Motivated by circle graphs, and the enumeration of Euler circuits, we define a one-variable ''interlace polynomial'' for any graph. The polynomial satisfies a beautiful and unexpected reduction relation, quite different from the cut and fuse reduction characterizing the Tutte polynomial.It emerges that the interlace graph polynomial may be viewed as a special case of the Martin polynomial of an isotropic system, which underlies its connections with the circuit partition polynomial and the Kauffman brackets of a link diagram. The graph polynomial, in addition to being perhaps more broadly accessible than the Martin polynomial for isotropic systems, also has a two-variable generalization that is unknown for the Martin polynomial. We consider extremal properties of the interlace polynomial, its values for various special graphs, and evaluations which relate to basic graph properties such as the component and independence numbers. r
We introduce a new graph polynomial in two variables. This "interlace" polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the significant differences are that our expansion is over vertex rather than edge subsets, and the rank and nullity employed are those of an adjacency matrix rather than an incidence matrix.The second computation is by a three-term reduction formula involving a graph pivot; the pivot arose previously in the study of interlacement and Euler circuits in four-regular graphs.We consider a few properties and specializations of the two-variable interlace polynomial. One specialization, the "vertex-nullity interlace polynomial", is the single-variable interlace graph polynomial we studied previously, closely related to the Tutte-Martin polynomial on isotropic systems previously considered by Bouchet. Another, the "vertex-rank interlace polynomial", is equally interesting. Yet another specialization of the two-variable polynomial is the independent-set polynomial.
ABSTRACT:With random inputs, certain decision problems undergo a "phase transition." We prove similar behavior in an optimization context. Given a conjunctive normal form (CNF) formula F on n variables and with m k-variable clauses, denote by max F the maximum number of clauses satisfiable by a single assignment of the variables. (Thus the decision problem k-SAT is to determine if max F is equal to m.) With the formula F chosen at random, the expectation of max F is trivially bounded by (3/4)m ޅ max F m. We prove that for random formulas with m ϭ cn clauses: for constants c Ͻ 1, ޅ max F is cn Ϫ ⌰(1/n); for large c, it approaches ((3/4)c ϩ ⌰( ͌ c))n; and in the "window" c ϭ 1 ϩ ⌰(n Ϫ1/3 ), it is cn Ϫ ⌰(1). Our full results are more detailed, but this already shows that the optimization problem MAX 2-SAT undergoes a phase transition just as the 2-SAT decision problem does, and at the same critical value c ϭ 1. Most of our results are established without reference to the analogous propositions for decision 2-SAT, and can be used to reproduce them.We consider "online" versions of MAX 2-SAT, and show that for one version the obvious greedy algorithm is optimal; all other natural questions remain open. We can extend only our simplest MAX 2-SAT results to MAX k-SAT, but we conjecture a "MAX k-SAT limiting function conjecture" analogous to the folklore "satisfiability threshold conjecture," but open even for k ϭ 2. Neither conjecture immediately implies the other, but it is natural to further conjecture a connection between them. We also prove analogous results for random MAX CUT.
The class Max (r, 2)-CSP, or simply Max 2-CSP, consists of constraint satisfaction problems with at most two r-valued variables per clause. For instances with n variables and m binary clauses, we present an O(nr 5+19m/100 )-time algorithm which is the fastest polynomialspace algorithm for many problems in the class, including Max Cut. The method also proves a treewidth bound tw(G) ≤ (13/75 + o(1))m, which gives a faster Max 2-CSP algorithm that uses exponential space: running in time O ⋆ (2 (13/75+o(1))m ), this is fastest for most problems in Max 2-CSP. Parametrizing in terms of n rather than m, for graphs of average degree d we show a simple algorithm running time O ⋆`2(1− 2 d+1 )n´, the fastest polynomial-space algorithm known. In combination with "Polynomial CSPs" introduced in a companion paper, these algorithms also allow (with an additional polynomial-factor overhead in space and time) counting and sampling, and the solution of problems like Max Bisection that escape the usual CSP framework.Linear programming is key to the design as well as the analysis of the algorithms.1.1. Results. The running times for our algorithms depend on the space allowed, and are summarized in Table 1. The O ⋆ (·) notation, which ignores leading polynomial factors, is defined in Section 2.1. For Max 2-CSP we give an O ⋆ (r 19m/100 )-time, linear-space algorithm. This is the fastest polynomial-space algorithm known for Max Cut, Max Dicut, Max 2-Lin, less common problems such as Max Ones 2-Sat, weighted versions of all these, and of course general Max 2-CSP; more efficient algorithms are known for only a few problems, such as Maximum Independent Set and Max 2-Sat. If exponential space is allowed, we give an algorithm running in time O ⋆ (r (13/75+o(1))m ) and space O ⋆ (r (1/9+o(1))m ); it is the fastest exponential-space algorithm known for most problems in Max 2-CSP (including those listed above for the polynomial-space algorithm).SS06c] 1.3. Literature survey. We are not sure where the class (a, b)-CSP was first introduced, but this model, where each variable has at most a possible colors and there are general constraints each involving at most b variables, is extensively exploited for example in Beigel and Eppstein's O ⋆ (1.3829 n )-time 3-coloring algorithm [BE05]. Finding relatively fast exponential-time algorithms for NP-hard problems is a field of endeavor that includes Schöning's famous randomized algorithm for 3-Sat [Sch99], taking time O ⋆ ((4/3) n ) for an instance on n variables. Narrowing the scope to Max 2-CSPs with time parametrized in m, we begin our history with an algorithm of Niedermeier and Rossmanith [NR00]: designed for Max Sat generally, it solves Max 2-Sat instances in time O ⋆ (2 0.348m ). The Max 2-Sat result was improved by Hirsch to O ⋆ (2 m/4 ) [Hir00]. Gramm, Hirsch, Niedermeier and Rossmanith showed how to solve Max 2-Sat in time O ⋆ (2 m/5 ), and used a transformation from Max Cut into Max 2-Sat to allow Max Cut's solution in time O ⋆ (2 m/3 ) [GHNR03]. Kulikov and Fedin showed how to solve Max Cut...
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