Exponential Time Complexity of the Permanent and the Tutte Polynomial

We show conditional lower bounds for well-studied #P-hard problems: (a) The number of satisfying assignments of a 2-CNF formula with n variables cannot be counted in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph. (b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed in time exp(o(n)). (c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs. Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting

Goldberg's Conjecture is true for random multigraphs

In the 70s, Goldberg, and independently Seymour, conjectured that for any multigraph $G$, the chromatic index $\chi'(G)$ satisfies $\chi'(G)\leq \max \{\Delta(G)+1, \lceil\rho(G)\rceil\}$, where $\rho(G)=\max \{\frac {e(G[S])}{\lfloor |S|/2\rfloor} \mid S\subseteq V \}$. We show that their conjecture (in a stronger form) is true for random multigraphs. Let $M(n,m)$ be the probability space consisting of all loopless multigraphs with $n$ vertices and $m$ edges, in which $m$ pairs from $[n]$ are chosen independently at random with repetitions. Our result states that, for a given $m:=m(n)$, $M\sim M(n,m)$ typically satisfies $\chi'(G)=\max\{\Delta(G),\lceil\rho(G)\rceil\}$. In particular, we show that if $n$ is even and $m:=m(n)$, then $\chi'(M)=\Delta(M)$ for a typical $M\sim M(n,m)$. Furthermore, for a fixed $\varepsilon>0$, if $n$ is odd, then a typical $M\sim M(n,m)$ has $\chi'(M)=\Delta(M)$ for $m\leq (1-\varepsilon)n^3\log n$, and $\chi'(M)=\lceil\rho(M)\rceil$ for $m\geq (1+\varepsilon)n^3\log n$.Comment: 26 page

Edge coloring multigraphs without small dense subsets

© 2015. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/One consequence of a long-standing conjecture of Goldberg and Seymour about the chromatic index of multigraphs would be the following statement. Suppose $G$ is a multigraph with maximum degree $\Delta$, such that no vertex subset $S$ of odd size at most $\Delta$ induces more than $(\Delta+1)(|S|-1)/2$ edges. Then $G$ has an edge coloring with $\Delta+1$ colors. Here we prove a weakened version of this statement.Natural Sciences and Engineering Research Counci

Proof of the Goldberg-Seymour Conjecture on Edge-Colorings of Multigraphs

Given a multigraph $G=(V,E)$, the {\em edge-coloring problem} (ECP) is to color the edges of $G$ with the minimum number of colors so that no two adjacent edges have the same color. This problem can be naturally formulated as an integer program, and its linear programming relaxation is called the {\em fractional edge-coloring problem} (FECP). In the literature, the optimal value of ECP (resp. FECP) is called the {\em chromatic index} (resp. {\em fractional chromatic index}) of $G$, denoted by $\chi'(G)$ (resp. $\chi^*(G)$). Let $\Delta(G)$ be the maximum degree of $G$ and let $$\Gamma(G)=\max \Big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\},$$ where $E(U)$ is the set of all edges of $G$ with both ends in $U$. Clearly, $\max\{\Delta(G), \, \lceil \Gamma(G) \rceil \}$ is a lower bound for $\chi'(G)$. As shown by Seymour, $\chi^*(G)=\max\{\Delta(G), \, \Gamma(G)\}$. In the 1970s Goldberg and Seymour independently conjectured that $\chi'(G) \le \max\{\Delta(G)+1, \, \lceil \Gamma(G) \rceil\}$. Over the past four decades this conjecture, a cornerstone in modern edge-coloring, has been a subject of extensive research, and has stimulated a significant body of work. In this paper we present a proof of this conjecture. Our result implies that, first, there are only two possible values for $\chi'(G)$, so an analogue to Vizing's theorem on edge-colorings of simple graphs, a fundamental result in graph theory, holds for multigraphs; second, although it is $NP$-hard in general to determine $\chi'(G)$, we can approximate it within one of its true value, and find it exactly in polynomial time when $\Gamma(G)>\Delta(G)$; third, every multigraph $G$ satisfies $\chi'(G)-\chi^*(G) \le 1$, so FECP has a fascinating integer rounding property

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz

The next few years will be exciting as prototype universal quantum processors emerge, enabling implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation, and which have the potential to significantly expand the breadth of quantum computing applications. A leading candidate is Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates between applying a cost-function-based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the Quantum Alternating Operator Ansatz, is the consideration of general parametrized families of unitaries rather than only those corresponding to the time-evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach to a wide variety of approximate optimization, exact optimization, and sampling problems. Here, we introduce the Quantum Alternating Operator Ansatz, lay out design criteria for mixing operators, detail mappings for eight problems, and provide brief descriptions of mappings for diverse problems.Comment: 51 pages, 2 figures. Revised to match journal pape

Approximability of Combinatorial Optimization Problems on Power Law Networks

One of the central parts in the study of combinatorial optimization is to classify the NP-hard optimization problems in terms of their approximability. In this thesis we study the Minimum Vertex Cover (Min-VC) problem and the Minimum Dominating Set (Min-DS) problem in the context of so called power law graphs. This family of graphs is motivated by recent findings on the degree distribution of existing real-world networks such as the Internet, the World-Wide Web, biological networks and social networks. In a power law graph the number of nodes yi of a given degree i is proportional to i-ß, that is, the distribution of node degrees follows a power law. The parameter ß > 0 is the so called power law exponent. With the aim of classifying the above combinatorial optimization problems, we are pursuing two basic approaches in this thesis. One is concerned with complexity theory and the other with the theory of algorithms. As a result, our main contributions to the classification of the problems Min-VC and Min-DS in the context of power law graphs are twofold: - Firstly, we give substantial improvements on the previously known approximation lower bounds for Min-VC and Min-DS in combinatorial power law graphs. More precisely, we are going to show the APX-hardness of Min-VC and Min-DS in connected power law graphs and give constant factor lower bounds for Min-VC and the first logarithmic lower bounds for Min-DS in this setting. The results are based on new approximation-preserving embedding reductions that embed certain instances of Min-VC and Min-DS into connected power law graphs. - Secondly, we design a new approximation algorithm for the Min-VC problem in random power law graphs with an expected approximation ratio strictly less than 2. The main tool is a deterministic rounding procedure that acts on a half-integral solution for Min-VC and produces a good approximation on the subset of low degree vertices. Moreover, for the case of Min-DS, we improve on the previously best upper bounds that rely on a greedy algorithm. The improvements are based on our new techniques for determining upper and lower bounds on the size and the volume of node intervals in power law graphs