our routine this product is linear in Δ, and this enables us to speed up the algorithm drastically.     

Solving the 2-Disjoint Connected Subgraphs Problem Faster than 2 n

The 2-Disjoint Connected Subgraphs problem, given a graph along with two disjoint sets of terminals Z ₁,Z ₂, asks whether it is possible to find disjoint sets A ₁,A ₂, such that Z ₁?A ₁, Z ₂?A ₂ and A ₁,A ₂ induce connected subgraphs. While the naive algorithm runs in O(2 ⁿ n ^O(1)) time, solutions with complexity of form O((2?ε) ⁿ ) have been found only for special graph classes (van ’t Hof et al. in Theor. Comput. Sci. 410(47–49):4834–4843, 2009; Paulusma and van Rooij in Theor. Comput. Sci. 412(48):6761–6769, 2011). In this paper we present an O(1.933 ⁿ ) algorithm for 2-Disjoint Connected Subgraphs in general case, thus breaking the 2 ⁿ barrier. As a counterpoise of this result we show that if we parameterize the problem by the number of non-terminal vertices, it is hard both to speed up the brute-force approach and to find a polynomial kernel.     

Minimizing the number of late jobs on a single machine under due date uncertainty

We study the problem of minimizing the number of late jobs on a single machine where job processing times are known precisely and due dates are uncertain. The uncertainty is captured through a set of scenarios. In this environment, an appropriate criterion to select a schedule is to find one with the best worst-case performance, which minimizes the maximum number of late jobs over all scenarios. For a variable number of scenarios and two distinct due dates over all scenarios, the problem is proved NP-hard in the strong sense and non-approximable in pseudo-polynomial time with approximation ratio less than 2. It is polynomially solvable if the number s of scenarios and the number v of distinct due dates over all scenarios are given constants. An O(nlog?n) time s-approximation algorithm is suggested for the general case, where n is the number of jobs, and a polynomial 3-approximation algorithm is suggested for the case of unit-time jobs and a constant number of scenarios. Furthermore, an O(n ^s+v?2/(v?1) ^v?2) time dynamic programming algorithm is presented for the case of unit-time jobs. The problem with unit-time jobs and the number of late jobs not exceeding a given constant value is solvable in polynomial time by an enumeration algorithm. The obtained results are related to a min-max assignment problem, an exact assignment problem and a multi-agent scheduling problem.     

Online capacity maximization in wireless networks

In this paper we study a dynamic version of capacity maximization in the physical model of wireless communication. In our model, requests for connections between pairs of points in Euclidean space of constant dimension d arrive iteratively over time. When a new request arrives, an online algorithm needs to decide whether or not to accept the request and to assign one out of k channels and a transmission power to the request. Accepted requests must satisfy constraints on the signal-to-interference-plus-noise (SINR) ratio. The objective is to maximize the number of accepted requests. Using competitive analysis we study algorithms using distance-based power assignments, for which the power of a request relies only on the distance between the points. Such assignments are inherently local and particularly useful in distributed settings. We first focus on the case of a single channel. For request sets with spatial lengths in [1,Δ] and duration in [1,Γ] we derive a lower bound of Ω(Γ?Δ ^d/2) on the competitive ratio of any deterministic online algorithm using a distance-based power assignment. Our main result is a near-optimal deterministic algorithm that is O(Γ?Δ^(d/2)+ε )-competitive, for any constant ε>0. Our algorithm for a single channel can be generalized to k channels. It can be adjusted to yield a competitive ratio of O(k?Γ ^1/k′?Δ^(d/2k″)+ε ) for any factorization (k′,k″) such that k′?k″=k. This illustrates the effectiveness of multiple channels when dealing with unknown request sequences. In particular, for Θ(log?Γ?log?Δ) channels this yields an O(log?Γ?log?Δ)-competitive algorithm. Additionally, we show how this approach can be turned into a randomized algorithm, which is O(log?Γ?log?Δ)-competitive even for a single channel.     

A simple factor-3 approximation for labeling points with circles

In this paper we present a simple factor-(3+ε), 0<ε<1, approximation algorithm, which runs in O(nlogn+n(1/ε)^O(1/ε2)log(D₃/εD₂)) time, for the problem of labeling a set P of n distinct points with uniform circles. (D₂ is the closest pair of P and D₃ is the minimum diameter of all subsets of P with size three.) This problem is known to be NP-hard. Our bound improves the previous factor of 3.6+ε.     

