Minimum-Weight Edge Discriminator in Hypergraphs

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

Fast Algorithms for Parameterized Problems with Relaxed Disjointness Constraints

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 $p$ into a given universe generalizes the Maximum Matching problem, which is recovered by taking $p=2$. 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 --- $r$-Simple $k$-Path --- that can be solved by a randomized algorithm with running time $O^*(2^{O(k \frac{\log r}{r})})$. That is, the complexity of the problem decreases with $r$. 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 $O^*(2^{O(k \frac{\log r}{r})})$ algorithm for $(r,k)$-Monomial Detection, which is the problem of finding a monomial of total degree $k$ and individual degrees at most $r$ 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 $r$-Simple $k$-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 $r$ 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

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

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 $2^d$ such as $2^d$-divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over $\mathbb{F}_2$ belong to the complexity class Polynomial Parity Argument (PPA)

A Characterization Theorem and An Algorithm for A Convex Hull Problem

Given $S= \{v_1, \dots, v_n\} \subset \mathbb{R} ^m$ and $p \in \mathbb{R} ^m$, testing if $p \in conv(S)$, the convex hull of $S$, 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: $p$ lies in $conv(S)$ if and only if for each $p' \in conv(S)$ there exists a {\it pivot}, $v_j \in S$ satisfying $d(p',v_j) \geq d(p,v_j)$. Equivalently, $p \not \in conv(S)$ if and only if there exists a {\it witness}, $p' \in conv(S)$ whose Voronoi cell relative to $p$ contains $S$. A witness separates $p$ from $conv(S)$ and approximate $d(p, conv(S))$ to within a factor of two. Next, we describe the {\it Triangle Algorithm}: given $\epsilon \in (0,1)$, an {\it iterate}, $p' \in conv(S)$, and $v \in S$, if $d(p, p') < \epsilon d(p,v)$, it stops. Otherwise, if there exists a pivot $v_j$, it replace $v$ with $v_j$ and $p'$ with the projection of $p$ onto the line $p'v_j$. Repeating this process, the algorithm terminates in $O(mn \min \{\epsilon^{-2}, c^{-1}\ln \epsilon^{-1} \})$ arithmetic operations, where $c$ is the {\it visibility factor}, a constant satisfying $c \geq \epsilon^2$ and $\sin (\angle pp'v_j) \leq 1/\sqrt{1+c}$, over all iterates $p'$. Additionally, (i) we prove a {\it strict distance duality} and a related minimax theorem, resulting in more effective pivots; (ii) describe $O(mn \ln \epsilon^{-1})$-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

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,