Optimal Robust Self-Testing by Binary Nonlocal XOR Games

Self-testing a quantum apparatus means verifying the existence of a certain quantum state as well as the effect of the associated measuring devices based only on the statistics of the measurement outcomes. Robust (i.e., error-tolerant) self-testing quantum apparatuses are critical building blocks for quantum cryptographic protocols that rely on imperfect or untrusted devices. We devise a general scheme for proving optimal robust self-testing properties for tests based on nonlocal binary XOR games. We offer some simplified proofs of known results on self-testing, and also prove some new results

Improved Online Algorithm for Weighted Flow Time

We discuss one of the most fundamental scheduling problem of processing jobs on a single machine to minimize the weighted flow time (weighted response time). Our main result is a $O(\log P)$-competitive algorithm, where $P$ is the maximum-to-minimum processing time ratio, improving upon the $O(\log^{2}P)$-competitive algorithm of Chekuri, Khanna and Zhu (STOC 2001). We also design a $O(\log D)$-competitive algorithm, where $D$ is the maximum-to-minimum density ratio of jobs. Finally, we show how to combine these results with the result of Bansal and Dhamdhere (SODA 2003) to achieve a $O(\log(\min(P,D,W)))$-competitive algorithm (where $W$ is the maximum-to-minimum weight ratio), without knowing $P,D,W$ in advance. As shown by Bansal and Chan (SODA 2009), no constant-competitive algorithm is achievable for this problem.Comment: 20 pages, 4 figure

Designing Networks with Good Equilibria under Uncertainty

We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of k terminal vertices or players need to establish connectivity with the root. The social optimum is the Minimum Steiner Tree. We are interested in situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying metric but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the designer's goal is to choose a single, universal protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is: to what extent can prior knowledge of the underlying metric help in the design? We first demonstrate that there exist graphs (outerplanar) where knowledge of the underlying metric can dramatically improve the performance of good network design. Then, in our main technical result, we show that there exist graph metrics, for which knowing the underlying metric does not help and any universal protocol has PoA of $\Omega(\log k)$, which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey Theory in high dimensional hypercubes. Then we switch to the stochastic model, where each player is independently activated. We show that there exists a randomized ordered protocol that achieves constant PoA. By using standard derandomization techniques, we produce a deterministic ordered protocol with constant PoA.Comment: This version has additional results about stochastic inpu

Dynamic Algorithms for Graph Coloring

We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for $(\Delta+1)$- vertex coloring and $(2\Delta-1)$-edge coloring in a graph with maximum degree $\Delta$. It is natural to ask if we can efficiently maintain such colorings in the dynamic setting as well. We get the following three results. (1) We present a randomized algorithm which maintains a $(\Delta+1)$-vertex coloring with $O(\log \Delta)$ expected amortized update time. (2) We present a deterministic algorithm which maintains a $(1+o(1))\Delta$-vertex coloring with $O(\text{poly} \log \Delta)$ amortized update time. (3) We present a simple, deterministic algorithm which maintains a $(2\Delta-1)$-edge coloring with $O(\log \Delta)$ worst-case update time. This improves the recent $O(\Delta)$-edge coloring algorithm with $\tilde{O}(\sqrt{\Delta})$ worst-case update time by Barenboim and Maimon.Comment: To appear in SODA 201

A Characterization of Undirected Graphs Admitting Optimal Cost Shares

In a seminal paper, Chen, Roughgarden and Valiant studied cost sharing protocols for network design with the objective to implement a low-cost Steiner forest as a Nash equilibrium of an induced cost-sharing game. One of the most intriguing open problems to date is to understand the power of budget-balanced and separable cost sharing protocols in order to induce low-cost Steiner forests. In this work, we focus on undirected networks and analyze topological properties of the underlying graph so that an optimal Steiner forest can be implemented as a Nash equilibrium (by some separable cost sharing protocol) independent of the edge costs. We term a graph efficient if the above stated property holds. As our main result, we give a complete characterization of efficient undirected graphs for two-player network design games: an undirected graph is efficient if and only if it does not contain (at least) one out of few forbidden subgraphs. Our characterization implies that several graph classes are efficient: generalized series-parallel graphs, fan and wheel graphs and graphs with small cycles.Comment: 60 pages, 69 figures, OR 2017 Berlin, WINE 2017 Bangalor