Bipartite Matching in the Semi-streaming Model

We present the first deterministic 1+ε approximation algorithm for finding a large matching in a bipartite graph in the semi-streaming model which requires only O((1/ε)⁵) passes over the input stream. In this model, the input graph G=(V,E) is given as a stream of its edges in some arbitrary order, and storage of the algorithm is bounded by O(npolylog?n) bits, where $n = \lvert {V}\rvert $ . The only previously known arbitrarily good approximation for general graphs is achieved by the randomized algorithm of McGregor (Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and Randomization and Computation, Berkeley, CA, USA, pp. 170–181, 2005), which uses Ω((1/ε)^1/ε ) passes. We show that even for bipartite graphs, McGregor’s algorithm needs Ω(1/ε) ^Ω(1/ε) passes, thus it is necessarily exponential in the approximation parameter. The design as well as the analysis of our algorithm require the introduction of some new techniques. A novelty of our algorithm is a new deterministic assignment of matching edges to augmenting paths which is responsible for the complexity reduction, and gets rid of randomization. We repeatedly grow an initial matching using augmenting paths up to a length of 2k+1 for k=?2/ε?. We terminate when the number of augmenting paths found in one iteration falls below a certain threshold also depending on k, that guarantees a 1+ε approximation. The main challenge is to find those augmenting paths without requiring an excessive number of passes. In each iteration, using multiple passes, we grow a set of alternating paths in parallel, considering each edge as a possible extension as it comes along in the stream. Backtracking is used on paths that fail to grow any further. Crucial are the so-called position limits: when a matching edge is the ith matching edge in a path and it is then removed by backtracking, it will only be inserted into a path again at a position strictly lesser than i. This rule strikes a balance between terminating quickly on the one hand and giving the procedure enough freedom on the other hand.     

Convergence of Map Seeking Circuits

The map-seeking circuit (MSC) is an explicit biologically-motivated computational mechanism which provides practical solution of problems in object recognition, image registration and stabilization, limb inverse-kinematics and other inverse problems which involve transformation discovery. We formulate this algorithm as discrete dynamical system on a set Δ=Π _?=1 ^L Δ^(?), where each Δ^(?) is a compact subset of a nonnegative orthant of ? ⁿ , and show that for an open and dense set of initial conditions in Δ the corresponding solutions converge to either a vector with unique nonzero element in each Δ^(?) or to a zero vector. The first result implies that the circuit finds a unique best mapping which relates reference pattern to a target pattern; the second result is interpreted as “no match found”. These results verify numerically observed behavior in numerous practical applications.     

Faster Algorithms for All-Pairs Small Stretch Distances in Weighted Graphs

Let G=(V,E) be a weighted undirected graph, with non-negative edge weights. We consider the problem of efficiently computing approximate distances between all pairs of vertices in?G. While many efficient algorithms are known for this problem in unweighted graphs, not many results are known for this problem in weighted graphs. Zwick?(J. Assoc. Comput. Mach. 49:289–317, 2002) showed that for any fixed ε>0, stretch 1+ε distances (a path in G between u,v∈V is said to be of stretch t if its length is at most t times the distance between u and v in G) between all pairs of vertices in a weighted directed graph on n vertices can be computed in $\tilde{O}(n^{\omega})$ time, where ω<2.376 is the exponent of matrix multiplication and n is the number of vertices. It is known that finding distances of stretch less than 2 between all pairs of vertices in G is at least as hard as Boolean matrix multiplication of two n×n matrices. Here we show that all pairs stretch 2+ε distances for any fixed ε>0 in G can be computed in expected time O(n ^9/4). This algorithm uses a fast rectangular matrix multiplication subroutine. We also present a combinatorial algorithm (that is, it does not use fast matrix multiplication) with expected running time O(n ^9/4) for computing all-pairs stretch 5/2 distances in?G. This combinatorial algorithm will serve as a key step in our all-pairs stretch 2+ε distances algorithm.     

An improved exact algorithm for the domatic number problem

The 3-domatic number problem asks whether a given graph can be partitioned into three dominating sets. We prove that this problem can be solved by a deterministic algorithm in time n^2.695 (up to polynomial factors) and in polynomial space. This result improves the previous bound of n^2.8805, which is due to Björklund and Husfeldt. To prove our result, we combine an algorithm by Fomin et al. with Yamamoto's algorithm for the satisfiability problem. In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree Δ(G) by a randomized polynomial-space algorithm, whose running time is better than the previous bound due to Riege and Rothe whenever Δ(G)?5. Our new randomized algorithm employs Schöning's approach to constraint satisfaction problems.     

