Sum of squares lower bounds for refuting any CSP

Let $P:\{0,1\}^k \to \{0,1\}$ be a nontrivial $k$-ary predicate. Consider a random instance of the constraint satisfaction problem $\mathrm{CSP}(P)$ on $n$ variables with $\Delta n$ constraints, each being $P$ applied to $k$ randomly chosen literals. Provided the constraint density satisfies $\Delta \gg 1$, such an instance is unsatisfiable with high probability. The \emph{refutation} problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate $P$ supports a $t$-\emph{wise uniform} probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree $d = \Theta(\frac{n}{\Delta^{2/(t-1)} \log \Delta})$ (which runs in time $n^{O(d)}$) \emph{cannot} refute a random instance of $\mathrm{CSP}(P)$. In particular, the polynomial-time SOS algorithm requires $\widetilde{\Omega}(n^{(t+1)/2})$ constraints to refute random instances of CSP$(P)$ when $P$ supports a $t$-wise uniform distribution on its satisfying assignments. Together with recent work of Lee et al. [LRS15], our result also implies that \emph{any} polynomial-size semidefinite programming relaxation for refutation requires at least $\widetilde{\Omega}(n^{(t+1)/2})$ constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate~$P$, they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of Allen et al. [AOW15] and Raghavendra et al. [RRS16], this full three-way tradeoff is \emph{tight}, up to lower-order factors.Comment: 39 pages, 1 figur

Phase Transition of the 2-Choices Dynamics on Core-Periphery Networks

Consider the following process on a network: Each agent initially holds either opinion blue or red; then, in each round, each agent looks at two random neighbors and, if the two have the same opinion, the agent adopts it. This process is known as the 2-Choices dynamics and is arguably the most basic non-trivial opinion dynamics modeling voting behavior on social networks. Despite its apparent simplicity, 2-Choices has been analytically characterized only on networks with a strong expansion property -- under assumptions on the initial configuration that establish it as a fast majority consensus protocol. In this work, we aim at contributing to the understanding of the 2-Choices dynamics by considering its behavior on a class of networks with core-periphery structure, a well-known topological assumption in social networks. In a nutshell, assume that a densely-connected subset of agents, the core, holds a different opinion from the rest of the network, the periphery. Then, depending on the strength of the cut between the core and the periphery, a phase-transition phenomenon occurs: Either the core's opinion rapidly spreads among the rest of the network, or a metastability phase takes place, in which both opinions coexist in the network for superpolynomial time. The interest of our result is twofold. On the one hand, by looking at the 2-Choices dynamics as a simplistic model of competition among opinions in social networks, our theorem sheds light on the influence of the core on the rest of the network, as a function of the core's connectivity towards the latter. On the other hand, to the best of our knowledge, we provide the first analytical result which shows a heterogeneous behavior of a simple dynamics as a function of structural parameters of the network. Finally, we validate our theoretical predictions with extensive experiments on real networks

Covering Many (Or Few) Edges with k Vertices in Sparse Graphs

We study the following two fixed-cardinality optimization problems (a maximization and a minimization variant). For a fixed ? between zero and one we are given a graph and two numbers k ? ? and t ? ?. The task is to find a vertex subset S of exactly k vertices that has value at least (resp. at most for minimization) t. Here, the value of a vertex set computes as ? times the number of edges with exactly one endpoint in S plus 1-? times the number of edges with both endpoints in S. These two problems generalize many prominent graph problems, such as Densest k-Subgraph, Sparsest k-Subgraph, Partial Vertex Cover, and Max (k,n-k)-Cut. In this work, we complete the picture of their parameterized complexity on several types of sparse graphs that are described by structural parameters. In particular, we provide kernelization algorithms and kernel lower bounds for these problems. A somewhat surprising consequence of our kernelizations is that Partial Vertex Cover and Max (k,n-k)-Cut not only behave in the same way but that the kernels for both problems can be obtained by the same algorithms

Multiwinner Analogues of Plurality Rule: Axiomatic and Algorithmic Perspectives

We characterize the class of committee scoring rules that satisfy the fixed-majority criterion. In some sense, the committee scoring rules in this class are multiwinner analogues of the single-winner Plurality rule, which is uniquely characterized as the only single-winner scoring rule that satisfies the simple majority criterion. We define top-$k$-counting committee scoring rules and show that the fixed majority consistent rules are a subclass of the top-$k$-counting rules. We give necessary and sufficient conditions for a top-$k$-counting rule to satisfy the fixed-majority criterion. We find that, for most of the rules in our new class, the complexity of winner determination is high (that is, the problem of computing the winners is NP-hard), but we also show examples of rules with polynomial-time winner determination procedures. For some of the computationally hard rules, we provide either exact FPT algorithms or approximate polynomial-time algorithms

Solving graph problems with single-photons and linear optics

An important challenge for current and near-term quantum devices is finding useful tasks that can be preformed on them. We first show how to efficiently encode a bounded $n \times n$ matrix $A$ into a linear optical circuit with $2n$ modes. We then apply this encoding to the case where $A$ is a matrix containing information about a graph $G$. We show that a photonic quantum processor consisting of single-photon sources, a linear optical circuit encoding $A$, and single-photon detectors can solve a range of graph problems including finding the number of perfect matchings of bipartite graphs, computing permanental polynomials, determining whether two graphs are isomorphic, and the $k$-densest subgraph problem. We also propose pre-processing methods to boost the probabilities of observing the relevant detection events and thus improve performance. Finally, we present various numerical simulations which validate our findings.Comment: 6 pages + 9 pages appendix. Comments Welcome

The Complexity of Finding Dense Subgraphs in Graphs with Large Cliques

The GapDensest-k-Subgraph(d) problem (GapDkS(d)) is defined as follows: given a graph G and parameters k,d, distinguish between the case that G contains a k-clique, and the case that every k-subgraph of G has density at most d. GapDkS(d) is a natural relaxation of the standard Clique problem, which is known to be NP-complete. For d very close to 1, the GapDkS(d) problem is equivalent to the Clique problem, and when d is very close to 0 the GapDkS(d) problem can easily be solved in polynomial time. However, despite much work on both the algorithmic and hardness front, the exact k and d parameter values for which GapDkS(d) can be solved in polynomial time are still unknown. In particular, the best polynomial-time algorithms can solve GapDkS(d) when d is an inverse polynomial in the number of vertices n, but there have been no NP-hardness results beyond the trivial result. This thesis attempts to understand the GapDkS(d) problem better by studying the case when k is restricted to be linear in n (where n is the number of vertices in G). In particular, we survey the GapDkS(d) algorithms and hardness results that can be best applied to this restriction in an attempt to determine the threshold for when the problem becomes NP-hard. With some modifications to the algorithms and proofs, we produce algorithms and hardness results for the GapDkS(d) problem with k linear in n. In addition, we study the connection between GapDkS(d) and MaxClique, and show that despite having strong hardness results for MaxClique, reductions from MaxClique do not give strong hardness bounds for GapDkS(d)