Convex Combinatorial Optimization

We introduce the convex combinatorial optimization problem, a far reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications

Letter to Secretary Guy Guernsey from Nettie Rothblum, 1913

Letter to Guy Guernsey, Secretary of Chicago-Kent College of Law, from Nettie Rothblum (1909), in 1913. Nettie Rothblum, one of the original founders of the Kappa Beta Pi Legal Sorority in 1908, was in Sinaloa during the Mexican Revolution, and included a short description of life in Mexico at the time along with payment for her sister Ruth\u27s tuition. Transcript: February 21, 1913. Dear Mr. Guernsey, Your cordial letter received and but for the fact that I have been extremely busy, would have answered it before this. With all the war news that you are getting in Chicago, it hardly behooves me to add anything, excepting the fact that this particular spot is quiet. At least, we expected to be besieged day before yesterday. Everyone was armed, and the home I live in was guarded, because it would be the centre of attack, but the revolutionists stopped at Mochicahui, a little town few miles north, thought better of it when they heard we were prepared to meet them, and didnt [sic] come that day, although we now expect them daily. In fact, their leader was at the house the morning the attack was expected, reconnoitred, and withdrew, and none of us were any the wiser until now. Inclosed [sic] herewith please find my check to order of Chicago-Kent to pay for Ruth\u27s tuition, the last semester. I have dated it March 4th, which is near enough to the beginning of the new semester, to be acceptable, I think. Please acknowledge receipt direct to my sister, as you did before. With kindest personal regards, I am, Sincerely yours, Nettie Rothblumhttps://scholarship.kentlaw.iit.edu/memorabilia/1009/thumbnail.jp

Efficient Batch Verification for UP

Consider a setting in which a prover wants to convince a verifier of the correctness of k NP statements. For example, the prover wants to convince the verifier that k given integers N_1,...,N_k are all RSA moduli (i.e., products of equal length primes). Clearly this problem can be solved by simply having the prover send the k NP witnesses, but this involves a lot of communication. Can interaction help? In particular, is it possible to construct interactive proofs for this task whose communication grows sub-linearly with k? Our main result is such an interactive proof for verifying the correctness of any k UP statements (i.e., NP statements that have a unique witness). The proof-system uses only a constant number of rounds and the communication complexity is k^delta * poly(m), where delta>0 is an arbitrarily small constant, m is the length of a single witness, and the poly term refers to a fixed polynomial that only depends on the language and not on delta. The (honest) prover strategy can be implemented in polynomial-time given access to the k (unique) witnesses. Our proof leverages "interactive witness verification" (IWV), a new type of proof-system that may be of independent interest. An IWV is a proof-system in which the verifier needs to verify the correctness of an NP statement using: (i) a sublinear number of queries to an alleged NP witness, and (ii) a short interaction with a powerful but untrusted prover. In contrast to the setting of PCPs and Interactive PCPs, here the verifier only has access to the raw NP witness, rather than some encoding thereof

Letter to Secretary Guy Guernsey from Nettie Rothblum, 1913

Letter to Guy Guernsey, Secretary of Chicago-Kent College of Law, from Nettie Rothblum (1909), in 1913. Nettie Rothblum, one of the original founders of the Kappa Beta Pi Legal Sorority in 1908, was in Sinaloa during the Mexican Revolution, and included a short description of life in Mexico at the time along with payment for her sister Ruth\u27s tuition. Transcript: February 21, 1913. Dear Mr. Guernsey, Your cordial letter received and but for the fact that I have been extremely busy, would have answered it before this. With all the war news that you are getting in Chicago, it hardly behooves me to add anything, excepting the fact that this particular spot is quiet. At least, we expected to be besieged day before yesterday. Everyone was armed, and the home I live in was guarded, because it would be the centre of attack, but the revolutionists stopped at Mochicahui, a little town few miles north, thought better of it when they heard we were prepared to meet them, and didnt [sic] come that day, although we now expect them daily. In fact, their leader was at the house the morning the attack was expected, reconnoitred, and withdrew, and none of us were any the wiser until now. Inclosed [sic] herewith please find my check to order of Chicago-Kent to pay for Ruth\u27s tuition, the last semester. I have dated it March 4th, which is near enough to the beginning of the new semester, to be acceptable, I think. Please acknowledge receipt direct to my sister, as you did before. With kindest personal regards, I am, Sincerely yours, Nettie Rothblumhttps://scholarship.kentlaw.iit.edu/memorabilia/1009/thumbnail.jp

