877 research outputs found
Shortest Superstring
In the Shortest Superstring problem (SS) one has to find a shortest string s containing given strings s_1,...,s_n as substrings. The problem is NP-hard, so a natural question is that of its approximability.
One natural approach to approximately solving SS is the following GREEDY heuristic: repeatedly merge two strings with the largest overlap until only a single string is left. This heuristic is conjectured to be a 2-approximation, but even after 30 years since the conjecture has been posed, we are still very far from proving it. The situation is better for non-greedy approximation algorithms, where several approaches yielding 2.5-approximation (and better) are known.
In this talk, we will survey the main results in the area, focusing on the fundamental ideas and intuitions
An Improved Algorithm For Online Min-Sum Set Cover
We study a fundamental model of online preference aggregation, where an
algorithm maintains an ordered list of elements. An input is a stream of
preferred sets . Upon seeing and without
knowledge of any future sets, an algorithm has to rerank elements (change the
list ordering), so that at least one element of is found near the list
front. The incurred cost is a sum of the list update costs (the number of swaps
of neighboring list elements) and access costs (position of the first element
of on the list). This scenario occurs naturally in applications such as
ordering items in an online shop using aggregated preferences of shop
customers. The theoretical underpinning of this problem is known as Min-Sum Set
Cover.
Unlike previous work (Fotakis et al., ICALP 2020, NIPS 2020) that mostly
studied the performance of an online algorithm ALG against the static optimal
solution (a single optimal list ordering), in this paper, we study an arguably
harder variant where the benchmark is the provably stronger optimal dynamic
solution OPT (that may also modify the list ordering). In terms of an online
shop, this means that the aggregated preferences of its user base evolve with
time. We construct a computationally efficient randomized algorithm whose
competitive ratio (ALG-to-OPT cost ratio) is and prove the existence
of a deterministic -competitive algorithm. Here, is the maximum
cardinality of sets . This is the first algorithm whose ratio does not
depend on : the previously best algorithm for this problem was -competitive and is a lower bound on the
performance of any deterministic online algorithm.Comment: Presented at AAAI 202
13/9-approximation for Graphic TSP
The Travelling Salesman Problem is one the most fundamental and most studied
problems in approximation algorithms. For more than 30 years, the best
algorithm known for general metrics has been Christofides's algorithm with
approximation factor of 3/2, even though the so-called Held-Karp LP relaxation
of the problem is conjectured to have the integrality gap of only 4/3. Very
recently, significant progress has been made for the important special case of
graphic metrics, first by Oveis Gharan et al., and then by Momke and Svensson.
In this paper, we provide an improved analysis for the approach introduced by
Momke and Svensson yielding a bound of 13/9 on the approximation factor, as
well as a bound of 19/12+epsilon for any epsilon>0 for a more general
Travelling Salesman Path Problem in graphic metrics
Approximation Algorithms for Union and Intersection Covering Problems
In a classical covering problem, we are given a set of requests that we need
to satisfy (fully or partially), by buying a subset of items at minimum cost.
For example, in the k-MST problem we want to find the cheapest tree spanning at
least k nodes of an edge-weighted graph. Here nodes and edges represent
requests and items, respectively.
In this paper, we initiate the study of a new family of multi-layer covering
problems. Each such problem consists of a collection of h distinct instances of
a standard covering problem (layers), with the constraint that all layers share
the same set of requests. We identify two main subfamilies of these problems: -
in a union multi-layer problem, a request is satisfied if it is satisfied in at
least one layer; - in an intersection multi-layer problem, a request is
satisfied if it is satisfied in all layers. To see some natural applications,
consider both generalizations of k-MST. Union k-MST can model a problem where
we are asked to connect a set of users to at least one of two communication
networks, e.g., a wireless and a wired network. On the other hand, intersection
k-MST can formalize the problem of connecting a subset of users to both
electricity and water.
We present a number of hardness and approximation results for union and
intersection versions of several standard optimization problems: MST, Steiner
tree, set cover, facility location, TSP, and their partial covering variants
Dynamic Beats Fixed: On Phase-Based Algorithms for File Migration
In this paper, we construct a deterministic 4-competitive algorithm for the online file migration problem, beating the currently best 20-year old, 4.086-competitive MTLM algorithm by Bartal et al. (SODA 1997). Like MTLM, our algorithm also operates in phases, but it adapts their lengths dynamically depending on the geometry of requests seen so far. The improvement was obtained by carefully analyzing a linear model (factor-revealing LP) of a single phase of the algorithm. We also show that if an online algorithm operates in phases of fixed length and the adversary is able to modify the graph between phases, no algorithm can beat the competitive ratio of 4.086
Equal-Subset-Sum Faster Than the Meet-in-the-Middle
In the Equal-Subset-Sum problem, we are given a set S of n integers and the problem is to decide if there exist two disjoint nonempty subsets A,B subseteq S, whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in O^*(3^(n/2)) <= O^*(1.7321^n) time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give O^*(1.7088^n) worst case Monte Carlo algorithm. This answers a question suggested by Woeginger in his inspirational survey.
Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in O^*(3^n) time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in O^*(2.6817^n) time and polynomial space
Superconductivity-induced features in electronic Raman spectrum of monolayer graphene
Using the continuum model, we investigate theoretically contribution of the
low-energy electronic excitations to the Raman spectrum of superconducting
monolayer graphene. We consider superconducting phases characterised by an
isotropic order parameter in a single valley and find a Raman peak at a shift
set by the size of the superconducting gap. The height of this peak is
proportional to the square root of the gap and the third power of the Fermi
level, and we estimate its quantum efficiency as .Comment: 7 pages, 2 figure
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