Reconfigurations of Combinatorial Problems: Graph Colouring and Hamiltonian Cycle

We explore algorithmic aspects of two known combinatorial problems, Graph Colouring and Hamiltonian Cycle, by examining properties of their solution space. One can model the set of solutions of a combinatorial problem $P$ by the solution graph $R(P)$, where vertices are solutions of $P$ and there is an edge between two vertices, when the two corresponding solutions satisfy an adjacency reconfiguration rule. For example, we can define the reconfiguration rule for graph colouring to be that two solutions are adjacent when they differ in colour in exactly one vertex. The exploration of the properties of the solution graph $R(P)$ can give rise to interesting questions. The connectivity of $R(P)$ is the most prominent question in this research area. This is reasonable, since the main motivation for modelling combinatorial solutions as a graph is to be able to transform one into the other in a stepwise fashion, by following paths between solutions in the graph. Connectivity questions can be made binary, that is expressed as decision problems which accept a 'yes' or 'no' answer. For example, given two specific solutions, is there a path between them? Is the graph of solutions $R(P)$ connected? In this thesis, we first show that the diameter of the solution graph $R_{l}(G)$ of vertex $l$-colourings of k-colourable chordal and chordal bipartite graphs $G$ is $O(n^2)$, where $l > k$ and n is the number of vertices of $G$. Then, we formulate a decision problem on the connectivity of the graph colouring solution graph, where we allow extra colours to be used in order to enforce a path between two colourings with no path between them. We give some results for general instances and we also explore what kind of graphs pose a challenge to determine the complexity of the problem for general instances. Finally, we give a linear algorithm which decides whether there is a path between two solutions of the Hamiltonian Cycle Problem for graphs of maximum degree five, and thus providing insights towards the complexity classification of the decision problem

Tree comparison: enumeration and application to cheminformatics

Graphs are a well-known data structure used in many application domains that rely on relationships between individual entities. Examples are social networks, where the users may be in friendship with each other, road networks, where one-way or bidirectional roads connect crossings, and work package assignments, where workers are assigned to tasks. In chem- and bioinformatics, molecules are often represented as molecular graphs, where vertices represent atoms, and bonds between them are represented by edges connecting the vertices. Since there is an ever-increasing amount of data that can be treated as graphs, fast algorithms are needed to compare such graphs. A well-researched concept to compare two graphs is the maximum common subgraph. On the one hand, this allows finding substructures that are common to both input graphs. On the other hand, we can derive a similarity score from the maximum common subgraph. A practical application is rational drug design which involves molecular similarity searches. In this thesis, we study the maximum common subgraph problem, which entails finding a largest graph, which is isomorphic to subgraphs of two input graphs. We focus on restrictions that allow polynomial-time algorithms with a low exponent. An example is the maximum common subtree of two input trees. We succeed in improving the previously best-known time bound. Additionally, we provide a lower time bound under certain assumptions. We study a generalization of the maximum common subtree problem, the block-and-bridge preserving maximum common induced subgraph problem between outerplanar graphs. This problem is motivated by the application to cheminformatics. First, the vast majority of drugs modeled as molecular graphs is outerplanar, and second, the blocks correspond to the ring structures and the bridges to atom chains or linkers. If we allow disconnected common subgraphs, the problem becomes NP-hard even for trees as input. We propose a second generalization of the maximum common subtree problem, which allows skipping vertices in the input trees while maintaining polynomial running time. Since a maximum common subgraph is not unique in general, we investigate the problem to enumerate all maximum solutions. We do this for both the maximum common subtree problem and the block-and-bridge preserving maximum common induced subgraph problem between outerplanar graphs. An arising subproblem which we analyze is the enumeration of maximum weight matchings in bipartite graphs. We support a weight function between the vertices and edges for all proposed common subgraph methods in this thesis. Thus the objective is to compute a common subgraph of maximum weight. The weights may be integral or real-valued, including negative values. A special case of using such a weight function is computing common subgraph isomorphisms between labeled graphs, where labels between mapped vertices and edges must be equal. An experimental study evaluates the practical running times and the usefulness of our block-and-bridge preserving maximum common induced subgraph algorithm against state of the art algorithms