Redundancy of minimal weight expansions in Pisot bases

Motivated by multiplication algorithms based on redundant number representations, we study representations of an integer n as a sum n=∑_kε_kU_k, where the digits ε_k are taken from a finite alphabet Σ and (U_k)_k is a linear recurrent sequence of Pisot type with U₀=1. The most prominent example of a base sequence (U_k)_k is the sequence of Fibonacci numbers. We prove that the representations of minimal weight ∑_k|ε_k| are recognised by a finite automaton and obtain an asymptotic formula for the average number of representations of minimal weight. Furthermore, we relate the maximal number of representations of a given integer to the joint spectral radius of a certain set of matrices.     

Preprocess, Set, Query!

Thorup and Zwick (J. ACM 52(1):1–24, 2005 and STOC’01) in their seminal work introduced the notion of distance oracles. Given an n-vertex weighted undirected graph with m edges, they show that for any integer k≥1 it is possible to preprocess the graph in $\tilde {O}(mn^{1/k})$ time and generate a compact data structure of size O(kn ^1+1/k ). For each pair of vertices, it is then possible to retrieve an estimated distance with multiplicative stretch 2k?1 in O(k) time. For k=2 this gives an oracle of O(n ^1.5) size that produces in constant time estimated distances with stretch 3. Recently, Pǎtra?cu and Roditty (In: Proc. of 51st FOCS, 2010) broke the theoretical status-quo in the field of distance oracles and obtained a distance oracle for sparse unweighted graphs of O(n ^5/3) size that produces in constant time estimated distances with stretch 2. In this paper we show that it is possible to break the stretch 2 barrier at the price of non-constant query time in unweighted undirected graphs. We present a data structure that produces estimated distances with 1+ε stretch. The size of the data structure is O(nm ^1?ε′) and the query time is $\tilde {O}(m^{1-\varepsilon '})$ . Using it for sparse unweighted graphs we can get a data structure of size O(n ^1.87) that can supply in O(n ^0.87) time estimated distances with multiplicative stretch 1.75.     

On the Largest Empty Axis-Parallel Box Amidst n Points

We give the first efficient (1?ε)-approximation algorithm for the following problem: Given an axis-parallel d-dimensional box R in ? ^d containing n points, compute a maximum-volume empty axis-parallel d-dimensional box contained in R. The minimum of this quantity over all such point sets is of the order $\Theta (\frac {1}{n} )$ . Our algorithm finds an empty axis-aligned box whose volume is at least (1?ε) of the maximum in O((8edε ^?2) ^d ?nlog ^d n) time. No previous efficient exact or approximation algorithms were known for this problem for d≥4. As the problem has been recently shown to be NP-hard in arbitrarily high dimensions (i.e., when d is part of the input), the existence of an efficient exact algorithm is unlikely. We also present a (1?ε)-approximation algorithm that, given an axis-parallel d-dimensional cube R in ? ^d containing n points, computes a maximum-volume empty axis-parallel hypercube contained in R. The minimum of this quantity over all such point sets is also shown to be of the order $\Theta (\frac{1}{n} )$ . A faster (1?ε)-approximation algorithm, with a milder dependence on d in the running time, is obtained in this case.     

A family of graphs with expensive depth-reduction

A family of directed acyclic graphs G_n with 2ⁿ⁺¹?1 nodes, n · 2ⁿ edges and depth 2ⁿ⁺¹?2 is constructed having the property: For any ε (0?ε<1) it is necessary to remove ω(n · 2ⁿ) edges in order to reduce the depth of G_n to (2ⁿ)^ε.     

Studies on the distribution of the shortest linear recurring sequences

The distribution of the shortest linear recurrence (SLR) sequences in the Z/(p) field and over the Z/(p^e) ring is studied. It is found that the length of the shortest linear recurrent (SLRL) is always equal to n/2, if n is even and n/2 + 1 if n is odd in the Z/(p) field, respectively. On the other hand, over the Z/(p^e) ring, the number of sequences with length n can also be calculated. The recurring distribution regulation of the shortest linear recurring sequences is also found. To solve the problem of calculating the SLRL, a new simple representation of the Berlekamp-Massey algorithm is developed as well.