Error-correcting codes are fundamental tools used to transmit digital information over unreliable channels. Their study goes back to the work of Hamming [Ham50] and Shannon [Sha48], who used them as the basis for the field of information theory. The problem of decoding the original information up to the full error-correcting potential of the system is often very complex, especially for modern c...

Structured prediction applications often involve complex inference problems that require the use of approximate methods. Approximations based on linear programming (LP) relaxations have proved particularly successful in this setting, with both theoretical and empirical support. Despite the general intractability of inference, it has been observed that in many real-world applications the LP rela...

Error-correcting codes are fundamental tools used to transmit digital information over unreliable channels. Their study goes back to the work of Hamming [Ham50] and Shannon [Sha48], who used them as the basis for the field of information theory. The problem of decoding the original information up to the full error-correcting potential of the system is often very complex, especially for modern c...

Karchmer, Kushilevitz and Nisan formulated the formula size problem as an integer programming problem called the rectangle bound and introduced a technique called the LP bound, which gives a formula size lower bound by showing a feasible solution of the dual problem of its LP-relaxation. As extensions of the LP bound, we introduce novel general techniques proving formula size lower bounds, name...

We study an extension of the precedence constrained knapsack problem where the knapsack can be filled in multiple periods. This problem is known in the mining industry as the open-pit mine production scheduling problem. We present a new algorithm for solving the LP relaxation of this problem and an LP-based heuristic to obtain feasible solutions. Computational experiments show that we can solve...

In the previous section we saw how we could use LP duality theory to develop an algorithm for the weighted matching problem in bipartite graphs. In this section, we’ll see how to extend that algorithm to handle general graphs. As in the unweighted case, blossom-shrinking plays a central role. However, in weighted graphs we will handle blossoms a bit differently. In particular, we will maintain ...

The cutting plane method solves a linear relaxation of the problem obtained by dropping the integrality constraint. Given an integer programming problem, it proceeds as follows: If the optimum solution found is not integral, then a cut inequality separating the optimum from all integer solutions is added to the LP and the LP is solved again; this two-step procedure is repeated till an integral ...

We reprove the hypergraphic generalization of the Tutte/Nash-Williams theorem, which gives sufficient conditions for a hypergraph to contain k disjoint connected hypergraphs. First we observe the theorem is equivalent to the natural LP relaxation having the integer decomposition property. Then we give a new proof of this property using LP uncrossing methods. We discover that “total dual laminar...

Recently, Linear Programming (LP)-based relaxations have been shown promising in boosting the performance of exact MAX-SAT solvers. We compare Semidefinite Programming (SDP) based relaxations with LP relaxations for MAX2SAT. We will show how SDP relaxations are surprisingly powerful, providing much tighter bounds than LP relaxations, across different constrainedness regions. SDP relaxations can...

The groundbreaking work of Rothvoß [2014] established that every linear program expressing the matching polytope has an exponential number of inequalities (formally, the matching polytope has exponential extension complexity). We generalize this result by deriving strong bounds on the LP inapproximability of the matching problem: for fixed 0 < ε < 1, every (1 − ε/n)-approximating LP requires an...