5,535 research outputs found

    Minimum-Weight Edge Discriminator in Hypergraphs

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    In this paper we introduce the concept of minimum-weight edge-discriminators in hypergraphs, and study its various properties. For a hypergraph H=(V,E)\mathcal H=(\mathcal V, \mathcal E), a function λ:VZ+{0}\lambda: \mathcal V\rightarrow \mathbb Z^{+}\cup\{0\} is said to be an {\it edge-discriminator} on H\mathcal H if vEiλ(v)>0\sum_{v\in E_i}{\lambda(v)}>0, for all hyperedges EiEE_i\in \mathcal E, and vEiλ(v)vEjλ(v)\sum_{v\in E_i}{\lambda(v)}\ne \sum_{v\in E_j}{\lambda(v)}, for every two distinct hyperedges Ei,EjEE_i, E_j \in \mathcal E. An {\it optimal edge-discriminator} on H\mathcal H, to be denoted by λH\lambda_\mathcal H, is an edge-discriminator on H\mathcal H satisfying vVλH(v)=minλvVλ(v)\sum_{v\in \mathcal V}\lambda_\mathcal H (v)=\min_\lambda\sum_{v\in \mathcal V}{\lambda(v)}, where the minimum is taken over all edge-discriminators on H\mathcal H. We prove that any hypergraph H=(V,E)\mathcal H=(\mathcal V, \mathcal E), with E=n|\mathcal E|=n, satisfies vVλH(v)n(n+1)/2\sum_{v\in \mathcal V} \lambda_\mathcal H(v)\leq n(n+1)/2, and equality holds if and only if the elements of E\mathcal E are mutually disjoint. For rr-uniform hypergraphs H=(V,E)\mathcal H=(\mathcal V, \mathcal E), it follows from results on Sidon sequences that vVλH(v)Vr+1+o(Vr+1)\sum_{v\in \mathcal V}\lambda_{\mathcal H}(v)\leq |\mathcal V|^{r+1}+o(|\mathcal V|^{r+1}), and the bound is attained up to a constant factor by the complete rr-uniform hypergraph. Next, we construct optimal edge-discriminators for some special hypergraphs, which include paths, cycles, and complete rr-partite hypergraphs. Finally, we show that no optimal edge-discriminator on any hypergraph H=(V,E)\mathcal H=(\mathcal V, \mathcal E), with E=n(3)|\mathcal E|=n (\geq 3), satisfies vVλH(v)=n(n+1)/21\sum_{v\in \mathcal V} \lambda_\mathcal H (v)=n(n+1)/2-1, which, in turn, raises many other interesting combinatorial questions.Comment: 22 pages, 5 figure

    Fast Algorithms for Parameterized Problems with Relaxed Disjointness Constraints

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    In parameterized complexity, it is a natural idea to consider different generalizations of classic problems. Usually, such generalization are obtained by introducing a "relaxation" variable, where the original problem corresponds to setting this variable to a constant value. For instance, the problem of packing sets of size at most pp into a given universe generalizes the Maximum Matching problem, which is recovered by taking p=2p=2. Most often, the complexity of the problem increases with the relaxation variable, but very recently Abasi et al. have given a surprising example of a problem --- rr-Simple kk-Path --- that can be solved by a randomized algorithm with running time O(2O(klogrr))O^*(2^{O(k \frac{\log r}{r})}). That is, the complexity of the problem decreases with rr. In this paper we pursue further the direction sketched by Abasi et al. Our main contribution is a derandomization tool that provides a deterministic counterpart of the main technical result of Abasi et al.: the O(2O(klogrr))O^*(2^{O(k \frac{\log r}{r})}) algorithm for (r,k)(r,k)-Monomial Detection, which is the problem of finding a monomial of total degree kk and individual degrees at most rr in a polynomial given as an arithmetic circuit. Our technique works for a large class of circuits, and in particular it can be used to derandomize the result of Abasi et al. for rr-Simple kk-Path. On our way to this result we introduce the notion of representative sets for multisets, which may be of independent interest. Finally, we give two more examples of problems that were already studied in the literature, where the same relaxation phenomenon happens. The first one is a natural relaxation of the Set Packing problem, where we allow the packed sets to overlap at each element at most rr times. The second one is Degree Bounded Spanning Tree, where we seek for a spanning tree of the graph with a small maximum degree

