Omid Amini

Hardness and approximation of traffic grooming

Traffic grooming is a central problem in optical networks. It refers to packing low rate signals into higher speed streams, in order to improve bandwidth utilization and reduce network cost. In WDM networks, the most accepted criterion is to minimize the number of electronic terminations, namely the number of SONET Add-Drop Multiplexers (ADMs). In this article we focus on ring and path topologies. On the one hand, we provide an inapproximability result for Traffic Grooming for fixed values of the grooming factor g, answering affirmatively the conjecture of Chow and Lin [T. Chow, P. Lin, The ring grooming problem, Networks 44 (2004), 194-202]. More precisely, we prove that Ring Traffic Grooming for fixed g>=1 and Path Traffic Grooming for fixed g>=2 are Apx-complete. That is, they do not accept a PTAS unless P=NP. Both results rely on the fact that finding the maximum number of edge-disjoint triangles in a tripartite graph (and more generally cycles of length 2g+1 in a (2g+1)-partite graph of girth 2g+1) is Apx-complete. On the other hand, we provide a polynomial-time approximation algorithm for Ring and Path Traffic Grooming, based on a greedy cover algorithm, with an approximation ratio independent of g. Namely, the approximation guarantee is O(n^1^/^3log^2n) for any g>=1, n being the size of the network. This is useful in practical applications, since in backbone networks the grooming factor is usually greater than the network size. Finally, we improve this approximation ratio under some extra assumptions about the request graph.

Lifting harmonic morphisms II: Tropical curves and metrized complexes

In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the non-vanishing of certain Hurwitz numbers, and we give various conditions under which this obstruction does vanish. In particular we show that any finite harmonic morphism of (non-augmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from C to the tropical projective line are equivalent. We study liftability of metrized complexes equipped with a finite group action, and use this to classify all augmented metric graphs arising as the tropicalization of a hyperelliptic curve. We prove that there exists a d-gonal tropical curve that does not lift to a d-gonal algebraic curve. This article is the second in a series of two.

Degree-Constrained Subgraph Problems: Hardness and Approximation Results

A general instance of a Degree-Constrained Subgraph problem consists of an edge-weighted or vertex-weighted graph G and the objective is to find an optimal weighted subgraph, subject to certain degree constraints on the vertices of the subgraph. This paper considers two natural Degree-Constrained Subgraph problems and studies their behavior in terms of approximation algorithms. These problems take as input an undirected graph G = (V,E), with |V| = n and |E| = m. Our results, together with the definition of the two problems, are listed below. The Maximum Degree-Bounded Connected Subgraph problem (MDBCS d ) takes as input a weight function

Counting Subgraphs via Homomorphisms

\omega : E \rightarrow \mathbb R^+

Submodular partition functions

and an integer d ≥ 2, and asks for a subset E′ ⊆ E such that the subgraph G′ = (V,E′) is connected, has maximum degree at most d, and ∑  e ∈ E′ ω(e) is maximized. This problem is one of the classical NP-hard problems listed by Garey and Johnson in [Computers and Intractability, W.H. Freeman, 1979], but there were no results in the literature except for d = 2. We prove that MDBCS d is not in Apx for any d ≥ 2 (this was known only for d = 2) and we provide a

Reduced divisors and embeddings of tropical curves

(\min \{m/ \log n,\ nd/(2 \log n)\})

Implicit branching and parameterized partial cover problems

-approximation algorithm for unweighted graphs, and a

On explosions in heavy-tailed branching random walks

(\min\{n/2,\ m/d\})

Parameterized complexity of finding small degree-constrained subgraphs

-approximation algorithm for weighted graphs. We also prove that when G has a low-degree spanning tree, in terms of d, MDBCS d can be approximated within a small constant factor in unweighted graphs. The Minimum Subgraph of Minimum Degree  ≥ d (MSMD d ) problem requires finding a smallest subgraph of G (in terms of number of vertices) with minimum degree at least d. We prove that MSMD d is not in Apx for any d ≥ 3 and we provide an

Guarding Art Galleries: The Extra Cost for Sculptures Is Linear

\mathcal{O}(n/\log n)