Abstracting Fairness: Oracles, Metrics, and Interpretability

It is well understood that classification algorithms, for example, for deciding on loan applications, cannot be evaluated for fairness without taking context into account. We examine what can be learned from a fairness oracle equipped with an underlying understanding of ``true'' fairness. The oracle takes as input a (context, classifier) pair satisfying an arbitrary fairness definition, and accepts or rejects the pair according to whether the classifier satisfies the underlying fairness truth. Our principal conceptual result is an extraction procedure that learns the underlying truth; moreover, the procedure can learn an approximation to this truth given access to a weak form of the oracle. Since every ``truly fair'' classifier induces a coarse metric, in which those receiving the same decision are at distance zero from one another and those receiving different decisions are at distance one, this extraction process provides the basis for ensuring a rough form of metric fairness, also known as individual fairness. Our principal technical result is a higher fidelity extractor under a mild technical constraint on the weak oracle's conception of fairness. Our framework permits the scenario in which many classifiers, with differing outcomes, may all be considered fair. Our results have implications for interpretablity -- a highly desired but poorly defined property of classification systems that endeavors to permit a human arbiter to reject classifiers deemed to be ``unfair'' or illegitimately derived.Comment: 17 pages, 1 figur

Preference-Informed Fairness

We study notions of fairness in decision-making systems when individuals have diverse preferences over the possible outcomes of the decisions. Our starting point is the seminal work of Dwork et al. which introduced a notion of individual fairness (IF): given a task-specific similarity metric, every pair of individuals who are similarly qualified according to the metric should receive similar outcomes. We show that when individuals have diverse preferences over outcomes, requiring IF may unintentionally lead to less-preferred outcomes for the very individuals that IF aims to protect. A natural alternative to IF is the classic notion of fair division, envy-freeness (EF): no individual should prefer another individual's outcome over their own. Although EF allows for solutions where all individuals receive a highly-preferred outcome, EF may also be overly-restrictive. For instance, if many individuals agree on the best outcome, then if any individual receives this outcome, they all must receive it, regardless of each individual's underlying qualifications for the outcome. We introduce and study a new notion of preference-informed individual fairness (PIIF) that is a relaxation of both individual fairness and envy-freeness. At a high-level, PIIF requires that outcomes satisfy IF-style constraints, but allows for deviations provided they are in line with individuals' preferences. We show that PIIF can permit outcomes that are more favorable to individuals than any IF solution, while providing considerably more flexibility to the decision-maker than EF. In addition, we show how to efficiently optimize any convex objective over the outcomes subject to PIIF for a rich class of individual preferences. Finally, we demonstrate the broad applicability of the PIIF framework by extending our definitions and algorithms to the multiple-task targeted advertising setting introduced by Dwork and Ilvento

Hard Properties with (Very) Short PCPPs and Their Applications

We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed ?, we construct a property P^(?)? {0,1}^n satisfying the following: Any testing algorithm for P^(?) requires ?(n) many queries, and yet P^(?) has a constant query PCPP whose proof size is O(n?log^(?)n), where log^(?) denotes the ? times iterated log function (e.g., log^(2)n = log log n). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was O(n?polylog(n)). As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed ?, we construct a property that has a constant-query tester, but requires ?(n/log^(?)(n)) queries for every tolerant or erasure-resilient tester