    Algorithms for the minimum sum coloring problem: a review

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    The Minimum Sum Coloring Problem (MSCP) is a variant of the well-known vertex coloring problem which has a number of AI related applications. Due to its theoretical and practical relevance, MSCP attracts increasing attention. The only existing review on the problem dates back to 2004 and mainly covers the history of MSCP and theoretical developments on specific graphs. In recent years, the field has witnessed significant progresses on approximation algorithms and practical solution algorithms. The purpose of this review is to provide a comprehensive inspection of the most recent and representative MSCP algorithms. To be informative, we identify the general framework followed by practical solution algorithms and the key ingredients that make them successful. By classifying the main search strategies and putting forward the critical elements of the reviewed methods, we wish to encourage future development of more powerful methods and motivate new applications

    Combinatorial Nullstellensatz modulo prime powers and the Parity Argument

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    We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime powers. We analyze computational search problems corresponding to these kinds of combinatorial questions and we prove that the problem of finding degree-constrained subgraphs modulo 2d2^d such as 2d2^d-divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over F2\mathbb{F}_2 belong to the complexity class Polynomial Parity Argument (PPA)

    A Characterization Theorem and An Algorithm for A Convex Hull Problem

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    Given S={v1,,vn}RmS= \{v_1, \dots, v_n\} \subset \mathbb{R} ^m and pRmp \in \mathbb{R} ^m, testing if pconv(S)p \in conv(S), the convex hull of SS, is a fundamental problem in computational geometry and linear programming. First, we prove a Euclidean {\it distance duality}, distinct from classical separation theorems such as Farkas Lemma: pp lies in conv(S)conv(S) if and only if for each pconv(S)p' \in conv(S) there exists a {\it pivot}, vjSv_j \in S satisfying d(p,vj)d(p,vj)d(p',v_j) \geq d(p,v_j). Equivalently, p∉conv(S)p \not \in conv(S) if and only if there exists a {\it witness}, pconv(S)p' \in conv(S) whose Voronoi cell relative to pp contains SS. A witness separates pp from conv(S)conv(S) and approximate d(p,conv(S))d(p, conv(S)) to within a factor of two. Next, we describe the {\it Triangle Algorithm}: given ϵ(0,1)\epsilon \in (0,1), an {\it iterate}, pconv(S)p' \in conv(S), and vSv \in S, if d(p,p)<ϵd(p,v)d(p, p') < \epsilon d(p,v), it stops. Otherwise, if there exists a pivot vjv_j, it replace vv with vjv_j and pp' with the projection of pp onto the line pvjp'v_j. Repeating this process, the algorithm terminates in O(mnmin{ϵ2,c1lnϵ1})O(mn \min \{\epsilon^{-2}, c^{-1}\ln \epsilon^{-1} \}) arithmetic operations, where cc is the {\it visibility factor}, a constant satisfying cϵ2c \geq \epsilon^2 and sin(ppvj)1/1+c\sin (\angle pp'v_j) \leq 1/\sqrt{1+c}, over all iterates pp'. Additionally, (i) we prove a {\it strict distance duality} and a related minimax theorem, resulting in more effective pivots; (ii) describe O(mnlnϵ1)O(mn \ln \epsilon^{-1})-time algorithms that may compute a witness or a good approximate solution; (iii) prove {\it generalized distance duality} and describe a corresponding generalized Triangle Algorithm; (iv) prove a {\it sensitivity theorem} to analyze the complexity of solving LP feasibility via the Triangle Algorithm. The Triangle Algorithm is practical and competitive with the simplex method, sparse greedy approximation and first-order methods.Comment: 42 pages, 17 figures, 2 tables. This revision only corrects minor typo

    A path following algorithm for the graph matching problem

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    We propose a convex-concave programming approach for the labeled weighted graph matching problem. The convex-concave programming formulation is obtained by rewriting the weighted graph matching problem as a least-square problem on the set of permutation matrices and relaxing it to two different optimization problems: a quadratic convex and a quadratic concave optimization problem on the set of doubly stochastic matrices. The concave relaxation has the same global minimum as the initial graph matching problem, but the search for its global minimum is also a hard combinatorial problem. We therefore construct an approximation of the concave problem solution by following a solution path of a convex-concave problem obtained by linear interpolation of the convex and concave formulations, starting from the convex relaxation. This method allows to easily integrate the information on graph label similarities into the optimization problem, and therefore to perform labeled weighted graph matching. The algorithm is compared with some of the best performing graph matching methods on four datasets: simulated graphs, QAPLib, retina vessel images and handwritten chinese characters. In all cases, the results are competitive with the state-of-the-art.Comment: 23 pages, 13 figures,typo correction, new results in sections 4,5,
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